WEBVTT

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People asked me if I'm
going to go over homework.

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Of course I will.

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Let me explain.

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Out of the four
hours you have, three

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should be more or
less lecture time.

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And the fourth hour, which
is the instructor's latitude,

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where they put it-- it's
applications, problems,

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homework like problems, all
sorts of practice for exams

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and so on.

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It's not a recitation.

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It's some sort of workshop that
the instructor conducts himself

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personally.

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All right.

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If you don't have
questions, I'm just

00:00:38.834 --> 00:00:42.030
going to go ahead and
review a little bit of what

00:00:42.030 --> 00:00:44.580
we discussed last time.

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Something new and exciting
was chapter 11, section 11.1.

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And we did 11.2.

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And what was that about?

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That was about functions
of several variables.

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And we discussed
several examples,

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but then we focused
our attention mainly

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to explicit functions, which
means z equals f of x, y,

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of two variables.

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And we call this a graph
because it is a graph.

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In 3D, it's a surface whose
domain is on the floor.

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And the altitude is z, and
that is the-- this is the-- OK.

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How many of you are
non-math majors?

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Can you raise hands?

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Oh, OK.

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So you know a little
bit about research

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from your own classes,
science classes

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or from science
fairs from school.

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These are the independent
variables, x, y.

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And z is the dependent variable.

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We don't use this kind of
terminology in this class.

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But so that you know-- we
discussed domain last time.

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This was about what?

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Domain, range.

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After range, what did we do?

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We talked about level curves.

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What is the level curve?

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Level curves are curves x,
y in the plane corresponding

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to f of x, y equals constant.

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These are called
level curves in plane,

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in the plane called x, y plane.

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What else have we discussed?

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We went straight into 11.2.

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In 11.2, we were very
happy to remember

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a little bit of Calculus
1, which was practically

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a review of limits from Calc 1.

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And what did we do?

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We did epsilon delta, which
was not covered in Calculus 1.

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And where is Aaron?

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OK.

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Thank you, Aaron.

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And today, I was thinking,
I want to show you actually

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an example that is quite
easy of how you use epsilon

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delta for continuity, to show
if the function is continuous,

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but for a function
of true variables.

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And that's not hard.

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You may think, oh, my god.

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That must be hard.

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That's not hard at all.

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I'm going to move on to the
second part of 11.2, which

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is continuity.

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11.2, second part.

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The first part was what?

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It was limits of
functions, right, guys?

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We discussed
properties of limits,

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algebraic properties of
adding sums and taking a limit

00:03:49.810 --> 00:03:53.750
of a sum, taking a limit
of a product of functions,

00:03:53.750 --> 00:03:58.500
taking the limit of a quotient
of function, when it exists,

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when it doesn't.

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Now the second part of
11.2 is called continuity.

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Continuity of what?

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Well, I'm too lazy
to right down,

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but it's continuity of functions
of two variables, right?

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Now in Calc 1-- you
reminded me last time.

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I tried to remind you.

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You tried to remind me.

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Let's remind each other.

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This is like a discussion.

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What was the meaning of f of x
being a continuous function x0,

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which is part of the domain?

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x0 has to be in the domain.

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This is if and only if what?

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Well, what kind of
function is that?

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A one variable
function, real value.

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It takes values on, let's say,
an interval on the real line.

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What was the group
of properties that

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have to be
simultaneously satisfied,

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satisfied at the same time?

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And you told me it has
to be at the same time.

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And I was very happy because
if one of the three conditions

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is missing, then
goodbye, continuity.

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One?

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STUDENT: It's defined
at that point.

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MAGDALENA TODA: Yes,
sir. f of x0 is defined.

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Actually, I said that
here in the domain.

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I'll remove it because
now I said it better.

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Two?

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STUDENT: The limit exists.

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MAGDALENA TODA: Very good.

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The limit, as I approach
x0 with any kind of value

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closer and closer,
exists and is finite.

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Let's give it a name.

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Let's call it L.

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STUDENT:
[? The following value ?]

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equals the limit.

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MAGDALENA TODA: Yes, sir.

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That's the last thing.

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And I'm glad I didn't
have to pull the truth out

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of your mouth.

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So the limit will-- the limit
of f of x when x goes to x0

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equals f of x0.

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No examples.

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You should know
Calc 1, and you do.

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I'm just going to
move on to Calc 3.

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And let's see what the
definition of continuity

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would mean for us in Calc 3.

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Can anybody mimic the properties
that-- well, f of x, y

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is said to be
continuous at x0, y0

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if and only if the following
conditions are-- my arm hurts.

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Are simultaneously satisfied.

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I don't like professors who
use PDF files or slides.

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Shh.

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OK.

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I don't want anything premade.

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The class is a
construction, is working,

00:08:00.810 --> 00:08:04.630
is something like
a work in progress.

00:08:04.630 --> 00:08:07.360
We are building things together.

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This is teamwork.

00:08:08.680 --> 00:08:11.110
If I come up with
some slides that were

00:08:11.110 --> 00:08:13.080
made at home or a PDF file.

00:08:13.080 --> 00:08:14.420
First of all, it means I'm lazy.

00:08:14.420 --> 00:08:17.035
Second of all, it
means that I'm not

00:08:17.035 --> 00:08:20.214
willing to take it
one step at a time

00:08:20.214 --> 00:08:24.584
and show you how
the idea's revealed.

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One.

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Who is telling me?

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I'm not going to say it.

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It's a work in progress.

00:08:36.332 --> 00:08:39.171


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STUDENT: [INAUDIBLE]

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MAGDALENA TODA: f of--

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STUDENT: [INAUDIBLE]

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MAGDALENA TODA: Of
x0, y0 is defined.

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And why not?

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Well, just to have
a silly [? pun ?].

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Two.

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Limit as the pair x, y
approaches x0, x0-- and guys,

00:09:02.922 --> 00:09:05.840
when you close your eyes--
no you close your eyes--

00:09:05.840 --> 00:09:09.410
and you imagine
x, y going to x0,

00:09:09.410 --> 00:09:16.030
y0 by any possible paths
in any possible way,

00:09:16.030 --> 00:09:21.570
it's not that you have a
predetermined path to x0, y0,

00:09:21.570 --> 00:09:23.650
because you may be trapped.

00:09:23.650 --> 00:09:26.170
You may have-- as you've
seen last time, you may have,

00:09:26.170 --> 00:09:28.540
coming from this direction,
the limit will exist,

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will be this one.

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Coming from that direction,
the limit will exist,

00:09:32.600 --> 00:09:34.600
would be another one.

00:09:34.600 --> 00:09:36.540
And then you don't
have overall limits.

00:09:36.540 --> 00:09:41.450
So the limit-- when I call that,
that means the overall limit

00:09:41.450 --> 00:09:51.250
exists, exists and
equals L. It's finite.

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That's what I mean.

00:09:53.430 --> 00:09:57.880
And three, the value
of the function at x0,

00:09:57.880 --> 00:10:04.510
y0 must be equal to the limit
of the function that value

00:10:04.510 --> 00:10:08.382
as you approach it, x0, y0.

00:10:08.382 --> 00:10:11.550
And equals L, of course.

00:10:11.550 --> 00:10:12.650
So great.

00:10:12.650 --> 00:10:16.810
So it's so obvious
that we are following

00:10:16.810 --> 00:10:19.358
exactly the same
type of definition,

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the same type of pattern.

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I'm going to ask you
to help me, to help

00:10:28.660 --> 00:10:34.720
me solve a harder problem
that involves continuity.

00:10:34.720 --> 00:10:38.558
And I'm asking you, if I
have the following function--

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I'm going to erase the
definition of continuity

00:10:40.950 --> 00:10:43.308
from Calc 1.

00:10:43.308 --> 00:10:45.720
I'm going to ask you, what if
I have this funny function?

00:10:45.720 --> 00:10:49.082
You've seen it before, and
I gave you a little bit

00:10:49.082 --> 00:10:50.384
of a warning about it.

00:10:50.384 --> 00:10:53.320


00:10:53.320 --> 00:11:00.860
Limit as x, y goes
to 0, 0 of x squared

00:11:00.860 --> 00:11:07.170
plus y squared times sine of 1
over x squared plus y squared.

00:11:07.170 --> 00:11:10.188


00:11:10.188 --> 00:11:11.176
Does that exist?

00:11:11.176 --> 00:11:16.120


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And also--

00:11:16.920 --> 00:11:20.320
STUDENT: It's actually--
so the limit is actually

00:11:20.320 --> 00:11:23.640
approaching a plane rather
than a set of [INAUDIBLE].

00:11:23.640 --> 00:11:26.850
MAGDALENA TODA: So
well, actually, it's

00:11:26.850 --> 00:11:28.080
not approaching a plane.

00:11:28.080 --> 00:11:29.800
Let's see what's
happening when--

00:11:29.800 --> 00:11:30.740
STUDENT: Sorry, sorry.

00:11:30.740 --> 00:11:32.417
Not a plane, a [? line. ?]

00:11:32.417 --> 00:11:33.250
MAGDALENA TODA: Yes.

00:11:33.250 --> 00:11:36.037
STUDENT: And is the z-axis--
the entire z-axis is 0, 0?

00:11:36.037 --> 00:11:37.620
MAGDALENA TODA: So
this is the z-axis.

00:11:37.620 --> 00:11:43.930
And that means exactly that
x and y-- it will be 0.

00:11:43.930 --> 00:11:47.030
Now I am just looking
at what happens

00:11:47.030 --> 00:11:50.530
in the plane, in the
floor plane x, y.

00:11:50.530 --> 00:11:53.730
The pairs x, y are wiggly.

00:11:53.730 --> 00:11:56.220
They are like
little wormy worms.

00:11:56.220 --> 00:12:00.680
And they float on the
water on the floor.

00:12:00.680 --> 00:12:03.340
And these squiggly
things approach

00:12:03.340 --> 00:12:05.870
x, y from any possible path.

00:12:05.870 --> 00:12:06.995
They go like this.

00:12:06.995 --> 00:12:09.040
They go like that.

00:12:09.040 --> 00:12:11.192
They go in every possible way.

00:12:11.192 --> 00:12:12.150
Let's see what happens.

00:12:12.150 --> 00:12:15.050


00:12:15.050 --> 00:12:17.550
Continuity-- is this continuous?

00:12:17.550 --> 00:12:19.840
Well, you say,
Magdalena, come on.

00:12:19.840 --> 00:12:21.960
You cannot have this
continuous at 0, 0,

00:12:21.960 --> 00:12:24.480
because it's undefined at 0, 0.

00:12:24.480 --> 00:12:24.980
Yes.

00:12:24.980 --> 00:12:27.410
But maybe I can extend
it by continuity.

00:12:27.410 --> 00:12:31.670
So let me introduce-- this
is my favorite, f of x, y.

00:12:31.670 --> 00:12:35.790
But I'll say, f of x, y
is not defined at 0, 0.

00:12:35.790 --> 00:12:46.775
But how about g of x, y as
being my f of x, y for any x,

00:12:46.775 --> 00:12:50.380
y different from 0, 0.

00:12:50.380 --> 00:12:55.540
And at the origin, at the
very origin, I will say,

00:12:55.540 --> 00:12:59.105
I want to have--
when x, y equals 0,

00:12:59.105 --> 00:13:00.430
0, I want to have a value.

00:13:00.430 --> 00:13:05.460
Which value do you
think might extend

00:13:05.460 --> 00:13:07.420
this function by continuity?

00:13:07.420 --> 00:13:08.890
STUDENT: The limit.

00:13:08.890 --> 00:13:10.650
MAGDALENA TODA: The
limit if it exists

00:13:10.650 --> 00:13:15.960
and if-- well, you know already,
I think, what the limit is

00:13:15.960 --> 00:13:18.780
because some of you
thought about this at home

00:13:18.780 --> 00:13:20.160
for extra credit.

00:13:20.160 --> 00:13:21.550
So it's not fair, right?

00:13:21.550 --> 00:13:22.640
No, I'm just kidding.

00:13:22.640 --> 00:13:26.690
So I claim that maybe--
if I put a 0 here,

00:13:26.690 --> 00:13:28.894
will this be continuous?

00:13:28.894 --> 00:13:31.284
Will g be continuous?

00:13:31.284 --> 00:13:35.600


00:13:35.600 --> 00:13:42.060
So prove, prove either way,
prove, justify your answer

00:13:42.060 --> 00:13:45.665
by a proof, a complete
proof with epsilon delta.

00:13:45.665 --> 00:13:46.425
Proof.

00:13:46.425 --> 00:13:48.300
OK.

00:13:48.300 --> 00:13:48.800
OK.

00:13:48.800 --> 00:13:51.330
So now is a worried face.

00:13:51.330 --> 00:13:52.660
Like, oh, my god.

00:13:52.660 --> 00:13:54.965
This guy is worried
because, oh, my god.

00:13:54.965 --> 00:13:55.786
Epsilon delta.

00:13:55.786 --> 00:13:57.530
Oh, my god.

00:13:57.530 --> 00:13:59.800
But the principle--
the intuition

00:13:59.800 --> 00:14:03.920
tells us that we should look
first at some sort of a graph,

00:14:03.920 --> 00:14:05.290
just like Ryan pointed out.

00:14:05.290 --> 00:14:09.240
One should close their eyes and
imagine a graph of a function

00:14:09.240 --> 00:14:16.800
with-- it's hard to visualize in
3D the graph of a function that

00:14:16.800 --> 00:14:18.740
is a surface.

00:14:18.740 --> 00:14:23.790
This is a surface. z
equals the whole shebang.

00:14:23.790 --> 00:14:29.650
But when I'm going to look
at the one dimensional case

00:14:29.650 --> 00:14:33.680
from last time, we
remember the sine of 1/x

00:14:33.680 --> 00:14:35.210
was a crazy function.

00:14:35.210 --> 00:14:39.320
We called it the harmonica,
well, 20-something years ago

00:14:39.320 --> 00:14:40.710
when I was in high school.

00:14:40.710 --> 00:14:42.736
I was in an advanced
calculus class.

00:14:42.736 --> 00:14:46.170
And our teacher was
not funny at all.

00:14:46.170 --> 00:14:49.260
He was also not teaching much,
gave us a lot of homework,

00:14:49.260 --> 00:14:50.580
very challenging.

00:14:50.580 --> 00:14:54.432
So in order to make our
life a little bit easier,

00:14:54.432 --> 00:14:57.228
we always worked in
groups, which was allowed.

00:14:57.228 --> 00:15:00.960
So we called it a harmonica
because it was oscillating

00:15:00.960 --> 00:15:02.800
like that to the point
that-- you've seen

00:15:02.800 --> 00:15:06.186
the harmonica-- the accordion.

00:15:06.186 --> 00:15:12.940
When you bring it back to
the-- harmonica came to my mind

00:15:12.940 --> 00:15:15.720
from the harmonic function.

00:15:15.720 --> 00:15:19.260
So the accordion is--
when you actually

00:15:19.260 --> 00:15:25.920
squeeze it, all that oscillation
things, the cusps are

00:15:25.920 --> 00:15:28.490
closer and closer to a line.

00:15:28.490 --> 00:15:33.920
So what you have here is
this kind of oscillation,

00:15:33.920 --> 00:15:37.630
very, very rapid
oscillation for sine of 1/x.

00:15:37.630 --> 00:15:41.806
When we want to multiply by
an x, what's going to happen?

00:15:41.806 --> 00:15:47.510
Well, this has not limit at 0
because it takes all the values

00:15:47.510 --> 00:15:49.600
infinitesimally close to 0.

00:15:49.600 --> 00:15:52.480
It keeps going through all the
values between minus 1 and 1,

00:15:52.480 --> 00:15:53.230
closer and closer.

00:15:53.230 --> 00:15:55.560
So that was no good.

00:15:55.560 --> 00:16:03.887
But if we take this guy--
that's going to go to-- well,

00:16:03.887 --> 00:16:05.348
I cannot do better.

00:16:05.348 --> 00:16:07.296
MATLAB can do better than me.

00:16:07.296 --> 00:16:09.244
Mathematica can do better.

00:16:09.244 --> 00:16:10.090
You can do that.

00:16:10.090 --> 00:16:12.600
In most engineering
classes, if you are--

00:16:12.600 --> 00:16:15.690
who is an electrical
engineering major?

00:16:15.690 --> 00:16:18.960
But even if you are
not, you are going

00:16:18.960 --> 00:16:21.250
to see this type
of function a lot.

00:16:21.250 --> 00:16:24.560
And you're going to see it
again in differential equations.

00:16:24.560 --> 00:16:27.150


00:16:27.150 --> 00:16:30.745
How can I imagine-- this
graph is hard to draw.

00:16:30.745 --> 00:16:34.060
Don't ask me to draw that.

00:16:34.060 --> 00:16:41.060
But ask me if I can use epsilon
delta to prove continuity.

00:16:41.060 --> 00:16:44.965
So what would it mean,
proving continuity?

00:16:44.965 --> 00:16:45.895
I have a feeling--

00:16:45.895 --> 00:16:47.290
STUDENT: Well, actually, if this
is-- going back to that graph,

00:16:47.290 --> 00:16:49.010
doesn't that graph look like--

00:16:49.010 --> 00:16:50.301
MAGDALENA TODA: This goes to 0.

00:16:50.301 --> 00:16:53.840
The limit exists for x
sine of 1/x, and it is 0.

00:16:53.840 --> 00:16:55.580
Why?

00:16:55.580 --> 00:16:57.170
Ryan?

00:16:57.170 --> 00:17:01.050
RYAN: Wouldn't the graph
with the x squared plus

00:17:01.050 --> 00:17:02.900
y squared times that
side-- wouldn't that

00:17:02.900 --> 00:17:06.040
just look like a ripple
in a circle going out

00:17:06.040 --> 00:17:07.372
from the center?

00:17:07.372 --> 00:17:09.079
MAGDALENA TODA: Yeah,
it will be ripples.

00:17:09.079 --> 00:17:10.680
STUDENT: Just like a
[INAUDIBLE] from an epicenter

00:17:10.680 --> 00:17:11.960
going outwards [INAUDIBLE].

00:17:11.960 --> 00:17:16.050
MAGDALENA TODA: And I think--
yes, we managed to-- you

00:17:16.050 --> 00:17:19.896
have a concentric image, right?

00:17:19.896 --> 00:17:20.480
STUDENT: Yeah.

00:17:20.480 --> 00:17:22.369
MAGDALENA TODA: Like those
ripples, exactly like--

00:17:22.369 --> 00:17:24.036
STUDENT: So that's
what that looks like?

00:17:24.036 --> 00:17:26.550
MAGDALENA TODA: --when you
throw a stone into the water,

00:17:26.550 --> 00:17:27.680
this kind of wave.

00:17:27.680 --> 00:17:30.730
But it's infinitesimally close.

00:17:30.730 --> 00:17:32.910
It's like acting weird.

00:17:32.910 --> 00:17:37.240
But then it sort
of shrinks here.

00:17:37.240 --> 00:17:40.786
And that-- it
imposes the limit 0.

00:17:40.786 --> 00:17:43.230
How come this goes
to 0, you say?

00:17:43.230 --> 00:17:46.150
Well, Magdalena, this
guy is crazy, right?

00:17:46.150 --> 00:17:49.155
Sine of 1/x goes
between minus 1 and 1

00:17:49.155 --> 00:17:51.500
infinitely many times
as I go close, close,

00:17:51.500 --> 00:17:57.930
closer and closer, more rapidly,
more and more rapidly close

00:17:57.930 --> 00:17:58.890
to 0.

00:17:58.890 --> 00:18:00.535
This will oscillate
more rapidly,

00:18:00.535 --> 00:18:03.190
more rapidly, and more rapidly.

00:18:03.190 --> 00:18:04.630
This is crazy, right?

00:18:04.630 --> 00:18:07.690
How does this guy, x-- how
is this guy taming this guy?

00:18:07.690 --> 00:18:10.445
STUDENT: Because
as 0 [INAUDIBLE].

00:18:10.445 --> 00:18:12.570
Something really small
times something [INAUDIBLE].

00:18:12.570 --> 00:18:14.069
MAGDALENA TODA:
Something very small

00:18:14.069 --> 00:18:17.900
that shrinks to 0 times
something bounded.

00:18:17.900 --> 00:18:20.880
Ryan brought the main idea.

00:18:20.880 --> 00:18:25.160
If something goes strongly to
0, and that multiplies something

00:18:25.160 --> 00:18:28.490
that's bounded, bounded
by a finite number,

00:18:28.490 --> 00:18:31.069
the whole problem will go to 0.

00:18:31.069 --> 00:18:32.961
Actually, you can prove
that as a theorem.

00:18:32.961 --> 00:18:34.860
And some of you did.

00:18:34.860 --> 00:18:36.630
In most honors
classes unfortunately,

00:18:36.630 --> 00:18:39.100
epsilon delta was not covered.

00:18:39.100 --> 00:18:43.350
So let's see how we prove
this with epsilon delta.

00:18:43.350 --> 00:18:45.120
And, oh, my god.

00:18:45.120 --> 00:18:52.825
Many of you read from the book
and may be able to help me.

00:18:52.825 --> 00:19:00.100
So what am I supposed to
show with epsilon delta?

00:19:00.100 --> 00:19:09.860
The limit of x squared plus
y squared sine of 1 over x

00:19:09.860 --> 00:19:14.560
squared plus y squared is
0 as I approach the origin

00:19:14.560 --> 00:19:19.735
with my pair, couple, x, y,
which can go any one path that

00:19:19.735 --> 00:19:20.340
approaches 0.

00:19:20.340 --> 00:19:23.780


00:19:23.780 --> 00:19:27.575
So you say, oh, well, Magdalena,
the Ryan principle-- this

00:19:27.575 --> 00:19:29.070
is the Ryan theorem.

00:19:29.070 --> 00:19:32.180
It's the same because
this guy will be

00:19:32.180 --> 00:19:34.100
bounded between minus 1 and 1.

00:19:34.100 --> 00:19:37.630
I multiplied with a guy
that very determinedly goes

00:19:37.630 --> 00:19:39.540
to 0 very strongly.

00:19:39.540 --> 00:19:41.160
And he knows where he's going.

00:19:41.160 --> 00:19:44.020
x squared plus y squared
says, I know what I'm doing.

00:19:44.020 --> 00:19:45.880
I'm not going to change my mind.

00:19:45.880 --> 00:19:49.480
This is like the guy who changes
his major too many times.

00:19:49.480 --> 00:19:52.210
And this guy knows
what he's doing.

00:19:52.210 --> 00:19:54.920
He's going there, and he's
a polynomial, goes to 0,

00:19:54.920 --> 00:19:56.290
0 very rapidly.

00:19:56.290 --> 00:20:00.520
Now it's clear what
happens intuitively.

00:20:00.520 --> 00:20:02.910
But I'm a mathematician.

00:20:02.910 --> 00:20:07.040
And if I don't publish
my proof, my article

00:20:07.040 --> 00:20:12.675
will be very nicely rejected
by all the serious journals

00:20:12.675 --> 00:20:13.970
on the market.

00:20:13.970 --> 00:20:17.470
This is how it goes
in mathematics.

00:20:17.470 --> 00:20:19.390
Even before journals
existed, mathematicians

00:20:19.390 --> 00:20:23.110
had to show a rigorous
proof of their work,

00:20:23.110 --> 00:20:25.720
of their conjecture.

00:20:25.720 --> 00:20:26.920
OK.

00:20:26.920 --> 00:20:35.080
So I go, for every epsilon
positive, no matter how small,

00:20:35.080 --> 00:20:41.040
there must exist a
delta positive, which

00:20:41.040 --> 00:20:51.990
depends on epsilon-- that
depends on epsilon-- such that

00:20:51.990 --> 00:20:58.550
as soon as-- how did
we write the distance?

00:20:58.550 --> 00:21:01.820
I'll write the distance
again because I'm lazy.

00:21:01.820 --> 00:21:05.590
The distance between the
point x, y and the origin

00:21:05.590 --> 00:21:07.931
is less than delta.

00:21:07.931 --> 00:21:16.720
It follows that the
absolute value--

00:21:16.720 --> 00:21:24.330
these are all real numbers--
of f of x, y or g of x,

00:21:24.330 --> 00:21:27.456
y-- g of x, y is the extension.

00:21:27.456 --> 00:21:32.150


00:21:32.150 --> 00:21:36.350
f of x, y minus 0, which
I claim to be the limit,

00:21:36.350 --> 00:21:39.135
will be less than epsilon.

00:21:39.135 --> 00:21:40.500
So you go, oh, my god.

00:21:40.500 --> 00:21:42.830
What is this woman doing?

00:21:42.830 --> 00:21:43.820
It's not hard.

00:21:43.820 --> 00:21:45.980
I need your help though.

00:21:45.980 --> 00:21:48.540
I need your help to do that.

00:21:48.540 --> 00:21:53.100
So it's hard to see how you
should-- you take any epsilon.

00:21:53.100 --> 00:21:58.300
You pick your favorite
epsilon, infinitesimally small,

00:21:58.300 --> 00:22:01.120
any small number, but
then you go, but then I

00:22:01.120 --> 00:22:03.400
have to show this delta exists.

00:22:03.400 --> 00:22:06.590
You have to grab that delta
and say, you are my delta.

00:22:06.590 --> 00:22:08.990
You cannot escape me.

00:22:08.990 --> 00:22:10.926
I tell you who you are.

00:22:10.926 --> 00:22:13.840
And that's the
hardest part in here,

00:22:13.840 --> 00:22:18.240
figuring out who that delta must
be as a function of epsilon.

00:22:18.240 --> 00:22:19.060
Is that hard?

00:22:19.060 --> 00:22:21.320
How do you build
such a construction?

00:22:21.320 --> 00:22:26.690
First of all,
understand what proof.

00:22:26.690 --> 00:22:30.385
"Choose any positive epsilon."

00:22:30.385 --> 00:22:32.885
Then forget about him,
because he's your friend,

00:22:32.885 --> 00:22:36.400
and he's going to do whatever
you want to do with him.

00:22:36.400 --> 00:22:40.120
Delta, chasing
after delta is going

00:22:40.120 --> 00:22:42.390
to be a little bit harder.

00:22:42.390 --> 00:22:56.010
"Chasing after delta
with that property."

00:22:56.010 --> 00:22:58.430
Dot, dot, dot, dot, dot.

00:22:58.430 --> 00:22:59.700
What is this distance?

00:22:59.700 --> 00:23:01.950
You guys have
helped me last time,

00:23:01.950 --> 00:23:04.670
you cannot let me down now.

00:23:04.670 --> 00:23:08.350
So as soon as this distance,
your gradient distance

00:23:08.350 --> 00:23:10.540
is less than delta,
you must have

00:23:10.540 --> 00:23:13.312
that f of x, y [INAUDIBLE].

00:23:13.312 --> 00:23:15.170
Could you tell me
what that would be?

00:23:15.170 --> 00:23:16.170
It was Euclidean, right?

00:23:16.170 --> 00:23:21.670
So I had squared
root of-- did I?

00:23:21.670 --> 00:23:30.396
Square root of x minus 0
squared plus y minus 0 squared.

00:23:30.396 --> 00:23:33.170
You say, but that's
silly, Magdalena.

00:23:33.170 --> 00:23:37.536
So you have to write
it down like that?

00:23:37.536 --> 00:23:38.970
STUDENT: It's the [INAUDIBLE].

00:23:38.970 --> 00:23:40.410
MAGDALENA TODA: Huh?

00:23:40.410 --> 00:23:42.290
Yeah.

00:23:42.290 --> 00:23:47.174
So square root of this
plus square root of that

00:23:47.174 --> 00:23:53.030
plus then delta,
that means what?

00:23:53.030 --> 00:24:00.351
If and only if x squared plus
y squared is less than delta

00:24:00.351 --> 00:24:00.850
squared.

00:24:00.850 --> 00:24:08.160


00:24:08.160 --> 00:24:11.110
And what do I want to do,
what do I want to build?

00:24:11.110 --> 00:24:15.090


00:24:15.090 --> 00:24:19.030
So we are thinking how
to set up all this thing.

00:24:19.030 --> 00:24:21.209
How to choose the delta.

00:24:21.209 --> 00:24:23.061
How to choose the delta.

00:24:23.061 --> 00:24:25.830


00:24:25.830 --> 00:24:28.350
OK, so what do I--
what am I after?

00:24:28.350 --> 00:24:34.253
"I am after having" double dot.

00:24:34.253 --> 00:24:39.830
F of x, y must be Mr. Ugly.

00:24:39.830 --> 00:24:40.820
This one.

00:24:40.820 --> 00:24:46.480
So absolute value of x squared
plus y squared, sine of 1

00:24:46.480 --> 00:24:51.241
over x squared plus
y squared minus 0.

00:24:51.241 --> 00:24:51.740
Duh.

00:24:51.740 --> 00:24:55.170
I'm not going to write it.

00:24:55.170 --> 00:24:59.260
We all know what that means.

00:24:59.260 --> 00:25:00.090
Less than epsilon.

00:25:00.090 --> 00:25:05.680
This is what must
follow as a conclusion.

00:25:05.680 --> 00:25:12.360
This is what must
follow, must happen.

00:25:12.360 --> 00:25:13.334
Must happen.

00:25:13.334 --> 00:25:16.260


00:25:16.260 --> 00:25:17.630
Now I'm getting excited.

00:25:17.630 --> 00:25:18.130
Why?

00:25:18.130 --> 00:25:21.050
Because I am thinking.

00:25:21.050 --> 00:25:23.040
I started thinking.

00:25:23.040 --> 00:25:26.420
Once I started thinking,
I'm dangerous, man.

00:25:26.420 --> 00:25:31.530
So here sine of 1 over x squared
plus y squared is your friend.

00:25:31.530 --> 00:25:34.210
Why is that your friend?

00:25:34.210 --> 00:25:37.250
Sine of 1 over x squared
plus y squared, this

00:25:37.250 --> 00:25:39.115
is always an absolute value.

00:25:39.115 --> 00:25:42.977
The absolute value of that
is always less than 1.

00:25:42.977 --> 00:25:43.476
OK?

00:25:43.476 --> 00:25:45.290
STUDENT: Can't it be 4?

00:25:45.290 --> 00:25:50.280
MAGDALENA TODA:
So-- so-- so what

00:25:50.280 --> 00:25:54.940
shall I take in terms of
delta-- this is my question.

00:25:54.940 --> 00:25:57.400
What shall I take
in terms of delta?

00:25:57.400 --> 00:26:03.590
"Delta equals 1 as a
function of epsilon

00:26:03.590 --> 00:26:20.160
in order to have the
conclusion satisfied."

00:26:20.160 --> 00:26:20.920
You say, OK.

00:26:20.920 --> 00:26:24.610
It's enough to choose delta
like that function of epsilon,

00:26:24.610 --> 00:26:28.840
and I'm done, because then
everything will be fine.

00:26:28.840 --> 00:26:33.510
So you chose your own epsilon,
positive, small, or God

00:26:33.510 --> 00:26:34.480
gave you an epsilon.

00:26:34.480 --> 00:26:37.040
You don't care how
you got the epsilon.

00:26:37.040 --> 00:26:38.423
The epsilon is arbitrary.

00:26:38.423 --> 00:26:40.910
You pick positive and small.

00:26:40.910 --> 00:26:44.610
Now, it's up to
you to find delta.

00:26:44.610 --> 00:26:48.820
So what delta would
satisfy everything?

00:26:48.820 --> 00:26:50.940
What delta would
be good enough--

00:26:50.940 --> 00:26:52.830
you don't care
for all the good--

00:26:52.830 --> 00:26:54.550
it's like when you get married.

00:26:54.550 --> 00:26:57.740
Do you care for all the
people who'd match you?

00:26:57.740 --> 00:27:00.596
Hopefully not, because
then you would probably

00:27:00.596 --> 00:27:05.416
have too large of a pool,
and it's hard to choose.

00:27:05.416 --> 00:27:13.285
You only need one that satisfies
that assumption, that satisfies

00:27:13.285 --> 00:27:14.974
all the conditions you have.

00:27:14.974 --> 00:27:18.780
So what is the delta that
satisfies all the conditions

00:27:18.780 --> 00:27:20.047
that I have?

00:27:20.047 --> 00:27:20.880
[INTERPOSING VOICES]

00:27:20.880 --> 00:27:22.270
MAGDALENA TODA: [INAUDIBLE].

00:27:22.270 --> 00:27:22.960
Who?

00:27:22.960 --> 00:27:25.350
[INTERPOSING VOICES]

00:27:25.350 --> 00:27:27.855
MAGDALENA TODA: For example,
delta equals epsilon.

00:27:27.855 --> 00:27:28.785
Would that satisfy?

00:27:28.785 --> 00:27:31.984


00:27:31.984 --> 00:27:33.860
Well, let's see.

00:27:33.860 --> 00:27:37.410
If I take delta to
be epsilon, then x

00:27:37.410 --> 00:27:40.410
squared plus y squared would
be less than epsilon squared.

00:27:40.410 --> 00:27:47.331
Now the question is is epsilon
squared less than epsilon?

00:27:47.331 --> 00:27:48.510
Not always.

00:27:48.510 --> 00:27:52.920
If epsilon is between 0
and 1, then epsilon squared

00:27:52.920 --> 00:27:54.200
is less then epsilon.

00:27:54.200 --> 00:27:59.200
But if I choose epsilon
to be greater than 1,

00:27:59.200 --> 00:28:00.220
then oh, my God.

00:28:00.220 --> 00:28:02.740
Then if it's greater than
1, then epsilon squared

00:28:02.740 --> 00:28:06.790
is greater than 1--
greater than it.

00:28:06.790 --> 00:28:14.700
So what if I choose
delta to be what?

00:28:14.700 --> 00:28:18.652


00:28:18.652 --> 00:28:19.595
STUDENT: 0?

00:28:19.595 --> 00:28:20.720
MAGDALENA TODA: No, no, no.

00:28:20.720 --> 00:28:22.090
Delta cannot be 0.

00:28:22.090 --> 00:28:26.330
So delta-- look, there exists
delta strictly bigger than 0,

00:28:26.330 --> 00:28:28.690
that depends on epsilon.

00:28:28.690 --> 00:28:33.673
Maybe if epsilon is very small,
in a way Alexander was right.

00:28:33.673 --> 00:28:37.350
But the delta [INAUDIBLE],
we don't go with epsilon

00:28:37.350 --> 00:28:38.280
greater than 1.

00:28:38.280 --> 00:28:39.000
Come on.

00:28:39.000 --> 00:28:39.500
Be serious.

00:28:39.500 --> 00:28:42.300
Epsilon is always
between 0 and 1.

00:28:42.300 --> 00:28:44.603
I mean, it's a lot
smaller than that.

00:28:44.603 --> 00:28:46.640
It's infinitesimal small.

00:28:46.640 --> 00:28:49.480
So in the end, yes, in
that case epsilon squared

00:28:49.480 --> 00:28:52.610
would be less than epsilon,
which would be OK for us

00:28:52.610 --> 00:28:54.590
and that would be fine.

00:28:54.590 --> 00:28:56.150
OK?

00:28:56.150 --> 00:28:58.380
So that would be a
possibility to say, hey,

00:28:58.380 --> 00:29:01.080
since epsilon-- Alexander,
if you write that as a proof

00:29:01.080 --> 00:29:01.815
I'll be OK.

00:29:01.815 --> 00:29:04.900
You say, I took my epsilon
to be a very small number,

00:29:04.900 --> 00:29:07.020
so anyway it's going
to be less than 1.

00:29:07.020 --> 00:29:09.190
So epsilon squared
is less than epsilon.

00:29:09.190 --> 00:29:14.090
So when I take
delta to be epsilon,

00:29:14.090 --> 00:29:18.210
for sure this guy will be less
than epsilon squared, which

00:29:18.210 --> 00:29:21.148
is less than epsilon,
so I'm satisfied.

00:29:21.148 --> 00:29:22.615
I'll give you a 100%.

00:29:22.615 --> 00:29:24.082
I'm happy.

00:29:24.082 --> 00:29:25.152
Is that the only way?

00:29:25.152 --> 00:29:26.527
STUDENT: But what
about the sine?

00:29:26.527 --> 00:29:27.505
What about [INAUDIBLE].

00:29:27.505 --> 00:29:28.483
STUDENT: Yeah.

00:29:28.483 --> 00:29:30.108
MAGDALENA TODA: So
this doesn't matter.

00:29:30.108 --> 00:29:32.395
Let me write it down.

00:29:32.395 --> 00:29:39.730
So note that x squared
plus y squared sine of 1

00:29:39.730 --> 00:29:42.650
over x squared plus
y square would always

00:29:42.650 --> 00:29:46.380
be less than absolute
value of x squared

00:29:46.380 --> 00:29:49.970
plus y, which is positive.

00:29:49.970 --> 00:29:52.145
Why is that?

00:29:52.145 --> 00:29:53.200
Is this true?

00:29:53.200 --> 00:29:54.080
Yeah.

00:29:54.080 --> 00:29:55.441
Why is that?

00:29:55.441 --> 00:29:58.150
STUDENT: Because the sine can
only be one of these negatives.

00:29:58.150 --> 00:30:00.445
MAGDALENA TODA: So
in absolute value,

00:30:00.445 --> 00:30:05.780
sine of 1 over x squared plus y
squared is always less than 1.

00:30:05.780 --> 00:30:08.525
STUDENT: Can't it equal 1?

00:30:08.525 --> 00:30:11.885
MAGDALENA TODA: Well,
when does it equal 1?

00:30:11.885 --> 00:30:14.310
STUDENT: Wouldn't it be x
squared plus y squared equals 1

00:30:14.310 --> 00:30:15.647
[INAUDIBLE]?

00:30:15.647 --> 00:30:17.230
MAGDALENA TODA: Less
than or equal to.

00:30:17.230 --> 00:30:18.480
For some values it will.

00:30:18.480 --> 00:30:19.160
STUDENT: Yeah.

00:30:19.160 --> 00:30:19.659
OK.

00:30:19.659 --> 00:30:21.870
MAGDALENA TODA: Now, will
that be a problem with us?

00:30:21.870 --> 00:30:22.120
No.

00:30:22.120 --> 00:30:23.060
Let's put it here.

00:30:23.060 --> 00:30:27.370
Less than or equal to x
squared plus y squared, which

00:30:27.370 --> 00:30:35.465
has to be less than epsilon
if and only if-- well,

00:30:35.465 --> 00:30:38.780
if delta is what?

00:30:38.780 --> 00:30:41.240
So, again, Alexander said,
well, but if I take delta

00:30:41.240 --> 00:30:42.820
to be epsilon, I'm done.

00:30:42.820 --> 00:30:45.760


00:30:45.760 --> 00:30:46.740
STUDENT: [INAUDIBLE].

00:30:46.740 --> 00:30:49.690
MAGDALENA TODA: How
about square root?

00:30:49.690 --> 00:30:52.150
Can I take delta to be
square root of epsilon.

00:30:52.150 --> 00:30:53.541
STUDENT: That's what I said.

00:30:53.541 --> 00:30:54.332
MAGDALENA TODA: No.

00:30:54.332 --> 00:30:55.694
You said epsilon.

00:30:55.694 --> 00:30:57.319
STUDENT: I said square
root of epsilon.

00:30:57.319 --> 00:30:58.270
MAGDALENA TODA: OK.

00:30:58.270 --> 00:31:01.290
If delta is square
root of epsilon,

00:31:01.290 --> 00:31:05.280
then everything will be perfect
and it will be a perfect match.

00:31:05.280 --> 00:31:05.997
In what case?

00:31:05.997 --> 00:31:07.705
STUDENT: If epsilon
is in between 0 and 1

00:31:07.705 --> 00:31:10.130
and if delta is equal
to bigger than epsilon.

00:31:10.130 --> 00:31:13.060


00:31:13.060 --> 00:31:17.740
MAGDALENA TODA: So that's
exactly the same assumption.

00:31:17.740 --> 00:31:22.480
Epsilon should be
made in less than.

00:31:22.480 --> 00:31:24.327
STUDENT: But I thought
delta was supposed

00:31:24.327 --> 00:31:25.910
to be less than
epsilon in every case.

00:31:25.910 --> 00:31:29.340
So if epsilon is between 0 and
1, the square root of epsilon

00:31:29.340 --> 00:31:31.800
is going to be [INAUDIBLE].

00:31:31.800 --> 00:31:38.330
MAGDALENA TODA: So when
both of them are small,

00:31:38.330 --> 00:31:45.040
delta squared will be-- if
I take delta-- so take delta

00:31:45.040 --> 00:31:47.698
to be square root of epsilon.

00:31:47.698 --> 00:31:50.118
STUDENT: Then anything less
than 1 and greater than 0,

00:31:50.118 --> 00:31:51.784
epsilon would be great
than [INAUDIBLE].

00:31:51.784 --> 00:31:54.958
MAGDALENA TODA: "Delta to
be square root of epsilon,

00:31:54.958 --> 00:32:01.570
then x squared plus y squared
less than delta squared equals

00:32:01.570 --> 00:32:03.780
epsilon."

00:32:03.780 --> 00:32:11.805
Then x squared plus
y squared sine of 1

00:32:11.805 --> 00:32:14.990
over x squared plus
y squared less than

00:32:14.990 --> 00:32:17.440
or equal to x squared
plus y squared.

00:32:17.440 --> 00:32:19.155
I dont' need the absolute value.

00:32:19.155 --> 00:32:20.380
I can [INAUDIBLE].

00:32:20.380 --> 00:32:23.320
Less than epsilon [INAUDIBLE].

00:32:23.320 --> 00:32:24.110
Qed.

00:32:24.110 --> 00:32:26.050
STUDENT: Well, but
you told us delta

00:32:26.050 --> 00:32:27.505
has to be less than epsilon.

00:32:27.505 --> 00:32:28.475
Well, if--

00:32:28.475 --> 00:32:31.390
MAGDALENA TODA: No,
I didn't say that.

00:32:31.390 --> 00:32:35.325
I didn't say that delta has
to be less than epsilon.

00:32:35.325 --> 00:32:35.825
Absolutely--

00:32:35.825 --> 00:32:36.408
STUDENT: Yeah.

00:32:36.408 --> 00:32:38.759
You said for all the values
of epsilon greater than 0,

00:32:38.759 --> 00:32:42.280
there's a value of delta that is
greater than 0 that [INAUDIBLE]

00:32:42.280 --> 00:32:45.648
such that as soon as the
distance between is less than

00:32:45.648 --> 00:32:46.959
delta-- I don't remember what--

00:32:46.959 --> 00:32:48.250
MAGDALENA TODA: OK, so, again--

00:32:48.250 --> 00:32:50.166
STUDENT: Such that the
distance is less than--

00:32:50.166 --> 00:32:52.255
MAGDALENA TODA: So again,
for epsilon positive,

00:32:52.255 --> 00:32:56.600
there is a delta
positive, very small.

00:32:56.600 --> 00:32:58.740
Very small means very small, OK?

00:32:58.740 --> 00:33:01.440
I'm not threatened by-- what?

00:33:01.440 --> 00:33:04.688
For epsilon greater
than 0, very small,

00:33:04.688 --> 00:33:07.128
there is a delta greater
than 0, very small,

00:33:07.128 --> 00:33:10.544
which depends on epsilon-- I
didn't say it cannot be equal

00:33:10.544 --> 00:33:21.100
to epsilon-- that depends on
epsilon such that whenever x,

00:33:21.100 --> 00:33:30.012
y is within delta
distance from origin,

00:33:30.012 --> 00:33:45.170
[INAUDIBLE] that f of x, y
is within epsilon of from l.

00:33:45.170 --> 00:33:47.840


00:33:47.840 --> 00:33:48.340
All right?

00:33:48.340 --> 00:33:52.720
And now I will actually give
you another example where

00:33:52.720 --> 00:33:55.970
maybe delta will be epsilon.

00:33:55.970 --> 00:33:59.430
And let me challenge you
with another problem that's

00:33:59.430 --> 00:34:00.530
not hard.

00:34:00.530 --> 00:34:01.295
OK?

00:34:01.295 --> 00:34:03.650
So let me give
you the function g

00:34:03.650 --> 00:34:16.460
of x, y equals x sine
of 1 over y as x, y.

00:34:16.460 --> 00:34:19.150


00:34:19.150 --> 00:34:29.782
y is equal [? to delta 0. ?]
And let's say 0 for the rest.

00:34:29.782 --> 00:34:35.500


00:34:35.500 --> 00:34:48.510
Can you show-- can you check
if g is continuous at 0, 0?

00:34:48.510 --> 00:34:55.510


00:34:55.510 --> 00:34:58.510
This is one of the
problems in your book.

00:34:58.510 --> 00:35:02.300
So how do you check
that with epsilon delta?

00:35:02.300 --> 00:35:04.130
Again, we recite the poetry.

00:35:04.130 --> 00:35:05.555
We have to say that.

00:35:05.555 --> 00:35:11.950
"For every epsilon
positive, small, very small,

00:35:11.950 --> 00:35:16.000
there is a delta
positive that depends

00:35:16.000 --> 00:35:33.830
on epsilon, such that as soon
as--" how is the distance?

00:35:33.830 --> 00:35:42.424
Square root of x squared plus
y squared is less than delta.

00:35:42.424 --> 00:35:46.861
This is the distance
between point and origin.

00:35:46.861 --> 00:36:09.260
"It follows that absolute value
of x sine of 1 over y minus--"

00:36:09.260 --> 00:36:12.247
so practically x, y no 0.

00:36:12.247 --> 00:36:16.223
x, y different from 0.

00:36:16.223 --> 00:36:17.720
OK?

00:36:17.720 --> 00:36:21.720
I"m careful here, because
if y is 0, then I blow up.

00:36:21.720 --> 00:36:23.080
And I don't want to blow up.

00:36:23.080 --> 00:36:25.850
So x sine of 1 over y minus who?

00:36:25.850 --> 00:36:30.810
Minus 0 is less than epsilon.

00:36:30.810 --> 00:36:32.850
So now you're thinking,
OK, you want me

00:36:32.850 --> 00:36:34.760
to prove there is such a delta?

00:36:34.760 --> 00:36:35.830
Yes.

00:36:35.830 --> 00:36:37.060
That depends on epsilon?

00:36:37.060 --> 00:36:38.850
Yes.

00:36:38.850 --> 00:36:40.430
And what would that delta be?

00:36:40.430 --> 00:36:43.710
The simplest choice you
can have in this case.

00:36:43.710 --> 00:36:45.080
So you go, oh, my God.

00:36:45.080 --> 00:36:46.070
How do I do that?

00:36:46.070 --> 00:36:48.470
You have to always
think backwards.

00:36:48.470 --> 00:36:58.790
So "we need to satisfy
absolute value of x sine of 1

00:36:58.790 --> 00:37:02.430
over y less than epsilon."

00:37:02.430 --> 00:37:05.730
Is this hard?

00:37:05.730 --> 00:37:10.295
What is your advantage here?

00:37:10.295 --> 00:37:13.570
Do you have any advantage?

00:37:13.570 --> 00:37:19.860
Remark absolute value
of x sine of 1 over y

00:37:19.860 --> 00:37:22.690
is smaller than who?

00:37:22.690 --> 00:37:26.670
Smaller than the product
of absolute values.

00:37:26.670 --> 00:37:27.580
Say it again?

00:37:27.580 --> 00:37:28.430
Yes?

00:37:28.430 --> 00:37:32.005
STUDENT: But, like, for
example, the only condition

00:37:32.005 --> 00:37:35.290
for that equation is that
y must not be equal to 0.

00:37:35.290 --> 00:37:38.520
What if you used
another point for x?

00:37:38.520 --> 00:37:43.170
Would the answer for
delta be different?

00:37:43.170 --> 00:37:45.240
MAGDALENA TODA:
Well, x is-- you can

00:37:45.240 --> 00:37:49.000
choose-- you were right here.

00:37:49.000 --> 00:37:52.990
You can say, OK, can you be
more restrictive, Magdelena,

00:37:52.990 --> 00:37:58.710
and say, for every point
of the type x equals 0

00:37:58.710 --> 00:38:01.470
and y not 0, it's still OK?

00:38:01.470 --> 00:38:03.490
Yes.

00:38:03.490 --> 00:38:07.000
So you could be a
professional mathematician.

00:38:07.000 --> 00:38:14.322
So practically all I care
about is x, y in the disk.

00:38:14.322 --> 00:38:15.410
What disk?

00:38:15.410 --> 00:38:16.810
What is this disk?

00:38:16.810 --> 00:38:24.150
Disk of radius 0 when--
what is the radius?

00:38:24.150 --> 00:38:31.930
Delta-- such that your
y should not be 0.

00:38:31.930 --> 00:38:35.990
So a more rigorous
point would be

00:38:35.990 --> 00:38:38.630
like take all the
couples that are

00:38:38.630 --> 00:38:43.600
in this small disk
of radius delta,

00:38:43.600 --> 00:38:46.020
except for those where y is 0.

00:38:46.020 --> 00:38:48.650
So what do you actually remove?

00:38:48.650 --> 00:38:54.860
You remove this stinking line.

00:38:54.860 --> 00:39:01.080
But everybody else in this
disk, every couple in this disk

00:39:01.080 --> 00:39:03.865
should be happy,
should be analyzed

00:39:03.865 --> 00:39:06.080
as part of this thread.

00:39:06.080 --> 00:39:08.100
Right?

00:39:08.100 --> 00:39:09.070
OK.

00:39:09.070 --> 00:39:13.180
x sine of 1 over y less
than-- is that true?

00:39:13.180 --> 00:39:16.120
Is that less than the
absolute value of x?

00:39:16.120 --> 00:39:16.930
STUDENT: Yeah.

00:39:16.930 --> 00:39:17.846
MAGDALENA TODA: Right.

00:39:17.846 --> 00:39:20.950
So it should be-- less
than should be made

00:39:20.950 --> 00:39:23.570
should be less than epsilon.

00:39:23.570 --> 00:39:27.040
When is this happening
on that occasion?

00:39:27.040 --> 00:39:28.320
If I take delta-- meh?

00:39:28.320 --> 00:39:29.570
STUDENT: When delta's epsilon.

00:39:29.570 --> 00:39:31.310
MAGDALENA TODA: So if
I take-- very good.

00:39:31.310 --> 00:39:35.570
So Alex saw that, hey,
Magdelena, your proof is over.

00:39:35.570 --> 00:39:37.700
And I mean it's over.

00:39:37.700 --> 00:39:42.990
Take delta, which is delta
of epsilon, to be epsilon.

00:39:42.990 --> 00:39:44.350
You're done.

00:39:44.350 --> 00:39:45.520
Why?

00:39:45.520 --> 00:39:47.590
Let me explain what
Alex wants, because he

00:39:47.590 --> 00:39:50.280
doesn't want to explain
much, but it's not his job.

00:39:50.280 --> 00:39:51.260
He's not your teacher.

00:39:51.260 --> 00:39:51.920
Right?

00:39:51.920 --> 00:39:54.346
So why is this working?

00:39:54.346 --> 00:40:02.850
Because in this case,
note that if I take delta

00:40:02.850 --> 00:40:05.650
to be exactly epsilon,
what's going to happen?

00:40:05.650 --> 00:40:08.620


00:40:08.620 --> 00:40:13.760
x, Mr. x, could be
positive or negative.

00:40:13.760 --> 00:40:15.950
See, x could be
positive or negative.

00:40:15.950 --> 00:40:18.810
Let's take this guy and
protect him in absolute value.

00:40:18.810 --> 00:40:23.350
He's always less than square
root of x square plus y

00:40:23.350 --> 00:40:25.650
squared.

00:40:25.650 --> 00:40:27.200
Why is that, guys?

00:40:27.200 --> 00:40:30.730
STUDENT: Because y can't be 0.

00:40:30.730 --> 00:40:34.482
MAGDALENA TODA: So this
is-- square it in your mind.

00:40:34.482 --> 00:40:36.690
You got x squared less than
x squared plus y squared.

00:40:36.690 --> 00:40:39.000
So this is always true.

00:40:39.000 --> 00:40:40.640
Always satisfied.

00:40:40.640 --> 00:40:44.940
But we chose this to
be less than delta,

00:40:44.940 --> 00:40:49.450
and if we choose delta to be
epsilon, that's our choice.

00:40:49.450 --> 00:40:54.310
So God gave us the epsilon,
but delta is our choice,

00:40:54.310 --> 00:40:57.090
because you have to prove
you can do something

00:40:57.090 --> 00:40:57.910
with your life.

00:40:57.910 --> 00:40:58.410
Right?

00:40:58.410 --> 00:41:00.700
So delta equals epsilon.

00:41:00.700 --> 00:41:02.670
If you take delta
equals epsilon,

00:41:02.670 --> 00:41:06.440
then you're done, because
in that case absolute value

00:41:06.440 --> 00:41:11.980
of x is less than epsilon, and
your conclusion, which is this,

00:41:11.980 --> 00:41:13.690
was satisfied.

00:41:13.690 --> 00:41:16.680
Now, if a student
is really smart--

00:41:16.680 --> 00:41:20.650
one time I had a student,
I gave him this proof.

00:41:20.650 --> 00:41:22.380
That was several
years ago in honors,

00:41:22.380 --> 00:41:24.620
because we don't do epsilon
delta in non-honors.

00:41:24.620 --> 00:41:28.200
And we very rarely do
it in honors as well.

00:41:28.200 --> 00:41:31.140
His proof consisted of this.

00:41:31.140 --> 00:41:34.060
Considering the fact that
absolute value of sine

00:41:34.060 --> 00:41:38.300
is less than 1, if I
take delta to be epsilon,

00:41:38.300 --> 00:41:39.870
that is sufficient.

00:41:39.870 --> 00:41:41.775
I'm done.

00:41:41.775 --> 00:41:44.470
And of course I gave
him 100%, because this

00:41:44.470 --> 00:41:46.030
is the essence of the proof.

00:41:46.030 --> 00:41:48.090
He didn't show any details.

00:41:48.090 --> 00:41:52.180
And I thought, this is the
kind of guy who is great.

00:41:52.180 --> 00:41:55.920
He's very smart, but he's not
going to make a good teacher.

00:41:55.920 --> 00:41:59.380
So he's probably going to
be the next researcher,

00:41:59.380 --> 00:42:04.810
the next astronaut, the next
something else, but not--

00:42:04.810 --> 00:42:11.040
And then, years later, he
took advanced calculus.

00:42:11.040 --> 00:42:13.635
He graduated with
a graduate degree

00:42:13.635 --> 00:42:17.600
in three years sponsored
by the Air Force.

00:42:17.600 --> 00:42:20.750
And he works right
now for the Air Force.

00:42:20.750 --> 00:42:24.420
He came out dressed
as a captain.

00:42:24.420 --> 00:42:28.840
He came and gave a talk this
year at Tech in a conference--

00:42:28.840 --> 00:42:29.770
he was rushed.

00:42:29.770 --> 00:42:32.150
I mean, if I talk
like that, my student

00:42:32.150 --> 00:42:33.810
wouldn't be able to follow me.

00:42:33.810 --> 00:42:38.320
But he was the same brilliant
student that I remember.

00:42:38.320 --> 00:42:46.280
So he's working on some very
important top secret projects.

00:42:46.280 --> 00:42:48.900
Very intelligent guy.

00:42:48.900 --> 00:42:52.520
And every now and than going
to give talks at conferences.

00:42:52.520 --> 00:42:58.170
Like, research talks
about what he's doing.

00:42:58.170 --> 00:43:01.958
In his class-- he took
advanced calculus with me,

00:43:01.958 --> 00:43:04.090
which was actually graduate
level [INAUDIBLE]--

00:43:04.090 --> 00:43:09.180
I explained epsilon delta, and
he had it very well understood.

00:43:09.180 --> 00:43:13.270
And after I left the classroom
he explained it to his peers,

00:43:13.270 --> 00:43:15.050
to his classmates.

00:43:15.050 --> 00:43:16.885
And he explained
it better than me.

00:43:16.885 --> 00:43:21.010
And I was there listening,
and I remember being jealous,

00:43:21.010 --> 00:43:23.050
because although
he was very rushed,

00:43:23.050 --> 00:43:27.430
he had a very clear
understanding of how

00:43:27.430 --> 00:43:31.140
you take an epsilon, no
matter how small, and then

00:43:31.140 --> 00:43:34.200
you take a little ball
here, radius delta.

00:43:34.200 --> 00:43:38.680
So the image of that little
ball will fit in that ball

00:43:38.680 --> 00:43:40.110
that you take here.

00:43:40.110 --> 00:43:43.510
So even if you
shrink on the image,

00:43:43.510 --> 00:43:46.020
you can take this
ball even smaller

00:43:46.020 --> 00:43:48.670
so the image will
still fit inside.

00:43:48.670 --> 00:43:51.490
And I was going, gosh,
this is the essence,

00:43:51.490 --> 00:43:54.660
but I wish I could convey
it, because no book

00:43:54.660 --> 00:43:58.550
will say it just-- or show you
how to do it with your hands.

00:43:58.550 --> 00:43:59.202


00:43:59.202 --> 00:44:00.035
STUDENT: [INAUDIBLE]

00:44:00.035 --> 00:44:00.951
MAGDALENA TODA: Right.

00:44:00.951 --> 00:44:04.985
So he was rushed, but he
had a very clear picture

00:44:04.985 --> 00:44:06.965
of what is going on.

00:44:06.965 --> 00:44:07.955
OK.

00:44:07.955 --> 00:44:10.925
11.3 is a completely new start.

00:44:10.925 --> 00:44:13.895
And you are gonna read
and be happy about that

00:44:13.895 --> 00:44:16.370
because that's
partial derivatives.

00:44:16.370 --> 00:44:20.360
And you say, Magdalena,
finally, this is piece of cake.

00:44:20.360 --> 00:44:22.700
You see, I know these things.

00:44:22.700 --> 00:44:25.760
I can do them in
my-- in my sleep.

00:44:25.760 --> 00:44:29.960
So f of x and y
is still a graph.

00:44:29.960 --> 00:44:33.115
And then you say,
how do we introduce

00:44:33.115 --> 00:44:37.740
the partial derivative with
respect to one variable only.

00:44:37.740 --> 00:44:39.768
You think, I draw the graph.

00:44:39.768 --> 00:44:41.640
OK.

00:44:41.640 --> 00:44:44.748
On this graph, I
pick a point x0, y0.

00:44:44.748 --> 00:44:53.950
And if I were to take x to
be 0, what is-- what is the z

00:44:53.950 --> 00:44:56.155
equals f of x0, y?

00:44:56.155 --> 00:45:02.940


00:45:02.940 --> 00:45:04.120
So I'll try to draw it.

00:45:04.120 --> 00:45:05.120
It's not easy.

00:45:05.120 --> 00:45:10.580


00:45:10.580 --> 00:45:16.410
This is x and y and z, and you
want your x0 to be a constant.

00:45:16.410 --> 00:45:17.457
STUDENT: [INAUDIBLE]

00:45:17.457 --> 00:45:19.540
MAGDALENA TODA: It's a
so-called coordinate curve.

00:45:19.540 --> 00:45:20.130
Very good.

00:45:20.130 --> 00:45:23.085
It's a curve, but I want to
be good enough to draw it.

00:45:23.085 --> 00:45:25.140
So you guys have
to wish me luck,

00:45:25.140 --> 00:45:28.296
because I don't-- didn't have
enough coffee and I don't feel

00:45:28.296 --> 00:45:30.060
like I can draw very well.

00:45:30.060 --> 00:45:33.530
x0 is here.

00:45:33.530 --> 00:45:39.580
So x is there, so you
cut with this board-- are

00:45:39.580 --> 00:45:40.880
you guys with me?

00:45:40.880 --> 00:45:44.115
You cut with this board
at the level x0 over here.

00:45:44.115 --> 00:45:45.920
You cut.

00:45:45.920 --> 00:45:49.290
When you cut with
this board-- you

00:45:49.290 --> 00:45:52.563
cut your surface
with this board--

00:45:52.563 --> 00:45:54.882
you get a curve like that.

00:45:54.882 --> 00:46:00.350
And we call that a
curve f of x0, y.

00:46:00.350 --> 00:46:05.838
Some people who are a little bit
in a hurry and smarter than me,

00:46:05.838 --> 00:46:07.782
they say x equals x0.

00:46:07.782 --> 00:46:09.726
That's called coordinate curve.

00:46:09.726 --> 00:46:16.550


00:46:16.550 --> 00:46:20.270
So, the thing is, this--
it's a curve in plane.

00:46:20.270 --> 00:46:21.324
This is the blue plane.

00:46:21.324 --> 00:46:22.490
I don't know how to call it.

00:46:22.490 --> 00:46:23.420
Pi.

00:46:23.420 --> 00:46:25.840
You know I love to call it pi.

00:46:25.840 --> 00:46:28.180
Since I'm in plane with
a point in a curve--

00:46:28.180 --> 00:46:33.570
a plane curve-- this curve
has a slope at x0, y0.

00:46:33.570 --> 00:46:35.180
Can I draw that slope?

00:46:35.180 --> 00:46:36.730
I'll try.

00:46:36.730 --> 00:46:38.540
The slope of the
blue line, though.

00:46:38.540 --> 00:46:39.976
Let me make it red.

00:46:39.976 --> 00:46:43.950
The slope of the red line--
now, if you don't have colors

00:46:43.950 --> 00:46:47.206
you can make it a dotted line.

00:46:47.206 --> 00:46:57.810
The slope of the dotted line
is-- who the heck is that?

00:46:57.810 --> 00:47:07.140
The derivative of f with respect
to y, because x0 is a constant.

00:47:07.140 --> 00:47:09.490
So how do we write that?

00:47:09.490 --> 00:47:12.800
Because x0 is sort of in
our way, driving us crazy.

00:47:12.800 --> 00:47:15.000
Although he was fixed.

00:47:15.000 --> 00:47:18.410
We keep him fixed by
keeping him in this plane.

00:47:18.410 --> 00:47:20.130
x0 is fixed.

00:47:20.130 --> 00:47:21.980
We have to write
another notation.

00:47:21.980 --> 00:47:24.210
We cannot say f prime.

00:47:24.210 --> 00:47:27.490
Because f depends
on two variables.

00:47:27.490 --> 00:47:31.600
f prime were for when we
were babies in calculus 1.

00:47:31.600 --> 00:47:32.934
We cannot use f prime anymore.

00:47:32.934 --> 00:47:33.850
We have two variables.

00:47:33.850 --> 00:47:36.310
Life became too complicated.

00:47:36.310 --> 00:47:37.393
So we have to say--

00:47:37.393 --> 00:47:38.184
STUDENT: Professor?

00:47:38.184 --> 00:47:40.701
MAGDALENA TODA: --instead
of df dy-- yes, sir.

00:47:40.701 --> 00:47:41.700
May you use a subscript?

00:47:41.700 --> 00:47:45.340
MAGDALENA TODA: You use--
yeah, you can do that as well.

00:47:45.340 --> 00:47:47.300
That's what I do.

00:47:47.300 --> 00:47:49.070
Let me do both.

00:47:49.070 --> 00:47:55.751
f sub y at-- who
was fixed? x0 and y.

00:47:55.751 --> 00:47:58.750
But this is my
favorite notation.

00:47:58.750 --> 00:48:01.190
I'm going to make a
face because I love it.

00:48:01.190 --> 00:48:02.760
This is what engineers love.

00:48:02.760 --> 00:48:04.820
This is what we physicists love.

00:48:04.820 --> 00:48:07.325
Mathematicians, though,
are crazy people.

00:48:07.325 --> 00:48:08.240
They are.

00:48:08.240 --> 00:48:09.330
All of them.

00:48:09.330 --> 00:48:12.930
And they invented
another notation.

00:48:12.930 --> 00:48:15.060
Do you remember
that Mr. Leibniz,

00:48:15.060 --> 00:48:19.030
because he had nothing better to
do, when he invented calculus,

00:48:19.030 --> 00:48:23.250
he did df dy, or df dx?

00:48:23.250 --> 00:48:24.250
What is that?

00:48:24.250 --> 00:48:27.140
That was the limit of
delta f, delta y, right?

00:48:27.140 --> 00:48:28.550
That's what Leibniz did.

00:48:28.550 --> 00:48:30.840
He introduced this
delta notation,

00:48:30.840 --> 00:48:34.510
and then he said if you have
delta space over delta time,

00:48:34.510 --> 00:48:38.080
then shrink both, and you
make a ratio in the limit,

00:48:38.080 --> 00:48:40.620
you should read-- you
should write it df dy.

00:48:40.620 --> 00:48:44.106
And that's the so-called
Leibniz notation, right?

00:48:44.106 --> 00:48:46.756
That was in calc 1.

00:48:46.756 --> 00:48:49.630
But I erased it because
that was calc 1.

00:48:49.630 --> 00:48:54.180
Now, mathematicians, to
imitate the Leibniz notation,

00:48:54.180 --> 00:48:57.870
they said, I cannot use df dy.

00:48:57.870 --> 00:49:00.610
So what the heck shall I use?

00:49:00.610 --> 00:49:02.650
After they thought
for about a year,

00:49:02.650 --> 00:49:05.268
and I was reading through
the history about how

00:49:05.268 --> 00:49:07.140
they invented this,
they said, let's take

00:49:07.140 --> 00:49:09.860
the Greek-- the Greek d.

00:49:09.860 --> 00:49:12.140
Which is the del.

00:49:12.140 --> 00:49:13.990
That's partial.

00:49:13.990 --> 00:49:19.400
The del f, del y, at x0, y.

00:49:19.400 --> 00:49:22.400
When I was 20--
no, I was 18 when

00:49:22.400 --> 00:49:27.100
I saw this the first time--
I had the hardest time making

00:49:27.100 --> 00:49:27.925
this sign.

00:49:27.925 --> 00:49:29.590
It's all in the wrist.

00:49:29.590 --> 00:49:32.160
It's very-- OK.

00:49:32.160 --> 00:49:32.950
Now.

00:49:32.950 --> 00:49:33.590
df dy.

00:49:33.590 --> 00:49:35.900
If you don't like it,
then what do you do?

00:49:35.900 --> 00:49:38.858
You can adopt this notation.

00:49:38.858 --> 00:49:41.710
And what is the meaning
of this by definition?

00:49:41.710 --> 00:49:45.222
You say, you haven't even
defined it, Magdalena.

00:49:45.222 --> 00:49:47.985
It has to be limit of
a difference quotient,

00:49:47.985 --> 00:49:49.066
just like here.

00:49:49.066 --> 00:49:53.320
But we have to be happy
and think of that.

00:49:53.320 --> 00:49:57.360
What is the delta f
versus the delta y?

00:49:57.360 --> 00:49:59.320
It has to be like that.

00:49:59.320 --> 00:50:02.710
f of Mr. x0 is fixed.

00:50:02.710 --> 00:50:07.060
x0, comma, y.

00:50:07.060 --> 00:50:09.850
We have an increment in y.

00:50:09.850 --> 00:50:16.310
y plus delta y. y plus
delta y minus-- that's

00:50:16.310 --> 00:50:18.220
the difference quotient.

00:50:18.220 --> 00:50:22.956
f of what-- the original
point was, well--

00:50:22.956 --> 00:50:24.450
STUDENT: x0, y0.

00:50:24.450 --> 00:50:26.800
MAGDALENA TODA:
x0-- let me put y0

00:50:26.800 --> 00:50:29.850
because our original
point was x0, y0.

00:50:29.850 --> 00:50:37.810
x0, y0 over-- over delta y.

00:50:37.810 --> 00:50:43.385
But if I am at x0, y0, I better
put x0, y0 fixed point here.

00:50:43.385 --> 00:50:46.800


00:50:46.800 --> 00:50:51.580
And I would like you to
photograph or put this thing--

00:50:51.580 --> 00:50:54.525
STUDENT: So is that a delta
that's in front of the f?

00:50:54.525 --> 00:50:56.400
MAGDALENA TODA: Let me
review the whole thing

00:50:56.400 --> 00:50:58.830
because it's very important.

00:50:58.830 --> 00:51:00.810
Where shall I start,
here, or here?

00:51:00.810 --> 00:51:01.860
It doesn't matter.

00:51:01.860 --> 00:51:02.960
So the limit--

00:51:02.960 --> 00:51:05.330
STUDENT: [INAUDIBLE] start at m.

00:51:05.330 --> 00:51:06.205
MAGDALENA TODA: At m?

00:51:06.205 --> 00:51:06.704
At m.

00:51:06.704 --> 00:51:07.930
OK, I'll start at m.

00:51:07.930 --> 00:51:13.070
The slopes of this line at
x0, y0, right at my point,

00:51:13.070 --> 00:51:18.640
will be, my favorite
notation is f sub y at x0,

00:51:18.640 --> 00:51:22.030
y0, which means partial
derivative of f with respect

00:51:22.030 --> 00:51:26.190
to y at the point--
fixed point x0, y0.

00:51:26.190 --> 00:51:30.670
Or, for most mathematicians,
df-- of del-- del f,

00:51:30.670 --> 00:51:34.310
del y at x0, y0.

00:51:34.310 --> 00:51:38.880
Which is by definition the limit
of this difference quotient.

00:51:38.880 --> 00:51:42.270
So x0 is held fixed
in both cases.

00:51:42.270 --> 00:51:45.480
y0 is allowed to
deviate a little bit.

00:51:45.480 --> 00:51:50.400
So y0 is fixed, but you
displace it by a little delta,

00:51:50.400 --> 00:51:53.626
or by a little-- how did we
denote that in calc 1, h?

00:51:53.626 --> 00:51:54.367
Little h?

00:51:54.367 --> 00:51:54.950
STUDENT: Yeah.

00:51:54.950 --> 00:51:56.616
MAGDALENA TODA: So
delta y, sometimes it

00:51:56.616 --> 00:51:58.440
was called little h.

00:51:58.440 --> 00:52:00.950
And this is the
same as little h.

00:52:00.950 --> 00:52:03.840
Over that h.

00:52:03.840 --> 00:52:07.370
Now you, without my
help, because you

00:52:07.370 --> 00:52:10.870
have all the knowledge
and you're smart,

00:52:10.870 --> 00:52:17.430
you should tell me how I
define f sub x at x0, y0,

00:52:17.430 --> 00:52:22.192
and shut up, Magdalena,
let people talk.

00:52:22.192 --> 00:52:23.947
This is hard.

00:52:23.947 --> 00:52:24.780
[INTERPOSING VOICES]

00:52:24.780 --> 00:52:26.090
MAGDALENA TODA: No.

00:52:26.090 --> 00:52:28.250
I hope not.

00:52:28.250 --> 00:52:31.515
As a limit of a
difference quotient,

00:52:31.515 --> 00:52:34.570
so it's gonna be an
instantaneous rate of change.

00:52:34.570 --> 00:52:37.062
That's the limit of a
difference quotient.

00:52:37.062 --> 00:52:38.771
Limit of what?

00:52:38.771 --> 00:52:39.271
Shut up.

00:52:39.271 --> 00:52:40.534
I will zip my lips.

00:52:40.534 --> 00:52:41.380
STUDENT: Delta x

00:52:41.380 --> 00:52:42.860
MAGDALENA TODA:
Delta x, excellent.

00:52:42.860 --> 00:52:44.492
Delta x going to 0.

00:52:44.492 --> 00:52:47.990
So you shrink-- you displace
by a small displacement

00:52:47.990 --> 00:52:50.254
only in the direction of x.

00:52:50.254 --> 00:52:52.220
STUDENT: So f.

00:52:52.220 --> 00:52:53.452
MAGDALENA TODA: f.

00:52:53.452 --> 00:52:57.260
STUDENT: [INAUDIBLE] this
time, x is changing, so--

00:52:57.260 --> 00:52:58.580
[INTERPOSING VOICES]

00:52:58.580 --> 00:53:04.890
MAGDALENA TODA: X0 plus
delta x, y0 is still fixed,

00:53:04.890 --> 00:53:11.183
minus f of x0, y0.

00:53:11.183 --> 00:53:13.115
Thank God this is always fixed.

00:53:13.115 --> 00:53:14.081
I love this guy.

00:53:14.081 --> 00:53:16.500
STUDENT: Delta--

00:53:16.500 --> 00:53:19.910
MAGDALENA TODA:
Delta x, which is

00:53:19.910 --> 00:53:23.200
like the h we were
talking about.

00:53:23.200 --> 00:53:24.680
Now in reality,
you never do that.

00:53:24.680 --> 00:53:28.770
You would die if for every
exercise, derivation exercise,

00:53:28.770 --> 00:53:31.500
you would have to compute a
limit of a difference quotient.

00:53:31.500 --> 00:53:33.050
You will go bananas.

00:53:33.050 --> 00:53:34.510
What we do?

00:53:34.510 --> 00:53:37.400
We do exactly the same thing.

00:53:37.400 --> 00:53:38.710
How can I draw?

00:53:38.710 --> 00:53:40.620
Can anybody help me draw?

00:53:40.620 --> 00:53:46.280
For y0, I would need to take
this other plane through y0.

00:53:46.280 --> 00:53:47.332
Where is y0?

00:53:47.332 --> 00:53:48.940
Here.

00:53:48.940 --> 00:53:50.200
Is my drawing good enough?

00:53:50.200 --> 00:53:51.598
I hope so.

00:53:51.598 --> 00:53:56.470
So it's something like
I have this plane with,

00:53:56.470 --> 00:53:57.920
oh, do you see that, guys?

00:53:57.920 --> 00:53:58.570
OK.

00:53:58.570 --> 00:54:02.842
So what is that, the other
curve, coordinate curve, look

00:54:02.842 --> 00:54:03.342
like?

00:54:03.342 --> 00:54:06.680


00:54:06.680 --> 00:54:08.702
Oh my God.

00:54:08.702 --> 00:54:10.260
Looks like that.

00:54:10.260 --> 00:54:13.070
Through the same point,
and then the slope

00:54:13.070 --> 00:54:18.370
of the line will be a
blue slope and the slope

00:54:18.370 --> 00:54:23.710
will be f sub-- well OK.

00:54:23.710 --> 00:54:28.130
So here I have in the red
one, which was the blue one,

00:54:28.130 --> 00:54:33.670
this is f sub y, and for
this one, this is f sub x.

00:54:33.670 --> 00:54:34.890
Right?

00:54:34.890 --> 00:54:42.240
So guys, don't look
at the picture.

00:54:42.240 --> 00:54:43.240
The picture's confusing.

00:54:43.240 --> 00:54:46.510
This is x coming
towards me, right?

00:54:46.510 --> 00:54:49.630
And y going there
and z is going up.

00:54:49.630 --> 00:54:52.150
This is the graph.

00:54:52.150 --> 00:54:54.650
When I do the
derivative with respect

00:54:54.650 --> 00:55:00.830
to what is this, y, the
derivative with respect to y,

00:55:00.830 --> 00:55:04.070
with respect to y, y
is my only variable,

00:55:04.070 --> 00:55:06.640
so the curve will be like that.

00:55:06.640 --> 00:55:11.480
And the slope will be for a
curve that depends on y only.

00:55:11.480 --> 00:55:14.355
When I do derivative
with respect to x,

00:55:14.355 --> 00:55:19.460
it's like I'm on top of a hill
and I decide to go skiing.

00:55:19.460 --> 00:55:22.310
And I'm-- and I point
my skis like that,

00:55:22.310 --> 00:55:27.100
and the slope is going down,
and that's the x direction.

00:55:27.100 --> 00:55:27.750
OK?

00:55:27.750 --> 00:55:30.910
And what I'm going to
describe as a skier

00:55:30.910 --> 00:55:34.130
will be a plane curve going
down in this direction.

00:55:34.130 --> 00:55:35.854
Zzzzsssshh, like that.

00:55:35.854 --> 00:55:40.900
And the slope at every
point, the slope of the line,

00:55:40.900 --> 00:55:44.720
of y trajectory, will
be the derivative.

00:55:44.720 --> 00:55:47.460
So I have a curve like
that, and a curve like this.

00:55:47.460 --> 00:55:50.140
And they're called
coordinate curves.

00:55:50.140 --> 00:55:51.480
Now this is hard.

00:55:51.480 --> 00:55:53.260
You'll see how
beautiful and easy

00:55:53.260 --> 00:55:57.160
it is when you actually
compute the partial derivatives

00:55:57.160 --> 00:55:59.890
of functions by hand.

00:55:59.890 --> 00:56:01.930
Examples?

00:56:01.930 --> 00:56:08.100
Let's take f of x, y to be
x squared plus y squared.

00:56:08.100 --> 00:56:12.550
I'm asking you, who
is f sub x at x, y?

00:56:12.550 --> 00:56:17.560
Who is f sub x at 1
minus 1, 1, 0, OK.

00:56:17.560 --> 00:56:20.630
Who is f sub y at x, y?

00:56:20.630 --> 00:56:26.130
And who is f sub y at 3 and 2.

00:56:26.130 --> 00:56:28.245
Since I make up my
example-- I don't

00:56:28.245 --> 00:56:30.460
want to copy the
examples from the book,

00:56:30.460 --> 00:56:35.020
because you are supposedly
going to read the book.

00:56:35.020 --> 00:56:40.230
This is-- should be another
example, just for you.

00:56:40.230 --> 00:56:44.370


00:56:44.370 --> 00:56:49.080
So who's gonna help me-- I'm
pausing a little bit-- who's

00:56:49.080 --> 00:56:50.760
gonna help me here?

00:56:50.760 --> 00:56:53.884
What's the answer here?

00:56:53.884 --> 00:56:55.810
So how do I think?

00:56:55.810 --> 00:57:00.060
I think I got-- when I
prime with respect to x, y

00:57:00.060 --> 00:57:01.465
is like a held constant.

00:57:01.465 --> 00:57:02.950
He's held prisoner.

00:57:02.950 --> 00:57:05.205
Poor guy cannot leave his cell.

00:57:05.205 --> 00:57:06.240
That's awful.

00:57:06.240 --> 00:57:09.450
So you prime with respect to x.

00:57:09.450 --> 00:57:11.350
Because x is the only variable.

00:57:11.350 --> 00:57:12.176
And he is--

00:57:12.176 --> 00:57:14.515
STUDENT: So then it's 2x plus y?

00:57:14.515 --> 00:57:17.068
MAGDALENA TODA: 2x plus 0.

00:57:17.068 --> 00:57:17.764
Plus 0.

00:57:17.764 --> 00:57:20.316
Because y is a constant and
when you prime a constant,

00:57:20.316 --> 00:57:22.074
you get 0.

00:57:22.074 --> 00:57:23.990
STUDENT: So when you
take partial derivatives,

00:57:23.990 --> 00:57:25.656
you-- when you're
taking it with respect

00:57:25.656 --> 00:57:28.610
to the first derivative, the
first variable [INAUDIBLE]

00:57:28.610 --> 00:57:30.110
MAGDALENA TODA: You
don't completely

00:57:30.110 --> 00:57:31.960
know because it
might be multiplied.

00:57:31.960 --> 00:57:33.440
But you view it as a constant.

00:57:33.440 --> 00:57:35.320
So for you-- very good, Ryan.

00:57:35.320 --> 00:57:38.090
So for you, it's like,
as if y would be 7.

00:57:38.090 --> 00:57:39.970
Imagine that y would be 7.

00:57:39.970 --> 00:57:44.154
And then you have x squared plus
7 squared prime is u, right?

00:57:44.154 --> 00:57:47.740
STUDENT: So then that means
f of 1-- or f x of 1,0

00:57:47.740 --> 00:57:48.479
is [INAUDIBLE]

00:57:48.479 --> 00:57:49.562
MAGDALENA TODA: Very good.

00:57:49.562 --> 00:57:50.870
STUDENT: OK.

00:57:50.870 --> 00:57:54.668
And in this case, f sub y,
what do you think it is?

00:57:54.668 --> 00:57:56.439
STUDENT: 2y.

00:57:56.439 --> 00:57:57.230
MAGDALENA TODA: 2y.

00:57:57.230 --> 00:57:59.955
And what is f y of 3, 2?

00:57:59.955 --> 00:58:01.092
STUDENT: 4.

00:58:01.092 --> 00:58:02.050
MAGDALENA TODA: It's 4.

00:58:02.050 --> 00:58:04.560
And you say, OK, that
makes sense, that was easy.

00:58:04.560 --> 00:58:06.580
Let's try something hard.

00:58:06.580 --> 00:58:09.385
I'm going to build them
on so many examples

00:58:09.385 --> 00:58:12.670
that you say, stop,
Magdalena, because I became

00:58:12.670 --> 00:58:15.580
an expert in partial
differentiation

00:58:15.580 --> 00:58:19.290
and I-- now everything is so
trivial that you have to stop.

00:58:19.290 --> 00:58:38.380
So example A, example B. A was f
of x, y [INAUDIBLE] x, y plus y

00:58:38.380 --> 00:58:39.929
sine x.

00:58:39.929 --> 00:58:41.470
And you say, wait,
wait, wait, you're

00:58:41.470 --> 00:58:44.240
giving me a little
bit of trouble.

00:58:44.240 --> 00:58:45.620
No, I don't mean to.

00:58:45.620 --> 00:58:47.020
It's very easy.

00:58:47.020 --> 00:58:50.260
Believe me guys,
very, very easy.

00:58:50.260 --> 00:58:55.354
We just have to
think how we do this.

00:58:55.354 --> 00:59:02.170
f sub x at 1 and 2, f
sub y at x, y in general,

00:59:02.170 --> 00:59:06.980
f sub y at 1 and
2, for God's sake.

00:59:06.980 --> 00:59:08.270
OK.

00:59:08.270 --> 00:59:09.885
All right.

00:59:09.885 --> 00:59:19.380
And now, while you're
staring at that,

00:59:19.380 --> 00:59:23.595
I take out my beautiful
colors that I paid $6 for.

00:59:23.595 --> 00:59:26.145


00:59:26.145 --> 00:59:31.620
The department told me that
they don't buy different colors,

00:59:31.620 --> 00:59:35.490
just two or three basic ones.

00:59:35.490 --> 00:59:35.990
All right?

00:59:35.990 --> 00:59:38.170
So what do we do?

00:59:38.170 --> 00:59:40.620
STUDENT: First
one will be the y.

00:59:40.620 --> 00:59:43.320
MAGDALENA TODA: It's like y
would be a constant 7, right,

00:59:43.320 --> 00:59:46.710
but you have to keep in
mind it's mister called y.

00:59:46.710 --> 00:59:48.580
Which for you is a constant.

00:59:48.580 --> 00:59:52.730
So you go, I'm priming this
with respect to x only--

00:59:52.730 --> 00:59:54.514
STUDENT: Then you get y.

00:59:54.514 --> 00:59:56.325
MAGDALENA TODA: Very good.

00:59:56.325 --> 00:59:56.824
Plus--

00:59:56.824 --> 00:59:59.910


00:59:59.910 --> 01:00:01.114
STUDENT: y cosine x.

01:00:01.114 --> 01:00:01.596
MAGDALENA TODA: y cosine x.

01:00:01.596 --> 01:00:02.096
Excellent.

01:00:02.096 --> 01:00:03.524
And stop.

01:00:03.524 --> 01:00:04.970
And stop.

01:00:04.970 --> 01:00:06.185
Because that's all I have.

01:00:06.185 --> 01:00:08.763
You see, it's not hard.

01:00:08.763 --> 01:00:11.710
Let me put here a y.

01:00:11.710 --> 01:00:13.443
OK.

01:00:13.443 --> 01:00:19.040
And then, I plug
a different color.

01:00:19.040 --> 01:00:21.520
I'm a girl, of course I
like different colors.

01:00:21.520 --> 01:00:26.700
So 1, 2. x is 1, and y is 2.

01:00:26.700 --> 01:00:30.270
2 plus 2 cosine 1.

01:00:30.270 --> 01:00:33.314
And you say, oh, wait a minute,
what is that cosine of 1?

01:00:33.314 --> 01:00:33.814
Never mind.

01:00:33.814 --> 01:00:34.790
Don't worry about it.

01:00:34.790 --> 01:00:37.230
It's like cosine
of 1, [INAUDIBLE]

01:00:37.230 --> 01:00:41.134
plug it in the
calculator, nobody cares.

01:00:41.134 --> 01:00:44.550
Well, in the final, you
don't have a calculator,

01:00:44.550 --> 01:00:47.966
so you leave it like that.

01:00:47.966 --> 01:00:49.430
Who cares?

01:00:49.430 --> 01:00:52.750
It's just the perfect--
I would actually hate it

01:00:52.750 --> 01:00:54.230
that you gave me--
because all you

01:00:54.230 --> 01:00:56.480
could give me would be an
approximation, a truncation,

01:00:56.480 --> 01:00:58.220
with two decimals.

01:00:58.220 --> 01:01:01.330
I prefer you give me the
precise answer, which

01:01:01.330 --> 01:01:03.960
is an exact answer like that.

01:01:03.960 --> 01:01:04.750
f sub y.

01:01:04.750 --> 01:01:07.500
Now, Mr. x is held prisoner.

01:01:07.500 --> 01:01:09.006
He is a constant.

01:01:09.006 --> 01:01:10.620
He cannot move.

01:01:10.620 --> 01:01:11.800
Mr. y can move.

01:01:11.800 --> 01:01:13.280
He has all the freedom.

01:01:13.280 --> 01:01:16.603
So prime with respect
to y, what do you have?

01:01:16.603 --> 01:01:17.102
STUDENT: x--

01:01:17.102 --> 01:01:18.026
[INTERPOSING VOICES]

01:01:18.026 --> 01:01:21.952
MAGDALENA TODA: x plus
sine x is a constant.

01:01:21.952 --> 01:01:25.200
So for God's sake,
I'll write it.

01:01:25.200 --> 01:01:30.680
So then I get 1,
plug in x equals 1. y

01:01:30.680 --> 01:01:31.960
doesn't appear in the picture.

01:01:31.960 --> 01:01:33.110
I don't care.

01:01:33.110 --> 01:01:35.132
1 plus sine 1.

01:01:35.132 --> 01:01:38.340


01:01:38.340 --> 01:01:39.820
And now comes-- don't erase.

01:01:39.820 --> 01:01:42.240
Now comes the-- I mean,
you cannot erase it.

01:01:42.240 --> 01:01:44.750
I can erase it.

01:01:44.750 --> 01:01:48.768
Comes this mean professor
who says, wait a minute,

01:01:48.768 --> 01:01:50.950
I want more.

01:01:50.950 --> 01:01:53.601
Mathematicians always want more.

01:01:53.601 --> 01:01:57.330
He goes, I want the
second derivative.

01:01:57.330 --> 01:02:01.040
f sub x x of x, y.

01:02:01.040 --> 01:02:03.530
And you say, what in
the world is that?

01:02:03.530 --> 01:02:06.290
Even some mathematicians,
they denote it

01:02:06.290 --> 01:02:13.295
as del 2 f dx 2, which
is d of-- d with respect

01:02:13.295 --> 01:02:16.520
to x sub d u with respect to x.

01:02:16.520 --> 01:02:17.811
What does it mean?

01:02:17.811 --> 01:02:20.640
You take the first derivative
and you derive it again.

01:02:20.640 --> 01:02:23.181
And don't drink and derive
because you'll be in trouble.

01:02:23.181 --> 01:02:23.680
Right?

01:02:23.680 --> 01:02:28.039
So you have d of dx primed
again, with-- differentiated

01:02:28.039 --> 01:02:30.410
again with respect to x.

01:02:30.410 --> 01:02:31.370
Is that hard?

01:02:31.370 --> 01:02:31.870
Uh-uh.

01:02:31.870 --> 01:02:32.950
What you do?

01:02:32.950 --> 01:02:36.065
In the-- don't do it here.

01:02:36.065 --> 01:02:37.430
You do it in general, right?

01:02:37.430 --> 01:02:43.000
With respect to x as a variable,
y is again held as a prisoner,

01:02:43.000 --> 01:02:44.570
constant.

01:02:44.570 --> 01:02:47.680
So when you prime
that y goes away.

01:02:47.680 --> 01:02:50.710
You're gonna get 0.

01:02:50.710 --> 01:02:54.580
I'll write 0 like a silly
because we are just starters.

01:02:54.580 --> 01:02:56.207
And what else?

01:02:56.207 --> 01:02:57.700
STUDENT: Negative y sine of x.

01:02:57.700 --> 01:02:59.900
MAGDALENA TODA:
Minus y sine of x.

01:02:59.900 --> 01:03:02.340
And I know you've gonna
love this process.

01:03:02.340 --> 01:03:04.940
You are becoming
experts in that.

01:03:04.940 --> 01:03:10.050
And in a way I'm a little
bit sorry it's so easy,

01:03:10.050 --> 01:03:13.180
but I guess not
everybody gets it.

01:03:13.180 --> 01:03:16.290
There are students who
don't get it the first time.

01:03:16.290 --> 01:03:17.980
So what do we get here?

01:03:17.980 --> 01:03:18.907
Minus--

01:03:18.907 --> 01:03:21.770
STUDENT: 0.

01:03:21.770 --> 01:03:25.821
MAGDALENA TODA: Please
tell me-- sine 1, 0.

01:03:25.821 --> 01:03:26.320
Good.

01:03:26.320 --> 01:03:30.002
I could do the same
thing for f y y.

01:03:30.002 --> 01:03:34.800
I could do this thing--
what is f sub x y?

01:03:34.800 --> 01:03:37.250
By definition f sub x y--

01:03:37.250 --> 01:03:39.975
STUDENT: Is that taking the
derivative of the derivative

01:03:39.975 --> 01:03:42.100
with respect-- is that
taking the second derivative

01:03:42.100 --> 01:03:44.058
with respect to y after
you take the derivative

01:03:44.058 --> 01:03:46.110
of the-- first derivative
with respect to x?

01:03:46.110 --> 01:03:47.026
MAGDALENA TODA: Right.

01:03:47.026 --> 01:03:49.510
So when I write like that,
because that's a little bit

01:03:49.510 --> 01:03:54.202
confusing, when students
ask me, which one is first?

01:03:54.202 --> 01:03:57.830
First you do f sub
x, and then you do y.

01:03:57.830 --> 01:04:02.766
And then f sub y x would be the
derivative with respect to y

01:04:02.766 --> 01:04:04.650
primed again with respect to x.

01:04:04.650 --> 01:04:07.480
Now, let me tell
you the good news.

01:04:07.480 --> 01:04:13.390
They-- the book doesn't call
it any name, because we don't

01:04:13.390 --> 01:04:14.886
like to call anybody names.

01:04:14.886 --> 01:04:15.840
I'm just kidding.

01:04:15.840 --> 01:04:23.710
It's called the
Schwartz principle,

01:04:23.710 --> 01:04:27.470
or the theorem of Schwartz.

01:04:27.470 --> 01:04:30.974
When I told my co-authors,
they said, who cares?

01:04:30.974 --> 01:04:34.950
Well I care, because I was a
student when my professors told

01:04:34.950 --> 01:04:38.290
me that this German
mathematician made

01:04:38.290 --> 01:04:41.040
this discovery, which
is so beautiful.

01:04:41.040 --> 01:04:55.380
If f is twice differentiable
with respect to x and y,

01:04:55.380 --> 01:04:58.820
and the partial derivatives--
the second partial

01:04:58.820 --> 01:05:14.750
derivatives-- are continuous,
then, now in English

01:05:14.750 --> 01:05:17.480
it would say it doesn't
matter in which order

01:05:17.480 --> 01:05:18.530
you differentiate.

01:05:18.530 --> 01:05:20.920
The mixed ones are
always the same.

01:05:20.920 --> 01:05:22.190
Say what?

01:05:22.190 --> 01:05:26.760
f sub x y equals f sub
y x for every point.

01:05:26.760 --> 01:05:31.590
For every-- do you remember
what I taught you for every x, y

01:05:31.590 --> 01:05:32.360
in the domain.

01:05:32.360 --> 01:05:36.110
Or for every x, y
where this happens.

01:05:36.110 --> 01:05:38.410
So what does this mean?

01:05:38.410 --> 01:05:41.219
That means that whether
you differentiate

01:05:41.219 --> 01:05:43.760
first with respect to x and then
with respect to, y, or first

01:05:43.760 --> 01:05:46.100
with respect to y and
then with respect to x,

01:05:46.100 --> 01:05:48.250
it doesn't matter.

01:05:48.250 --> 01:05:50.900
The mixed partial
derivatives are the same.

01:05:50.900 --> 01:05:52.140
Which is wonderful.

01:05:52.140 --> 01:05:55.110
I mean, this is one
of the best things

01:05:55.110 --> 01:05:58.100
that ever happened to us.

01:05:58.100 --> 01:06:01.306
Let's see if this
is true in our case.

01:06:01.306 --> 01:06:03.864
I mean, of course it's true
because it's a theorem,

01:06:03.864 --> 01:06:06.154
if it weren't true
I wouldn't teach it,

01:06:06.154 --> 01:06:11.050
but let's verify it on a baby.

01:06:11.050 --> 01:06:14.343
Not on a real baby,
on a baby example.

01:06:14.343 --> 01:06:15.230
Right?

01:06:15.230 --> 01:06:21.040
So, f sub x is y plus y
equals sine x primed again

01:06:21.040 --> 01:06:22.782
with respect to y.

01:06:22.782 --> 01:06:27.630
And what do we get out of it?

01:06:27.630 --> 01:06:29.250
Cosine of x.

01:06:29.250 --> 01:06:31.081
Are you guys with me?

01:06:31.081 --> 01:06:35.130
So f sub x was y
plus y equals sine x.

01:06:35.130 --> 01:06:39.078
Take this guy
again, put it here,

01:06:39.078 --> 01:06:42.667
squeeze them up a little
bit, divide by-- no.

01:06:42.667 --> 01:06:47.780
Time with respect to y, x is
a constant, what do you think?

01:06:47.780 --> 01:06:48.794
Cosine of x, am I right?

01:06:48.794 --> 01:06:49.960
STUDENT: 1 plus [INAUDIBLE].

01:06:49.960 --> 01:06:52.389


01:06:52.389 --> 01:06:54.180
MAGDALENA TODA: That's
what it starts with.

01:06:54.180 --> 01:06:56.700
Plus [INAUDIBLE].

01:06:56.700 --> 01:07:02.030
So cosine of x, [INAUDIBLE]
a constant, plus 1.

01:07:02.030 --> 01:07:04.965
Another way to have done
it is, like, wait a minute,

01:07:04.965 --> 01:07:10.810
at this point I go, constant
out-- are you with me?--

01:07:10.810 --> 01:07:14.630
constant out, prime with respect
to y, equals sine x plus 1.

01:07:14.630 --> 01:07:16.885
Thank you.

01:07:16.885 --> 01:07:17.385
All right.

01:07:17.385 --> 01:07:20.822


01:07:20.822 --> 01:07:26.370
F sub yx is going to be f sub y.

01:07:26.370 --> 01:07:32.336
x plus sine x, but I have
to take it from here,

01:07:32.336 --> 01:07:38.416
and I prime again with respect
to x, and I get the same thing.

01:07:38.416 --> 01:07:39.790
I don't know,
maybe I'm dyslexic,

01:07:39.790 --> 01:07:43.450
I go from the right to the
left, what's the matter with me.

01:07:43.450 --> 01:07:47.305
Instead of saying 1 plus,
I go cosine of x plus 1.

01:07:47.305 --> 01:07:53.024


01:07:53.024 --> 01:07:54.476
So it's the same thing.

01:07:54.476 --> 01:07:55.444
Yes, sir.

01:07:55.444 --> 01:07:58.611
STUDENT:I'm looking at
the f of xy from the--

01:07:58.611 --> 01:08:00.486
MAGDALENA TODA: Which
one are you looking at?

01:08:00.486 --> 01:08:01.460
Show me.

01:08:01.460 --> 01:08:03.518
STUDENT: It's in the purple.

01:08:03.518 --> 01:08:05.150
MAGDALENA TODA: It
is in the purple.

01:08:05.150 --> 01:08:05.680
STUDENT: It's that
one right there.

01:08:05.680 --> 01:08:06.140
So--

01:08:06.140 --> 01:08:06.725
MAGDALENA TODA: This one?

01:08:06.725 --> 01:08:07.308
STUDENT: Mmhm.

01:08:07.308 --> 01:08:10.550
So, I'm looking at
the y plus y cosine x.

01:08:10.550 --> 01:08:12.503
You got that from f of x.

01:08:12.503 --> 01:08:14.196
MAGDALENA TODA: I
got this from f of x,

01:08:14.196 --> 01:08:16.640
and I prime it again,
with respect to y.

01:08:16.640 --> 01:08:18.920
The whole thing.

01:08:18.920 --> 01:08:21.510
STUDENT: OK, so you're not
writing that as a derivative?

01:08:21.510 --> 01:08:25.274
You're just substituting
that in for f of x?

01:08:25.274 --> 01:08:27.475
MAGDALENA TODA: So,
let me write it better,

01:08:27.475 --> 01:08:30.783
because I was a little bit
rushed, and I don't know,

01:08:30.783 --> 01:08:32.167
silly or something.

01:08:32.167 --> 01:08:35.135
When I prime this
with respect to y--

01:08:35.135 --> 01:08:38.274
STUDENT: Then you get
the cosine of x plus 1.

01:08:38.274 --> 01:08:39.149
MAGDALENA TODA: Yeah.

01:08:39.149 --> 01:08:42.426
I could say, I can take
out all the constants.

01:08:42.426 --> 01:08:43.160
STUDENT: OK.

01:08:43.160 --> 01:08:46.210
MAGDALENA TODA: And that
constant is this plus 1.

01:08:46.210 --> 01:08:47.455
And that's all I'm left with.

01:08:47.455 --> 01:08:47.955
Right?

01:08:47.955 --> 01:08:51.609
It's the same thing
as 1 plus cosine x,

01:08:51.609 --> 01:08:53.879
which is a constant times y.

01:08:53.879 --> 01:08:57.238
Prime this with respect
to y, I get the constant.

01:08:57.238 --> 01:09:04.020
It's the same principal as when
you have bdy of 7y equals 7.

01:09:04.020 --> 01:09:06.752
Right?

01:09:06.752 --> 01:09:08.703
OK.

01:09:08.703 --> 01:09:10.130
Is this too easy?

01:09:10.130 --> 01:09:13.420
I'll give you a nicer function.

01:09:13.420 --> 01:09:28.760
I'm imitating the one
in WeBWorK [INAUDIBLE]

01:09:28.760 --> 01:09:31.444
To make it harder for you.

01:09:31.444 --> 01:09:34.354
Nothing I can make at this
point is hard for you,

01:09:34.354 --> 01:09:39.250
because you're becoming experts
in partial differentiation,

01:09:39.250 --> 01:09:41.720
and I cannot
challenge you on that.

01:09:41.720 --> 01:09:54.113


01:09:54.113 --> 01:09:57.048
I'm just trying to
make it harder for you.

01:09:57.048 --> 01:09:59.004
And I'm trying to
look up something.

01:09:59.004 --> 01:10:02.930


01:10:02.930 --> 01:10:03.970
OK, how about that?

01:10:03.970 --> 01:10:06.670


01:10:06.670 --> 01:10:09.110
This is harder than the
ones you have in WeBWorK.

01:10:09.110 --> 01:10:11.970
But that was kind of
the idea-- that when

01:10:11.970 --> 01:10:15.620
you go home, and open
those WeBWorK problem sets,

01:10:15.620 --> 01:10:17.430
that's a piece of cake.

01:10:17.430 --> 01:10:20.600
What we did in class was harder.

01:10:20.600 --> 01:10:23.960
When I was a graduate
student, one professor said,

01:10:23.960 --> 01:10:27.320
the easy examples are the
ones that the professor's

01:10:27.320 --> 01:10:29.740
supposed to write in
class, on the board.

01:10:29.740 --> 01:10:31.440
The hard examples
are the ones that

01:10:31.440 --> 01:10:34.290
are left for the
students' homework.

01:10:34.290 --> 01:10:35.750
I disagree.

01:10:35.750 --> 01:10:37.760
I think it should be
the other way around.

01:10:37.760 --> 01:10:40.260
So f sub x.

01:10:40.260 --> 01:10:43.380


01:10:43.380 --> 01:10:50.552
That means bfdx for
the pair xy, any xy.

01:10:50.552 --> 01:10:53.560
I'm not specifying an x and a y.

01:10:53.560 --> 01:10:56.180
I'm not making them a constant.

01:10:56.180 --> 01:10:58.970
What am I going to
have in this case?

01:10:58.970 --> 01:11:03.850
Chain -- if I catch you
not knowing the chain rule,

01:11:03.850 --> 01:11:05.370
you fail the final.

01:11:05.370 --> 01:11:12.590
Not really, but, OK,
you get some penalty.

01:11:12.590 --> 01:11:13.730
You know it.

01:11:13.730 --> 01:11:16.110
Just pay attention
to what you do.

01:11:16.110 --> 01:11:18.340
I make my own
mistakes sometimes.

01:11:18.340 --> 01:11:21.160
So 1 over.

01:11:21.160 --> 01:11:23.590
What do you do here
when you differentiate

01:11:23.590 --> 01:11:24.350
with respect to x?

01:11:24.350 --> 01:11:31.600
You think, OK, from the outside
to the inside, one at a time.

01:11:31.600 --> 01:11:36.130
1 over the variable
squared plus 1, right?

01:11:36.130 --> 01:11:42.215
Whatever that variable,
it's like you call variable

01:11:42.215 --> 01:11:44.890
of the argument xy, right?

01:11:44.890 --> 01:11:47.240
STUDENT: [INAUDIBLE]

01:11:47.240 --> 01:11:49.558
MAGDALENA TODA: Square plus 1.

01:11:49.558 --> 01:11:56.710
Times-- cover it with your
hand-- prime with respect to x.

01:11:56.710 --> 01:11:59.000
y, right?

01:11:59.000 --> 01:12:00.395
Good!

01:12:00.395 --> 01:12:01.325
And you're done.

01:12:01.325 --> 01:12:02.720
You see how easy it was.

01:12:02.720 --> 01:12:07.510
Just don't forget something
because it can cost you points.

01:12:07.510 --> 01:12:09.640
Are you guys with me?

01:12:09.640 --> 01:12:13.380
So, once we are done with
saying, 1 over argument

01:12:13.380 --> 01:12:16.130
squared plus 1, I cover
this with my hand,

01:12:16.130 --> 01:12:20.110
xy prime with
respect to 2x is y.

01:12:20.110 --> 01:12:22.380
And I'm done.

01:12:22.380 --> 01:12:23.250
And I'm done.

01:12:23.250 --> 01:12:26.250
And here, pause.

01:12:26.250 --> 01:12:29.680
What's the easiest
way to do that?

01:12:29.680 --> 01:12:32.010
You look at it like,
she wants me to get

01:12:32.010 --> 01:12:34.310
caught in the quotient rule.

01:12:34.310 --> 01:12:37.310
She wants to catch me
not knowing this rule,

01:12:37.310 --> 01:12:40.250
while I can do better.

01:12:40.250 --> 01:12:43.460
One way to do it would
be numerator prime plus

01:12:43.460 --> 01:12:47.740
denominator, minus
numerator [INAUDIBLE] What's

01:12:47.740 --> 01:12:50.410
the easier way to do it?

01:12:50.410 --> 01:12:52.870
STUDENT: x squared plus
y squared, all of it

01:12:52.870 --> 01:12:53.855
to the negative one.

01:12:53.855 --> 01:12:54.771
MAGDALENA TODA: Right.

01:12:54.771 --> 01:12:56.890
So you say, hey,
you cannot catch me,

01:12:56.890 --> 01:13:00.680
I'm the gingerbread man.

01:13:00.680 --> 01:13:01.280
Good!

01:13:01.280 --> 01:13:03.100
That was a good idea.

01:13:03.100 --> 01:13:10.330
Chain rule, and
minus 1/2, times--

01:13:10.330 --> 01:13:11.760
who tells me what's next?

01:13:11.760 --> 01:13:13.210
I'm not going to say a word.

01:13:13.210 --> 01:13:15.258
STUDENT: 2x plus y squared.

01:13:15.258 --> 01:13:19.170
No, it's 2x.

01:13:19.170 --> 01:13:20.637
x squared plus y squared.

01:13:20.637 --> 01:13:22.595
MAGDALENA TODA: From the
outside to the inside.

01:13:22.595 --> 01:13:25.340
From the outside-- to the what?

01:13:25.340 --> 01:13:27.186
STUDENT: [INAUDIBLE]

01:13:27.186 --> 01:13:28.061
MAGDALENA TODA: Good.

01:13:28.061 --> 01:13:28.970
And now I'm done.

01:13:28.970 --> 01:13:31.220
I don't see that anymore.

01:13:31.220 --> 01:13:33.860
I focus to the core.

01:13:33.860 --> 01:13:35.650
2x.

01:13:35.650 --> 01:13:38.640
Times 2x.

01:13:38.640 --> 01:13:42.390
And that is plenty.

01:13:42.390 --> 01:13:45.250
OK, now, let me
ask you a question.

01:13:45.250 --> 01:13:51.340
What if you would
ask a smart kid,

01:13:51.340 --> 01:13:56.760
I don't know, somebody
who knows that,

01:13:56.760 --> 01:14:01.620
can you pose the f sub y of xy
without doing the whole thing

01:14:01.620 --> 01:14:03.370
all over again?

01:14:03.370 --> 01:14:06.350
Can you sort of figure
out what it would be?

01:14:06.350 --> 01:14:08.700
The beautiful
thing about x and y

01:14:08.700 --> 01:14:11.042
is that these are
symmetric polynomials.

01:14:11.042 --> 01:14:12.750
What does it mean,
symmetric polynomials?

01:14:12.750 --> 01:14:19.260
That means, if you swap x
and y, and you swap x and y,

01:14:19.260 --> 01:14:20.810
it's the same thing.

01:14:20.810 --> 01:14:23.300
Just think of that--
swapping x and y.

01:14:23.300 --> 01:14:25.230
Swapping the roles of x and y.

01:14:25.230 --> 01:14:28.250
So what do you think
you're going to get?

01:14:28.250 --> 01:14:31.130
OK, one student said,
this is for smart people,

01:14:31.130 --> 01:14:32.564
not for people like me.

01:14:32.564 --> 01:14:34.890
And I said, well, OK,
what's the matter with you?

01:14:34.890 --> 01:14:36.240
I'm a hard worker.

01:14:36.240 --> 01:14:39.910
I'm the kind of guy who takes
the whole thing again, and does

01:14:39.910 --> 01:14:42.180
the derivation from scratch.

01:14:42.180 --> 01:14:45.325
And thinking back in high
school, I think, even

01:14:45.325 --> 01:14:47.810
for symmetric polynomials,

01:14:47.810 --> 01:14:49.570
I'm sure that being
smart and being

01:14:49.570 --> 01:14:53.510
able to guess the
whole thing-- but I

01:14:53.510 --> 01:14:56.000
did the computation
many times mechanically,

01:14:56.000 --> 01:14:59.440
just in the same way,
because I was a hard worker.

01:14:59.440 --> 01:15:01.220
So what do you
have in that case?

01:15:01.220 --> 01:15:09.810
1/xy squared plus 1 times x
plus-- the same kind of thing.

01:15:09.810 --> 01:15:14.338
Attention, this is the symmetric
polynomial, and I go to that.

01:15:14.338 --> 01:15:17.280
And then times 2y.

01:15:17.280 --> 01:15:20.610
So, see-- that kind
of easy, fast thing.

01:15:20.610 --> 01:15:24.340
Why is this a good
observation when

01:15:24.340 --> 01:15:26.015
you have symmetric polynomials?

01:15:26.015 --> 01:15:28.710
If you are on the final and
you don't have that much time,

01:15:28.710 --> 01:15:33.540
or on any kind of exam when
you are in a time-crunch.

01:15:33.540 --> 01:15:36.120
Now, we want those
exams so you are not

01:15:36.120 --> 01:15:38.024
going to be in a time-crunch.

01:15:38.024 --> 01:15:41.665
If there is something I hate,
I hate a final of 2 hours

01:15:41.665 --> 01:15:44.640
and a half with 25
serious problems,

01:15:44.640 --> 01:15:48.260
and you know nobody can do that.

01:15:48.260 --> 01:15:50.710
So, it happens a lot.

01:15:50.710 --> 01:15:55.880
I see that-- one of my jobs
is also to look at the finals

01:15:55.880 --> 01:15:58.586
after people wrote
them, and I still

01:15:58.586 --> 01:16:05.256
do that every semester-- I see
too many people making finals.

01:16:05.256 --> 01:16:06.880
The finals are not
supposed to be long.

01:16:06.880 --> 01:16:10.990
The finals are supposed
to be comprehensive, cover

01:16:10.990 --> 01:16:16.300
everything, but not extensive.

01:16:16.300 --> 01:16:21.060
So maybe you'll have 15
problems that cover practically

01:16:21.060 --> 01:16:22.911
the material entirely.

01:16:22.911 --> 01:16:23.410
Why?

01:16:23.410 --> 01:16:29.080
Because every little problem
can have two short questions.

01:16:29.080 --> 01:16:30.530
You were done with
a section, you

01:16:30.530 --> 01:16:34.690
shot half of a chapter
only one question.

01:16:34.690 --> 01:16:39.660
This is one example just--
not involving [INAUDIBLE]

01:16:39.660 --> 01:16:41.210
of an expression like that, no.

01:16:41.210 --> 01:16:43.310
That's too time-consuming.

01:16:43.310 --> 01:16:47.600
But maybe just tangent of
x-squared plus y-squared,

01:16:47.600 --> 01:16:49.964
find the partial derivatives.

01:16:49.964 --> 01:16:53.380
That's a good exam
question, and that's enough

01:16:53.380 --> 01:16:55.895
when it comes to
testing partials.

01:16:55.895 --> 01:16:58.130
By the way, how
much-- what is that?

01:16:58.130 --> 01:17:00.832
And I'm going to let
you go right now.

01:17:00.832 --> 01:17:01.816
Use the bathroom.

01:17:01.816 --> 01:17:05.240
And when you come back from the
bathroom, we'll fill in this.

01:17:05.240 --> 01:17:10.515
You know I am horrible in the
sense that I want-- I'm greedy.

01:17:10.515 --> 01:17:12.055
I need extra time.

01:17:12.055 --> 01:17:15.340
I want to use more time.

01:17:15.340 --> 01:17:17.850
I will do your
problems from now on,

01:17:17.850 --> 01:17:22.061
and you can use the bathroom,
eat something, wash your hands.

01:17:22.061 --> 01:17:26.498


01:17:26.498 --> 01:17:28.470
I'll start in
about five minutes.

01:17:28.470 --> 01:17:29.456
Don't worry.

01:17:29.456 --> 01:17:32.907


01:17:32.907 --> 01:17:33.893
Alexander?

01:17:33.893 --> 01:17:35.372
Are you here?

01:17:35.372 --> 01:17:37.837
Come get this.

01:17:37.837 --> 01:17:40.350
I apologize.

01:17:40.350 --> 01:17:42.426
This is long due back to you.

01:17:42.426 --> 01:17:43.420
STUDENT: Oh.

01:17:43.420 --> 01:17:43.920
Thank you.

01:17:43.920 --> 01:17:47.406


01:17:47.406 --> 01:17:49.900
STUDENT: Is there an
attendance sheet today?

01:17:49.900 --> 01:17:53.110
MAGDALENA TODA: I will--
I'm making up one.

01:17:53.110 --> 01:17:56.750
There is already on
one side attendance.

01:17:56.750 --> 01:17:58.650
Let's use the other side.

01:17:58.650 --> 01:18:01.566
Put today's date.

01:18:01.566 --> 01:18:02.066
[INAUDIBLE]

01:18:02.066 --> 01:18:44.396


01:18:44.396 --> 01:18:48.380
[SIDE CONVERSATIONS]

01:18:48.380 --> 01:18:56.348


01:18:56.348 --> 01:18:58.340
MAGDALENA TODA: They
are spoiling me.

01:18:58.340 --> 01:19:03.334
They give me new
sprays every week.

01:19:03.334 --> 01:19:05.298
[INAUDIBLE] take care of this.

01:19:05.298 --> 01:19:09.226
[SIDE CONVERSATIONS]

01:19:09.226 --> 01:19:14.136


01:19:14.136 --> 01:19:17.082
MAGDALENA TODA: So I'm
going to ask you something.

01:19:17.082 --> 01:19:20.530
And you respond honestly.

01:19:20.530 --> 01:19:24.810
Which chapter-- we already
browsed through three chapters.

01:19:24.810 --> 01:19:26.840
I mean, Chapter 9
was vector spaces,

01:19:26.840 --> 01:19:29.540
and it was all review
from-- from what?

01:19:29.540 --> 01:19:30.540
From Calc 2.

01:19:30.540 --> 01:19:34.610
Chapter 10 was curves in
[INAUDIBLE] and curves

01:19:34.610 --> 01:19:36.602
in space, practically.

01:19:36.602 --> 01:19:40.540


01:19:40.540 --> 01:19:46.626
And Chapter 11 is functions
of several variables.

01:19:46.626 --> 01:19:48.640
Now you have a flavor
of all of them,

01:19:48.640 --> 01:19:50.475
which one was hardest for you?

01:19:50.475 --> 01:19:51.476
STUDENT: 9 and 10, both.

01:19:51.476 --> 01:19:52.725
MAGDALENA TODA: 9 and 10 both.

01:19:52.725 --> 01:19:53.710
STUDENT: [INAUDIBLE].

01:19:53.710 --> 01:19:56.420
MAGDALENA TODA: This is so
much better than the other.

01:19:56.420 --> 01:20:00.660
No, I think you guys
actually-- it looks better,

01:20:00.660 --> 01:20:06.696
because you've seen a lot more
vectors and vector functions.

01:20:06.696 --> 01:20:08.592
STUDENT: I didn't
understand any of 9 or 10.

01:20:08.592 --> 01:20:09.540
STUDENT: [INAUDIBLE].

01:20:09.540 --> 01:20:10.370
MAGDALENA TODA: Yes, ma'am.

01:20:10.370 --> 01:20:12.370
STUDENT: Could you go over
parametrization [INAUDIBLE]?

01:20:12.370 --> 01:20:14.370
MAGDALENA TODA: I will
go over that again.

01:20:14.370 --> 01:20:18.370
And I will go over some
other parametrizations today.

01:20:18.370 --> 01:20:24.500
And I promised that at the
end, in those 20 minutes,

01:20:24.500 --> 01:20:28.380
I will do that problem that
gave a few of you trouble.

01:20:28.380 --> 01:20:29.181
Yes, sir?

01:20:29.181 --> 01:20:30.805
STUDENT: Do we take
the same final exam

01:20:30.805 --> 01:20:33.230
as all the other [INAUDIBLE]
classes? [INAUDIBLE]?

01:20:33.230 --> 01:20:36.500
MAGDALENA TODA: Well, that's
what I was asked yesterday.

01:20:36.500 --> 01:20:43.150
So practically, it's at the
latitude of the instructor who

01:20:43.150 --> 01:20:45.320
teaches honors if they
write their own final,

01:20:45.320 --> 01:20:48.620
and in general make
it harder, or they

01:20:48.620 --> 01:20:51.060
take the general final
like everybody else.

01:20:51.060 --> 01:20:53.988
For your formative
purposes, and as a study,

01:20:53.988 --> 01:20:57.892
I would like you to
take the general final,

01:20:57.892 --> 01:21:01.350
because I want to see
where you stand compared

01:21:01.350 --> 01:21:02.780
to the rest of the population.

01:21:02.780 --> 01:21:06.848
So you are my sample, and
they are the entire student

01:21:06.848 --> 01:21:08.394
population of Calc
3, I want to make

01:21:08.394 --> 01:21:14.494
the statistical analysis of your
performance compared to them.

01:21:14.494 --> 01:21:16.446
STUDENT: So we'll
take the regular one?

01:21:16.446 --> 01:21:17.422
MAGDALENA TODA: Yeah.

01:21:17.422 --> 01:21:19.005
For this one, I just
have to make sure

01:21:19.005 --> 01:21:22.302
that they also have that
extra credit added in.

01:21:22.302 --> 01:21:25.840
Because if I have too much
extra credit in there,

01:21:25.840 --> 01:21:27.310
well they also count that.

01:21:27.310 --> 01:21:28.780
So that's what that means.

01:21:28.780 --> 01:21:30.250
So we can [INAUDIBLE].

01:21:30.250 --> 01:21:34.180


01:21:34.180 --> 01:21:35.690
All right.

01:21:35.690 --> 01:21:37.015
Let me finish this exercise.

01:21:37.015 --> 01:21:41.000
And then [? stop ?]
[INAUDIBLE] and go

01:21:41.000 --> 01:21:45.635
over some homework problems and
some parametrization problems.

01:21:45.635 --> 01:21:48.605
And I will see what else.

01:21:48.605 --> 01:21:55.535
So tangent of [INAUDIBLE].

01:21:55.535 --> 01:21:59.495


01:21:59.495 --> 01:22:00.485
Is this hard?

01:22:00.485 --> 01:22:01.970
No, it's [INAUDIBLE].

01:22:01.970 --> 01:22:05.930
But you have to
remind me, because I

01:22:05.930 --> 01:22:09.420
pretend that I
forgot-- let me pretend

01:22:09.420 --> 01:22:14.250
that I forgot what the
derivative [INAUDIBLE] notation

01:22:14.250 --> 01:22:17.722
of tangent of t was.

01:22:17.722 --> 01:22:19.786
STUDENT: Secant squared.

01:22:19.786 --> 01:22:23.237
MAGDALENA TODA: You guys love
that secant squared thingy.

01:22:23.237 --> 01:22:26.195


01:22:26.195 --> 01:22:30.632
Why do you like secant squared?

01:22:30.632 --> 01:22:34.083
I, as a student, I didn't
like expressing it like that.

01:22:34.083 --> 01:22:35.562
I liked [INAUDIBLE].

01:22:35.562 --> 01:22:37.041
Of course, it's the same thing.

01:22:37.041 --> 01:22:40.492
But I always like it like
1 over cosine [INAUDIBLE].

01:22:40.492 --> 01:22:45.422


01:22:45.422 --> 01:22:47.720
And of course, I have
to ask you something,

01:22:47.720 --> 01:22:52.224
because I'm curious to
see what you remember.

01:22:52.224 --> 01:22:55.212
And you say yeah,
curiosity killed the cat.

01:22:55.212 --> 01:23:00.192
But where did the
derivative exist?

01:23:00.192 --> 01:23:06.720
Because maybe was
that tangent of T--

01:23:06.720 --> 01:23:08.300
STUDENT: Wasn't
it a quotient rule

01:23:08.300 --> 01:23:10.260
of sine and [? cosine x? ?]

01:23:10.260 --> 01:23:11.730
MAGDALENA TODA: Good.

01:23:11.730 --> 01:23:15.170
I'm proud of you.

01:23:15.170 --> 01:23:17.430
That is the answer.

01:23:17.430 --> 01:23:23.170
So [? my ?] [? have ?] this
blowing up, this blows up--

01:23:23.170 --> 01:23:29.520
blows up where cosine
T was zero, right?

01:23:29.520 --> 01:23:32.380
So where did that blow up?

01:23:32.380 --> 01:23:36.780
[INAUDIBLE] blow up of
cosine and zero [INAUDIBLE].

01:23:36.780 --> 01:23:40.740
The cosine was the
shadow on the x-axis.

01:23:40.740 --> 01:23:43.992
So here you blow up here, you
blow up here, you blow up here,

01:23:43.992 --> 01:23:44.700
you blow up here.

01:23:44.700 --> 01:23:49.155


01:23:49.155 --> 01:23:51.630
So [? what does ?] [INAUDIBLE].

01:23:51.630 --> 01:23:53.115
It should not be what?

01:23:53.115 --> 01:23:55.496
STUDENT: Pi over 2.

01:23:55.496 --> 01:23:56.370
MAGDALENA TODA: Yeah.

01:23:56.370 --> 01:23:59.630
And can we express
that OK, among 0pi,

01:23:59.630 --> 01:24:03.320
let's say you go in
between 0 and 2pi only.

01:24:03.320 --> 01:24:08.300
I get rid of pi over
2 and 3pi over 2.

01:24:08.300 --> 01:24:11.940
But if I express that in
general for [INAUDIBLE] T

01:24:11.940 --> 01:24:15.000
not restricted to 0
to T, what do I say?

01:24:15.000 --> 01:24:15.960
STUDENT: It's k.

01:24:15.960 --> 01:24:18.820
STUDENT: So it can
[? never be ?] pi over 2

01:24:18.820 --> 01:24:19.320
plus pi?

01:24:19.320 --> 01:24:21.240
MAGDALENA TODA: 2k plus 1.

01:24:21.240 --> 01:24:23.952
2k plus 1.

01:24:23.952 --> 01:24:25.398
Odd number over--

01:24:25.398 --> 01:24:26.243
STUDENT: Pi over 2.

01:24:26.243 --> 01:24:27.326
MAGDALENA TODA: Pi over 2.

01:24:27.326 --> 01:24:28.290
Odd number, pi over 2.

01:24:28.290 --> 01:24:30.200
And all the odd
numbers are 2k plus 1.

01:24:30.200 --> 01:24:30.700
Right?

01:24:30.700 --> 01:24:32.146
All right.

01:24:32.146 --> 01:24:38.990
So you have a not
existence and-- OK.

01:24:38.990 --> 01:24:39.640
Coming back.

01:24:39.640 --> 01:24:42.410
I'm just playing, because
we are still in the break.

01:24:42.410 --> 01:24:44.290
Now we are ready.

01:24:44.290 --> 01:24:50.195
What is dfdx, del f, del x, xy.

01:24:50.195 --> 01:24:51.920
And what is del f, del y?

01:24:51.920 --> 01:24:55.600
I'm not going to ask you for
the second partial derivative.

01:24:55.600 --> 01:24:57.380
We've had enough of that.

01:24:57.380 --> 01:25:05.230
We also agreed that we have
important results in that.

01:25:05.230 --> 01:25:08.980
What is the final answer here?

01:25:08.980 --> 01:25:13.940
STUDENT: [INAUDIBLE] plus
x-squared [INAUDIBLE].

01:25:13.940 --> 01:25:15.490
MAGDALENA TODA: 1
over [INAUDIBLE].

01:25:15.490 --> 01:25:17.980
I love this one, OK?

01:25:17.980 --> 01:25:20.480
Don't tell me what I
want to [INAUDIBLE].

01:25:20.480 --> 01:25:22.376
I'm just kidding.

01:25:22.376 --> 01:25:23.940
[INAUDIBLE] squared times--

01:25:23.940 --> 01:25:24.805
STUDENT: 2x.

01:25:24.805 --> 01:25:26.450
MAGDALENA TODA: 2x, good.

01:25:26.450 --> 01:25:27.700
How about the other one?

01:25:27.700 --> 01:25:28.440
The same thing.

01:25:28.440 --> 01:25:34.320


01:25:34.320 --> 01:25:36.670
Times 2y.

01:25:36.670 --> 01:25:41.840


01:25:41.840 --> 01:25:43.300
OK.

01:25:43.300 --> 01:25:46.280
I want to tell you something
that I will repeat.

01:25:46.280 --> 01:25:49.365
But you will see it
all through the course.

01:25:49.365 --> 01:25:52.180
There is a certain
notion that Alexander,

01:25:52.180 --> 01:25:54.435
who is not talking--
I'm just kidding,

01:25:54.435 --> 01:25:58.400
you can talk-- he
reminded me of gradient.

01:25:58.400 --> 01:26:02.960
We don't talk about gradient
until a few sections from now.

01:26:02.960 --> 01:26:05.120
But I'd like to
anticipate a little bit.

01:26:05.120 --> 01:26:08.440
So the gradient of
a function, wherever

01:26:08.440 --> 01:26:15.080
the partial derivatives exist,
with the partial derivative--

01:26:15.080 --> 01:26:21.215
that is, f sub x
and f sub y exist--

01:26:21.215 --> 01:26:26.572
I'm going to have that
delta f-- nabla f.

01:26:26.572 --> 01:26:29.494
nabla is a [INAUDIBLE].

01:26:29.494 --> 01:26:34.340
Nable f at xy represents what?

01:26:34.340 --> 01:26:34.850
The vector.

01:26:34.850 --> 01:26:37.400


01:26:37.400 --> 01:26:39.020
And I know you love vectors.

01:26:39.020 --> 01:26:45.580
And that's why I'm going back
to the vector notation f sub x

01:26:45.580 --> 01:26:51.508
at xy times i, i being
the standard vector i

01:26:51.508 --> 01:26:59.130
unit along the x axis,
f sub y at xy times j.

01:26:59.130 --> 01:27:03.450
STUDENT: So it's just like
the notation of [INAUDIBLE]?

01:27:03.450 --> 01:27:05.322
MAGDALENA TODA: Just
the vector notation.

01:27:05.322 --> 01:27:08.010
How else could I write it?

01:27:08.010 --> 01:27:13.280
Angular bracket, f sub x x
at xy, comma, f sub y at xy.

01:27:13.280 --> 01:27:16.935
And you know-- people who
saw my videos, colleagues

01:27:16.935 --> 01:27:19.770
who teach Calc 3
at the same time

01:27:19.770 --> 01:27:25.110
said I have a tendency of not
going by the book notations

01:27:25.110 --> 01:27:28.260
all the time, and just give you
the [? round ?] parentheses.

01:27:28.260 --> 01:27:28.960
It's OK.

01:27:28.960 --> 01:27:31.340
I mean, different books,
different notations.

01:27:31.340 --> 01:27:35.460
But what I mean is to represent
the vector in the standard way

01:27:35.460 --> 01:27:37.430
[INAUDIBLE].

01:27:37.430 --> 01:27:38.010
All right.

01:27:38.010 --> 01:27:38.510
OK.

01:27:38.510 --> 01:27:41.760
Can you have this
notion for something

01:27:41.760 --> 01:27:44.840
like a function of
three variables?

01:27:44.840 --> 01:27:45.870
Absolutely.

01:27:45.870 --> 01:27:48.430
Now I'll give you an easy one.

01:27:48.430 --> 01:27:50.920
Suppose that you have
x-squared plus y-squared

01:27:50.920 --> 01:27:54.402
plus z-squared equals 1.

01:27:54.402 --> 01:28:00.130
And that is called-- let's
call it names-- f of x, y, z.

01:28:00.130 --> 01:28:16.800
Compute the gradient nabla f
at any point x, y, z for f.

01:28:16.800 --> 01:28:20.705
Find the meaning of that
gradient-- of that-- find

01:28:20.705 --> 01:28:29.460
the geometric meaning of it.

01:28:29.460 --> 01:28:34.210
For this case, not in
general, for this case.

01:28:34.210 --> 01:28:35.860
So you say, wait,
wait, Magdalena.

01:28:35.860 --> 01:28:38.846
A-dah-dah, you're confusing me.

01:28:38.846 --> 01:28:39.941
This is the gradient.

01:28:39.941 --> 01:28:40.440
Hmm.

01:28:40.440 --> 01:28:43.360
Depends on how many
variables you have.

01:28:43.360 --> 01:28:47.680
So you have to show a vector
whose coordinates represent

01:28:47.680 --> 01:28:50.800
the partial derivatives with
respect to all the variables.

01:28:50.800 --> 01:28:56.065
If I have n variables, I have
f sub x1 comma f sub x2 comma

01:28:56.065 --> 01:28:58.770
f sub x3 comma f
sub xn, and stop.

01:28:58.770 --> 01:28:59.570
Yes, sir.

01:28:59.570 --> 01:29:03.610
STUDENT: If the formula
was just f of xy,

01:29:03.610 --> 01:29:05.160
wouldn't that be implicit?

01:29:05.160 --> 01:29:06.640
MAGDALENA TODA:
That is implicit.

01:29:06.640 --> 01:29:08.510
That's exactly what I meant.

01:29:08.510 --> 01:29:12.220
What's the geometric
meaning of this animal?

01:29:12.220 --> 01:29:13.920
Forget about the left hand side.

01:29:13.920 --> 01:29:15.510
I'm going to clean it quickly.

01:29:15.510 --> 01:29:16.740
What is that animal?

01:29:16.740 --> 01:29:19.640
That is a hippopotamus.

01:29:19.640 --> 01:29:20.500
What is that?

01:29:20.500 --> 01:29:22.148
STUDENT: It's a sphere.

01:29:22.148 --> 01:29:23.398
MAGDALENA TODA: It's a sphere.

01:29:23.398 --> 01:29:24.847
But what kind of sphere?

01:29:24.847 --> 01:29:27.950
Center 0, 0, 0 with radius 1.

01:29:27.950 --> 01:29:30.150
What do we call that?

01:29:30.150 --> 01:29:30.670
Unit sphere.

01:29:30.670 --> 01:29:33.594
Do you know what notation
that mathematicians

01:29:33.594 --> 01:29:36.516
use for that object?

01:29:36.516 --> 01:29:40.380
You don't know but I'll
tell you. s1 is the sphere.

01:29:40.380 --> 01:29:42.340
We have s2, I'm
sorry, the sphere

01:29:42.340 --> 01:29:45.190
of dimension 2, which
means the surface.

01:29:45.190 --> 01:29:47.127
s1 is the circle.

01:29:47.127 --> 01:29:49.115
s1 is a circle.

01:29:49.115 --> 01:29:51.580
s2 is a sphere.

01:29:51.580 --> 01:29:54.860
So what is this number
here for a mathematician?

01:29:54.860 --> 01:29:59.050
That's the dimension of
that kind of manifold.

01:29:59.050 --> 01:30:02.310
So if I have just a
circle, we call it s1

01:30:02.310 --> 01:30:05.510
because there is only a one
independent variable, which

01:30:05.510 --> 01:30:08.000
is time, and we parameterize.

01:30:08.000 --> 01:30:09.135
Why go clockwise?

01:30:09.135 --> 01:30:09.745
Shame on me.

01:30:09.745 --> 01:30:12.190
Go counterclockwise.

01:30:12.190 --> 01:30:13.030
All right.

01:30:13.030 --> 01:30:14.110
That's s1.

01:30:14.110 --> 01:30:16.260
For s2, I have two
degrees of freedom.

01:30:16.260 --> 01:30:18.770
It's a surface.

01:30:18.770 --> 01:30:23.005
On earth, what are those
two degrees of freedom?

01:30:23.005 --> 01:30:25.980
It's a riddle.

01:30:25.980 --> 01:30:27.065
No extra credit.

01:30:27.065 --> 01:30:30.320
STUDENT: The latitude
and longitude?

01:30:30.320 --> 01:30:31.882
MAGDALENA TODA: Who said it?

01:30:31.882 --> 01:30:33.532
Who said it first?

01:30:33.532 --> 01:30:35.300
STUDENT: [INAUDIBLE].

01:30:35.300 --> 01:30:40.185
MAGDALENA TODA: How many of
you said it at the same time?

01:30:40.185 --> 01:30:40.935
Alexander said it.

01:30:40.935 --> 01:30:42.726
STUDENT: I know there
was one other person.

01:30:42.726 --> 01:30:43.910
I wasn't the only one.

01:30:43.910 --> 01:30:44.660
STUDENT: I didn't.

01:30:44.660 --> 01:30:47.490


01:30:47.490 --> 01:30:48.882
STUDENT: [INAUDIBLE], sorry.

01:30:48.882 --> 01:30:49.715
[INTERPOSING VOICES]

01:30:49.715 --> 01:30:52.420
MAGDALENA TODA: I
don't have enough.

01:30:52.420 --> 01:30:54.340
STUDENT: I'll take
the credit for it.

01:30:54.340 --> 01:30:56.170
MAGDALENA TODA:
[INAUDIBLE] extra credit.

01:30:56.170 --> 01:30:58.620
OK, you choose.

01:30:58.620 --> 01:30:59.560
These are good.

01:30:59.560 --> 01:31:01.950
They are Valentine's hearts,
chocolate [INAUDIBLE].

01:31:01.950 --> 01:31:04.791


01:31:04.791 --> 01:31:05.290
Wilson.

01:31:05.290 --> 01:31:09.385


01:31:09.385 --> 01:31:12.040
I heard you saying Wilson.

01:31:12.040 --> 01:31:13.008
I have more.

01:31:13.008 --> 01:31:13.976
I have more.

01:31:13.976 --> 01:31:17.364
These are cough drops,
so I'm [INAUDIBLE].

01:31:17.364 --> 01:31:20.489
You set it right
next time, Alexander.

01:31:20.489 --> 01:31:21.968
STUDENT: [INAUDIBLE].

01:31:21.968 --> 01:31:22.954
MAGDALENA TODA: OK.

01:31:22.954 --> 01:31:23.940
Anybody else?

01:31:23.940 --> 01:31:26.398
Anybody needing cough drops?

01:31:26.398 --> 01:31:26.898
OK.

01:31:26.898 --> 01:31:27.884
I'll leave them here.

01:31:27.884 --> 01:31:29.363
Just let me see.

01:31:29.363 --> 01:31:31.830
Do I have more chocolate?

01:31:31.830 --> 01:31:32.986
Eh, next time.

01:31:32.986 --> 01:31:35.160
I'm going to get some
before-- we have-- we

01:31:35.160 --> 01:31:37.090
need before Valentine's, right?

01:31:37.090 --> 01:31:37.970
So it's Thursday.

01:31:37.970 --> 01:31:41.060
I'm going to bring
you a lot more.

01:31:41.060 --> 01:31:46.650
So in that case, what
is the gradient of f?

01:31:46.650 --> 01:31:47.970
An x, y, z.

01:31:47.970 --> 01:31:48.470
Aha.

01:31:48.470 --> 01:31:50.170
I have three variables.

01:31:50.170 --> 01:31:52.520
What's the gradient?

01:31:52.520 --> 01:31:56.050
I can write it as a
bracket, angular notation.

01:31:56.050 --> 01:31:58.070
Am I right?

01:31:58.070 --> 01:32:02.790
Or I can write it 2xi
plus 2ij plus 2zk.

01:32:02.790 --> 01:32:06.590
Can anybody tell me why?

01:32:06.590 --> 01:32:09.640
What in the world are
these, 2x, 2y, 2z?

01:32:09.640 --> 01:32:11.697
STUDENT: Those are the
partial derivatives.

01:32:11.697 --> 01:32:14.030
MAGDALENA TODA: They are
exactly the partial derivatives

01:32:14.030 --> 01:32:17.810
with respect to x, with respect
to y, with respect to z.

01:32:17.810 --> 01:32:19.410
Does this have a
geometric meaning?

01:32:19.410 --> 01:32:20.540
I don't know.

01:32:20.540 --> 01:32:21.870
I have to draw.

01:32:21.870 --> 01:32:24.210
And maybe when I
draw, I get an idea.

01:32:24.210 --> 01:32:29.105


01:32:29.105 --> 01:32:31.526
Is this a unit vector?

01:32:31.526 --> 01:32:32.482
Uh-uh.

01:32:32.482 --> 01:32:33.980
It's not.

01:32:33.980 --> 01:32:35.770
Nabla s, right.

01:32:35.770 --> 01:32:36.490
In a way it is.

01:32:36.490 --> 01:32:37.800
It's not a unit vector.

01:32:37.800 --> 01:32:41.040
But if I were to
[? uniterize ?] it--

01:32:41.040 --> 01:32:43.710
and you know very well what it
means to [? uniterize it ?].

01:32:43.710 --> 01:32:44.732
It means to--

01:32:44.732 --> 01:32:45.690
STUDENT: Divide it by--

01:32:45.690 --> 01:32:47.440
MAGDALENA TODA: Divide
it by its magnitude

01:32:47.440 --> 01:32:51.075
and make it a unit vector
that would have a meaning.

01:32:51.075 --> 01:32:52.180
This is the sphere.

01:32:52.180 --> 01:32:56.016


01:32:56.016 --> 01:32:57.670
What if I make like this?

01:32:57.670 --> 01:33:04.145
n equals nabla f over
a magnitude of f.

01:33:04.145 --> 01:33:10.050
And what is the meaning
of that going to be?

01:33:10.050 --> 01:33:12.066
Can you tell me what
I'm going to get here?

01:33:12.066 --> 01:33:18.830


01:33:18.830 --> 01:33:24.870
In your head,
compute the magnitude

01:33:24.870 --> 01:33:29.430
and divide by the magnitude,
and you have exactly 15 seconds

01:33:29.430 --> 01:33:31.420
to tell me what it is.

01:33:31.420 --> 01:33:32.845
STUDENT: [INAUDIBLE].

01:33:32.845 --> 01:33:34.470
MAGDALENA TODA:
[? Ryan, ?] [? Ryan, ?]

01:33:34.470 --> 01:33:36.170
you are in a Twilight Zone.

01:33:36.170 --> 01:33:39.798
But I'm sure once I tell you,
once I tell you, [INAUDIBLE].

01:33:39.798 --> 01:33:41.548
STUDENT: 1 divided by
the square root of 2

01:33:41.548 --> 01:33:42.922
for the [? i controller. ?]

01:33:42.922 --> 01:33:43.797
STUDENT: [INAUDIBLE].

01:33:43.797 --> 01:33:47.610


01:33:47.610 --> 01:33:49.730
MAGDALENA TODA: Well, OK.

01:33:49.730 --> 01:33:51.125
Say it again, somebody.

01:33:51.125 --> 01:33:52.990
STUDENT: x plus y plus z.

01:33:52.990 --> 01:33:58.112
MAGDALENA TODA: xi plus yj
plus zk, not x plus x, y,

01:33:58.112 --> 01:33:59.820
z because that
would be a mistake.

01:33:59.820 --> 01:34:03.325
It would be a scalar function.
[INAUDIBLE] has to be a vector.

01:34:03.325 --> 01:34:07.190
If I am to draw this vector,
how am I going to draw it?

01:34:07.190 --> 01:34:10.030
Well, this is the
position vector.

01:34:10.030 --> 01:34:11.220
Say it again.

01:34:11.220 --> 01:34:12.720
This is the position vector.

01:34:12.720 --> 01:34:15.881
When I have a point on this
stinking earth, whatever

01:34:15.881 --> 01:34:21.100
it is, x, y, z, the
position vector is x, y, z.

01:34:21.100 --> 01:34:26.260
It's xi plus yj plus zk.

01:34:26.260 --> 01:34:28.690
I have this identification
between the point

01:34:28.690 --> 01:34:29.639
and the vector.

01:34:29.639 --> 01:34:30.430
This is our vector.

01:34:30.430 --> 01:34:33.300
So I'm going to draw these
needles, all these needles,

01:34:33.300 --> 01:34:41.880
all these vectors whose tips
are exactly on the sphere.

01:34:41.880 --> 01:34:42.960
So why?

01:34:42.960 --> 01:34:43.870
You say, OK.

01:34:43.870 --> 01:34:46.470
I understand that is
the position vector,

01:34:46.470 --> 01:34:48.735
but why did you put an n here?

01:34:48.735 --> 01:34:52.900
And anybody who answers
that gets a cough drops.

01:34:52.900 --> 01:34:54.737
STUDENT: [INAUDIBLE].

01:34:54.737 --> 01:34:56.070
MAGDALENA TODA: Because that is?

01:34:56.070 --> 01:34:58.235
STUDENT: The normal
to the surface.

01:34:58.235 --> 01:34:59.360
MAGDALENA TODA: You get a--

01:34:59.360 --> 01:35:00.824
STUDENT: Yeah, cough drop.

01:35:00.824 --> 01:35:02.288
MAGDALENA TODA: Two of them.

01:35:02.288 --> 01:35:03.157
STUDENT: Aw, yeah.

01:35:03.157 --> 01:35:04.240
MAGDALENA TODA: All right.

01:35:04.240 --> 01:35:07.840
So that's the normal
to the surface, which

01:35:07.840 --> 01:35:11.087
would be a continuation
of the position vector.

01:35:11.087 --> 01:35:11.670
You see, guys?

01:35:11.670 --> 01:35:14.450
So imagine you take
your position vector.

01:35:14.450 --> 01:35:15.840
This is the sphere.

01:35:15.840 --> 01:35:17.580
It's like an egg.

01:35:17.580 --> 01:35:20.550
And these tips
are on the sphere.

01:35:20.550 --> 01:35:24.950
If you continue from
sitting on the sphere,

01:35:24.950 --> 01:35:29.240
another radius vector
colinear to that,

01:35:29.240 --> 01:35:31.280
that would be the
normal to the sphere.

01:35:31.280 --> 01:35:36.480
So in topology, we
have a name for that.

01:35:36.480 --> 01:35:38.914
We call that the hairy ball.

01:35:38.914 --> 01:35:41.970
The hairy ball in
mathematics, I'm not kidding,

01:35:41.970 --> 01:35:44.410
it's a concentrated notations.

01:35:44.410 --> 01:35:47.620
You see it in graduate
courses, if you're

01:35:47.620 --> 01:35:50.157
going to become a graduate
student in mathematics,

01:35:50.157 --> 01:35:51.990
or you want to do a
dual degree or whatever,

01:35:51.990 --> 01:35:55.690
you're going to see the hairy
ball, all those normal vectors

01:35:55.690 --> 01:35:58.820
of length 1.

01:35:58.820 --> 01:36:01.620
It's also called
the normal field.

01:36:01.620 --> 01:36:04.526
So if you ask Dr.
Ibragimov, because he

01:36:04.526 --> 01:36:08.900
is in this kind of field
theory, [INAUDIBLE] normal field

01:36:08.900 --> 01:36:10.080
to a surface.

01:36:10.080 --> 01:36:13.010
But for the topologists
or geometers,

01:36:13.010 --> 01:36:15.300
they say, oh, that's
the hairy ball.

01:36:15.300 --> 01:36:18.860
So if you ask him what the
hairy ball is, he will say,

01:36:18.860 --> 01:36:21.800
why are you talking
nonsense to me?

01:36:21.800 --> 01:36:22.780
Right.

01:36:22.780 --> 01:36:24.250
Exactly.

01:36:24.250 --> 01:36:30.745
So here's where we stopped
our intrusion in chapter 11.

01:36:30.745 --> 01:36:33.129
It's going to be as
fun as it was today

01:36:33.129 --> 01:36:34.420
with these partial derivatives.

01:36:34.420 --> 01:36:35.590
You're going to love them.

01:36:35.590 --> 01:36:39.860
You have a lot of computations
like the ones we did today.

01:36:39.860 --> 01:36:42.590
Let's go back to
something you hated,

01:36:42.590 --> 01:36:45.630
which is the parameterizations.

01:36:45.630 --> 01:36:48.590
So one of you--
no, three of you--

01:36:48.590 --> 01:36:51.606
asked me to redo one
problem like the one

01:36:51.606 --> 01:36:54.366
with the parameterization
of a circle.

01:36:54.366 --> 01:36:58.280
But now I have to pay
attention to the data

01:36:58.280 --> 01:36:59.920
that I come up with.

01:36:59.920 --> 01:37:14.258
So write the parameterization
of a circle of radius.

01:37:14.258 --> 01:37:17.240


01:37:17.240 --> 01:37:20.888
Do you want specific
data or you want letters?

01:37:20.888 --> 01:37:21.763
STUDENT: [INAUDIBLE].

01:37:21.763 --> 01:37:25.561


01:37:25.561 --> 01:37:26.352
MAGDALENA TODA: OK.

01:37:26.352 --> 01:37:30.480
Let's do it [INAUDIBLE] r,
and then I'll give an example.

01:37:30.480 --> 01:37:43.230
And center x0, y0 in plane
where-- what is the point?

01:37:43.230 --> 01:37:57.200
Where is the particle
moving for time t equals 0?

01:37:57.200 --> 01:37:59.260
Where is it located?

01:37:59.260 --> 01:38:00.310
All right.

01:38:00.310 --> 01:38:02.746
So review.

01:38:02.746 --> 01:38:15.624
We had frame that we always
picked at the origin.

01:38:15.624 --> 01:38:23.490
That was bad because we could
pick x0, y0 as a center,

01:38:23.490 --> 01:38:25.115
and that has a separate radius.

01:38:25.115 --> 01:38:31.905


01:38:31.905 --> 01:38:39.050
And now, they want me to write
a parameterization of a circle.

01:38:39.050 --> 01:38:41.020
How do you achieve it?

01:38:41.020 --> 01:38:49.400
You say the circle is x minus x0
squared plus y minus y0 squared

01:38:49.400 --> 01:38:50.980
equals r squared.

01:38:50.980 --> 01:38:53.740
And one of you
asked me by email--

01:38:53.740 --> 01:38:56.650
and that was a good
question-- you said, come on.

01:38:56.650 --> 01:38:58.920
Look, it was [INAUDIBLE].

01:38:58.920 --> 01:39:02.760
So you said, I was
quite good in math.

01:39:02.760 --> 01:39:04.050
I was smart.

01:39:04.050 --> 01:39:09.550
Why didn't I know the equations,
the parametric equations,

01:39:09.550 --> 01:39:11.490
or even this?

01:39:11.490 --> 01:39:13.730
I'll tell you why.

01:39:13.730 --> 01:39:15.850
This used to be
covered in high school.

01:39:15.850 --> 01:39:18.056
It's something called
college algebra.

01:39:18.056 --> 01:39:21.460
We had a chapter,
either trigonometry

01:39:21.460 --> 01:39:22.272
or college algebra.

01:39:22.272 --> 01:39:24.520
We had a chapter called
analytic geometry.

01:39:24.520 --> 01:39:26.340
This is analytic geometry.

01:39:26.340 --> 01:39:28.530
It's the same chapter
in which you guys

01:39:28.530 --> 01:39:33.510
covered conics, [INAUDIBLE],
ellipse, [INAUDIBLE], parabola.

01:39:33.510 --> 01:39:36.120
It's no longer covered
in most high schools.

01:39:36.120 --> 01:39:37.030
I asked around.

01:39:37.030 --> 01:39:39.920
The teachers told
me that we reduced

01:39:39.920 --> 01:39:41.810
the geometric
applications a lot,

01:39:41.810 --> 01:39:47.920
according to the general
standards that are imposed.

01:39:47.920 --> 01:39:51.603
That's a pity, because you
really need this in college.

01:39:51.603 --> 01:39:52.590
All right.

01:39:52.590 --> 01:39:55.520
So how do you come up
with a parameterization?

01:39:55.520 --> 01:40:01.060
You say, I would like to
parameterize in such way

01:40:01.060 --> 01:40:03.490
that this would be
easy to understand

01:40:03.490 --> 01:40:06.300
this for Pythagorean theorem.

01:40:06.300 --> 01:40:07.450
Oh, OK.

01:40:07.450 --> 01:40:10.395
So what is the Pythagorean
theorem telling me?

01:40:10.395 --> 01:40:14.240
It's telling you that if you are
in a unit circle practically,

01:40:14.240 --> 01:40:19.005
then this is cosine and
theta and this is sine theta,

01:40:19.005 --> 01:40:21.637
and the sum of
cosine theta squared

01:40:21.637 --> 01:40:24.050
plus sine theta squared is 1.

01:40:24.050 --> 01:40:26.778
This is 1, so that is
the Pythagorean theorem

01:40:26.778 --> 01:40:28.722
[INAUDIBLE].

01:40:28.722 --> 01:40:38.230
So xy plus x0 should be cosine
of theta times an R. Why an R?

01:40:38.230 --> 01:40:41.920
Because I want, when I square,
I want the R squared up.

01:40:41.920 --> 01:40:46.230
And here, this guy inside
will be our sine [? thing. ?]

01:40:46.230 --> 01:40:47.550
Am I going to be in good shape?

01:40:47.550 --> 01:40:51.450
Yes, because when I
square this fellow squared

01:40:51.450 --> 01:40:54.660
plus this fellow squared will
give me exactly R squared.

01:40:54.660 --> 01:40:58.300
And here is my
[INAUDIBLE] smiley face.

01:40:58.300 --> 01:41:01.260
So I want to understand
what I'm doing.

01:41:01.260 --> 01:41:05.440
x minus x0 must
be R cosine theta.

01:41:05.440 --> 01:41:09.106
y minus y0 is R sine theta.

01:41:09.106 --> 01:41:13.860
Theta in general is an
angular velocity, [INAUDIBLE].

01:41:13.860 --> 01:41:17.250
But it's also time, right?

01:41:17.250 --> 01:41:19.290
It has the meaning
of time parameter.

01:41:19.290 --> 01:41:22.980
So when we wrote those--
and some of you are bored,

01:41:22.980 --> 01:41:25.620
but I think it's not
going to harm anybody

01:41:25.620 --> 01:41:27.240
that I do this again.

01:41:27.240 --> 01:41:36.406
R cosine of t plus x0 y is R
sine t plus x0, or plus y0.

01:41:36.406 --> 01:41:41.490
Now note, all those
examples in web work,

01:41:41.490 --> 01:41:43.840
they were not very imaginative.

01:41:43.840 --> 01:41:47.580
They didn't mean for
you to try other things.

01:41:47.580 --> 01:41:53.630
Like if one would put here
cosine of 5t or sine of 5t,

01:41:53.630 --> 01:41:56.830
that person would move five
times faster on the circle.

01:41:56.830 --> 01:42:00.240
And instead of being back
at 2 pi, in time 2 pi,

01:42:00.240 --> 01:42:02.970
they would be there
in time 2 pi over 5.

01:42:02.970 --> 01:42:06.860
All the examples-- and each of
you, it was randomized somehow.

01:42:06.860 --> 01:42:09.730
Each of you has a
different data set.

01:42:09.730 --> 01:42:11.970
Different R,
different x0 with 0,

01:42:11.970 --> 01:42:15.570
and a different place where
the particle is moving.

01:42:15.570 --> 01:42:18.580
But no matter what
they gave you,

01:42:18.580 --> 01:42:21.910
it's a response to
the same problem.

01:42:21.910 --> 01:42:26.930
And at time t equals
0, you have M. Do

01:42:26.930 --> 01:42:28.650
you want me to call it M0?

01:42:28.650 --> 01:42:33.090
Yes, from my initial-- M0.

01:42:33.090 --> 01:42:41.040
For t equals 0, you're
going to have R plus x0.

01:42:41.040 --> 01:42:44.670
And for t equals 0, you have y0.

01:42:44.670 --> 01:42:50.425
So for example, Ryan had-- Ryan,
I don't remember what you had.

01:42:50.425 --> 01:42:54.232
You had some where theta R was--

01:42:54.232 --> 01:42:54.940
STUDENT: 4 and 8.

01:42:54.940 --> 01:42:57.057
MAGDALENA TODA: 7.

01:42:57.057 --> 01:42:58.015
You, what did you have?

01:42:58.015 --> 01:43:00.400
STUDENT: No, R was 7
and x was 3, y was 1.

01:43:00.400 --> 01:43:03.510
MAGDALENA TODA: R
was 7 and x0 was--

01:43:03.510 --> 01:43:05.740
STUDENT: 3, 1.

01:43:05.740 --> 01:43:11.570
MAGDALENA TODA: 3, 1 was x0, y0
so in that case, the point they

01:43:11.570 --> 01:43:15.820
gave here was 7 plus 3.

01:43:15.820 --> 01:43:16.825
Am I right, Ryan?

01:43:16.825 --> 01:43:17.700
You can always check.

01:43:17.700 --> 01:43:18.200
I remember.

01:43:18.200 --> 01:43:22.010
It was 10 and God
knows, and 10 and 1.

01:43:22.010 --> 01:43:25.810
So all of the data that
you had in that problem

01:43:25.810 --> 01:43:30.480
was created so that you
have these equations.

01:43:30.480 --> 01:43:36.392
And at time 0, you were exactly
at the time t equals 0 replaced

01:43:36.392 --> 01:43:37.181
the t.

01:43:37.181 --> 01:43:38.130
All right.

01:43:38.130 --> 01:43:39.230
OK.

01:43:39.230 --> 01:43:40.315
STUDENT: What's the M0?

01:43:40.315 --> 01:43:42.280
What is--

01:43:42.280 --> 01:43:45.240
MAGDALENA TODA: M0
is Magdalena times 0.

01:43:45.240 --> 01:43:46.820
I don't know.

01:43:46.820 --> 01:43:51.045
I mean, it's the
point where you are.

01:43:51.045 --> 01:43:55.140
I couldn't come up
with a better name.

01:43:55.140 --> 01:44:01.569
So I'm going to
erase here and I'll

01:44:01.569 --> 01:44:08.280
get to another problem, which
gave you guys a big headache.

01:44:08.280 --> 01:44:16.570
And it's not so hard, but this
is the computational problem,

01:44:16.570 --> 01:44:18.046
very pretty in itself.

01:44:18.046 --> 01:44:24.934


01:44:24.934 --> 01:44:35.706
[INAUDIBLE] cosine t i plus
e to the 3t sine t j plus e

01:44:35.706 --> 01:44:36.540
to the 3tk.

01:44:36.540 --> 01:44:40.040


01:44:40.040 --> 01:44:43.850
And I think this was more or
less in everybody's homework

01:44:43.850 --> 01:44:45.480
the same.

01:44:45.480 --> 01:44:51.680
There's a position vector
given as parameterized form.

01:44:51.680 --> 01:44:54.330
So since you love
parameterization so much,

01:44:54.330 --> 01:45:00.523
I'm going to remind you what
that means for x and y and zr.

01:45:00.523 --> 01:45:03.481
And what did they want from you?

01:45:03.481 --> 01:45:07.920
I forget what number of
the problem that was.

01:45:07.920 --> 01:45:16.270
They wanted the length of
the arc of a curve from t

01:45:16.270 --> 01:45:18.604
equals-- I don't know.

01:45:18.604 --> 01:45:19.580
STUDENT: 2 to 5.

01:45:19.580 --> 01:45:21.044
MAGDALENA TODA: 2 to 5.

01:45:21.044 --> 01:45:22.020
Thank you.

01:45:22.020 --> 01:45:23.972
[INAUDIBLE] t equals 5.

01:45:23.972 --> 01:45:30.316
So this is the beginning and the
end of the curve, the beginning

01:45:30.316 --> 01:45:32.268
and the end of a curve.

01:45:32.268 --> 01:45:35.630
So what is that going
to be [INAUDIBLE]?

01:45:35.630 --> 01:45:40.500
How does [INAUDIBLE],
which we have

01:45:40.500 --> 01:45:46.828
to write down 2 to 5
magnitude of r prime at t, dt.

01:45:46.828 --> 01:45:50.140


01:45:50.140 --> 01:45:53.000
And I don't know.

01:45:53.000 --> 01:45:56.980
But I want to review this
because-- so what in the world?

01:45:56.980 --> 01:45:59.540
Maybe I put this
on the midterm or I

01:45:59.540 --> 01:46:03.530
make it a little bit easier,
but the same what I don't like,

01:46:03.530 --> 01:46:05.070
it's time consuming.

01:46:05.070 --> 01:46:07.780
But I can give you
something a lot easier

01:46:07.780 --> 01:46:10.570
that tests the
concept, the idea, not

01:46:10.570 --> 01:46:13.090
the computational power.

01:46:13.090 --> 01:46:20.210
So r prime of t here with
a little bit of attention,

01:46:20.210 --> 01:46:25.100
of course, most of you
computing this correctly.

01:46:25.100 --> 01:46:28.010
You are just a little bit scared
of what happened after that,

01:46:28.010 --> 01:46:30.467
and you should not be scared
because now I'll tell you

01:46:30.467 --> 01:46:32.860
why you shouldn't be scared.

01:46:32.860 --> 01:46:34.850
Chain rule, product rule.

01:46:34.850 --> 01:46:37.890
So I have first prime--

01:46:37.890 --> 01:46:38.900
STUDENT: 3.

01:46:38.900 --> 01:46:42.490
MAGDALENA TODA: 3 into the
3e second and [? time ?]

01:46:42.490 --> 01:46:46.730
cosine t plus-- I'm
going to do that later.

01:46:46.730 --> 01:46:48.320
I know what you're thinking.

01:46:48.320 --> 01:46:49.792
STUDENT: e 3t.

01:46:49.792 --> 01:46:53.264
MAGDALENA TODA: e to
the 3t minus sine.

01:46:53.264 --> 01:46:55.600
I'm not worried
about this minus now.

01:46:55.600 --> 01:46:57.430
I'll take care of that later.

01:46:57.430 --> 01:46:58.010
Times i.

01:46:58.010 --> 01:47:00.710


01:47:00.710 --> 01:47:03.390
Now with your
permission-- when you

01:47:03.390 --> 01:47:08.150
say, why is she not writing the
whole thing in continuation?

01:47:08.150 --> 01:47:09.470
Because I don't want to.

01:47:09.470 --> 01:47:09.970
No.

01:47:09.970 --> 01:47:13.170
Because I want to help
you see what's going on.

01:47:13.170 --> 01:47:16.310
You do the same kind of stuff
for this individual one.

01:47:16.310 --> 01:47:17.804
I want to put it
right underneath.

01:47:17.804 --> 01:47:21.290
If I put it right underneath,
it's going to [? agree ?].

01:47:21.290 --> 01:47:23.966
Otherwise it's not
going to [? agree ?].

01:47:23.966 --> 01:47:32.371
E to the 3t times sine t
plus e to the 3t cosine t.

01:47:32.371 --> 01:47:34.111
You didn't have a
problem because you

01:47:34.111 --> 01:47:36.000
know how to differentiate.

01:47:36.000 --> 01:47:40.780
You started having the
problem from this point on.

01:47:40.780 --> 01:47:44.412
3 into the 3tk.

01:47:44.412 --> 01:47:47.010
The problem came when
you were supposed

01:47:47.010 --> 01:47:55.510
to identify the coordinates and
square them and squeeze them

01:47:55.510 --> 01:47:57.280
under the same square root.

01:47:57.280 --> 01:48:01.250
And that drove you crazy
when you have enough.

01:48:01.250 --> 01:48:04.210
Let me put the minus here to
make it more obvious what's

01:48:04.210 --> 01:48:06.360
going to happen.

01:48:06.360 --> 01:48:08.120
When you're going
to have problems

01:48:08.120 --> 01:48:09.770
like that in
differential equations,

01:48:09.770 --> 01:48:14.670
you better have the eye
for it, [INAUDIBLE].

01:48:14.670 --> 01:48:18.870
You should be able to recognize
this is like a pattern.

01:48:18.870 --> 01:48:26.585
Have you seen the
movie A Beautiful Mind?

01:48:26.585 --> 01:48:27.210
STUDENT: Yeah.

01:48:27.210 --> 01:48:28.620
MAGDALENA TODA:
OK, so Nash, when

01:48:28.620 --> 01:48:34.180
he was writing with the finger
on everything, on the walls

01:48:34.180 --> 01:48:39.835
at Princeton, on the window,
he was thinking of patterns.

01:48:39.835 --> 01:48:42.270
He's actually
trying to-- and it's

01:48:42.270 --> 01:48:44.130
hard to visualize
without drawing,

01:48:44.130 --> 01:48:48.374
but this is what most of us
recognize all the time when

01:48:48.374 --> 01:48:51.212
a mathematician writes
down some computations

01:48:51.212 --> 01:48:52.905
in a different way.

01:48:52.905 --> 01:48:58.420
All we hope for is to get a
few steps behind that board

01:48:58.420 --> 01:48:59.850
and see a pattern.

01:48:59.850 --> 01:49:02.340
And when you do that,
you see the pattern.

01:49:02.340 --> 01:49:05.580
This is an a minus b
and that's an a plus b.

01:49:05.580 --> 01:49:08.685
And then you say, OK, if
I'm going to square them,

01:49:08.685 --> 01:49:10.630
what's going to happen?

01:49:10.630 --> 01:49:15.330
When you square an a minus
b and you square an a plus b

01:49:15.330 --> 01:49:18.810
and you have this giggly
guy there-- leave him there.

01:49:18.810 --> 01:49:21.980
He's having too much fun.

01:49:21.980 --> 01:49:28.194
You actually develop these
guys and you put them one

01:49:28.194 --> 01:49:31.530
under the other
and say wow, what

01:49:31.530 --> 01:49:34.280
a beautiful simplification.

01:49:34.280 --> 01:49:36.760
When I'm going to
add these guys,

01:49:36.760 --> 01:49:40.340
this thing in the middle
will simply will cancel out,

01:49:40.340 --> 01:49:44.700
but the a squared will double
and the b squared will double.

01:49:44.700 --> 01:49:46.670
And that's the beauty
of seeing pattern.

01:49:46.670 --> 01:49:50.970
You see how there is
something symmetric and magic

01:49:50.970 --> 01:49:56.470
in mathematics that make
the answer simplified.

01:49:56.470 --> 01:50:01.360
And that allows you to compress
your equations that originally

01:50:01.360 --> 01:50:05.740
seemed to be a mess
into something that's

01:50:05.740 --> 01:50:08.595
more easily expressed.

01:50:08.595 --> 01:50:11.450
So when you're going
to compute this r

01:50:11.450 --> 01:50:17.792
prime of t magic absolute
value of the magnitude, that's

01:50:17.792 --> 01:50:21.664
going to be square root of--
instead of writing all the

01:50:21.664 --> 01:50:25.190
[INAUDIBLE], I hate writing
and rewriting the whole thing

01:50:25.190 --> 01:50:28.804
squared plus the whole thing
squared plus this squared.

01:50:28.804 --> 01:50:32.970
If I love to write so much,
I'd be in humanities and not

01:50:32.970 --> 01:50:34.780
in mathematics.

01:50:34.780 --> 01:50:41.175
So as a mathematician, how
am I going to write that?

01:50:41.175 --> 01:50:44.379
As a mathematician, I'm going
to use some sort of-- like the U

01:50:44.379 --> 01:50:44.920
substitution.

01:50:44.920 --> 01:50:48.910
So I say, I call this Mr.
A, and I call this Mr. B.

01:50:48.910 --> 01:50:50.966
And that's A minus B,
and that's A plus B.

01:50:50.966 --> 01:50:53.810
And that's somebody else.

01:50:53.810 --> 01:50:57.470
So when I square
the first guy, and I

01:50:57.470 --> 01:51:00.668
square the second component, and
I square the third component,

01:51:00.668 --> 01:51:09.780
and I add them together,
I'm going to get what?

01:51:09.780 --> 01:51:15.810
Square root of 2A
squared plus 2B squared.

01:51:15.810 --> 01:51:19.010
Because I know that
these are the first two.

01:51:19.010 --> 01:51:21.290
This guy squared
plus this guy squared

01:51:21.290 --> 01:51:23.430
is going to be
exactly 2A squared

01:51:23.430 --> 01:51:25.974
plus 2B squared,
nothing in the middle.

01:51:25.974 --> 01:51:28.830
These guys cancel out.

01:51:28.830 --> 01:51:30.636
STUDENT: A and B
are not the same.

01:51:30.636 --> 01:51:34.210


01:51:34.210 --> 01:51:42.030
MAGDALENA TODA: Well,
yeah, you're right.

01:51:42.030 --> 01:51:45.869
Let me call-- you're
right, this is the same,

01:51:45.869 --> 01:51:47.630
but these are different.

01:51:47.630 --> 01:51:51.995
So let me call them
A prime plus B prime.

01:51:51.995 --> 01:51:53.680
No, that's derivative.

01:51:53.680 --> 01:51:56.244
Let me call them C
and D-- very good,

01:51:56.244 --> 01:52:03.810
thank you-- C squared
plus 2CD plus D squared.

01:52:03.810 --> 01:52:06.370


01:52:06.370 --> 01:52:08.130
But the principle is the same.

01:52:08.130 --> 01:52:11.492
So I'm going to have A
squared plus C squared.

01:52:11.492 --> 01:52:12.940
This goes away.

01:52:12.940 --> 01:52:14.660
Why?

01:52:14.660 --> 01:52:18.530
Because this times that is
the same as this times that.

01:52:18.530 --> 01:52:19.836
Say it again.

01:52:19.836 --> 01:52:22.692
If we look in the
middle, the middle term

01:52:22.692 --> 01:52:28.298
will have 3e to the 3t cosine
t times e to the 3t sine t.

01:52:28.298 --> 01:52:33.030
Middle term here is 3e to the
3t e to the 3t sine and cosine.

01:52:33.030 --> 01:52:36.440
So they will cancel
out, this and that.

01:52:36.440 --> 01:52:40.191
So here I have the
sum of the square of A

01:52:40.191 --> 01:52:45.907
plus the square
of C. And here I'm

01:52:45.907 --> 01:52:50.680
going to have the square
of B plus the square of D.

01:52:50.680 --> 01:52:54.388
OK, now when I square this
and that, what do I get?

01:52:54.388 --> 01:52:57.220


01:52:57.220 --> 01:53:00.970
The beauty of that-- let me
write it down then explicitly.

01:53:00.970 --> 01:53:06.910
9e to the 3t cosine squared
t remains from this guy.

01:53:06.910 --> 01:53:08.790
Plus from the square
of that, we'll

01:53:08.790 --> 01:53:19.588
have 9e to the 3t-- no, just 3,
9 to the 6t, 9 to the 6t sine

01:53:19.588 --> 01:53:21.540
squared.

01:53:21.540 --> 01:53:22.870
So I take this guy.

01:53:22.870 --> 01:53:23.545
I square it.

01:53:23.545 --> 01:53:24.712
I take this guy.

01:53:24.712 --> 01:53:26.920
I square it.

01:53:26.920 --> 01:53:30.050
The middle terms will
disappear, thank god.

01:53:30.050 --> 01:53:33.490
Then I have this guy, I square
it, that guy, I square it,

01:53:33.490 --> 01:53:34.840
good.

01:53:34.840 --> 01:53:41.290
Plus another parenthesis-- e
to the 6t sine squared t plus e

01:53:41.290 --> 01:53:44.266
to the 6t cosine squared t.

01:53:44.266 --> 01:53:47.110


01:53:47.110 --> 01:53:50.340
So even if they don't
double because they're not

01:53:50.340 --> 01:53:52.820
the same thing, what
is the principle

01:53:52.820 --> 01:53:54.390
that will make my life easier?

01:53:54.390 --> 01:53:58.840
The same pattern
of simplification.

01:53:58.840 --> 01:54:00.730
What is that same pattern
of simplification?

01:54:00.730 --> 01:54:03.720
Look at the beauty
of this guy and look

01:54:03.720 --> 01:54:05.110
at the beauty of this guy.

01:54:05.110 --> 01:54:06.790
And then there is
something missing,

01:54:06.790 --> 01:54:12.560
the happy guy that was quiet
because I told him to be quiet.

01:54:12.560 --> 01:54:17.162
That's 9e to the 6t.

01:54:17.162 --> 01:54:18.488
He was there in the corner.

01:54:18.488 --> 01:54:22.330
And you had to square this
guy and square this guy

01:54:22.330 --> 01:54:26.188
and square this guy and
add them on top together.

01:54:26.188 --> 01:54:27.649
Now what is the pattern?

01:54:27.649 --> 01:54:35.441
The pattern is 9e to the 6t
with 9e to the 6t, same guy.

01:54:35.441 --> 01:54:38.190
The orange guys-- that's
why I love the colors.

01:54:38.190 --> 01:54:40.595
Cosine squared cosine
squared will be 1.

01:54:40.595 --> 01:54:47.400
Another pattern like that, I
have e to the 6t, to the 6t,

01:54:47.400 --> 01:54:52.260
and the same happy guys sine
squared t, sine squared t,

01:54:52.260 --> 01:54:54.660
add them together is 1.

01:54:54.660 --> 01:55:00.505
So all in all, this mess
is not a mess anymore.

01:55:00.505 --> 01:55:11.290
So it becomes 9e to the 6t plus
e to the 6t plus 9e to the 6t.

01:55:11.290 --> 01:55:12.630
Are you guys with me?

01:55:12.630 --> 01:55:17.960
All right, now how many
e to the 6t's do we have?

01:55:17.960 --> 01:55:25.850
9 plus 9 plus 1, 19, square
root of 19 e to the 6t.

01:55:25.850 --> 01:55:29.900
So when we integrate,
we go integral

01:55:29.900 --> 01:55:33.410
from 2 to 5 square root of 19.

01:55:33.410 --> 01:55:34.850
Kick him out of your life.

01:55:34.850 --> 01:55:36.990
He's just making
your life harder.

01:55:36.990 --> 01:55:40.065
And then you have square root
of e to the 6t e to the 3t.

01:55:40.065 --> 01:55:42.910


01:55:42.910 --> 01:55:47.930
So after you kick
the guy out, you

01:55:47.930 --> 01:55:55.060
have e to the 3t divided
by 3 between t equals 2

01:55:55.060 --> 01:55:58.170
and t equals 5.

01:55:58.170 --> 01:56:03.230
Actually, I took it right off
the WeBWorK problem you had.

01:56:03.230 --> 01:56:06.104
So if you type this
in your WeBWorK--

01:56:06.104 --> 01:56:12.000
you probably already did-- you
should get exactly the answer

01:56:12.000 --> 01:56:13.202
as being correct.

01:56:13.202 --> 01:56:17.800


01:56:17.800 --> 01:56:24.160
On the exam, do not
expect anything that long.

01:56:24.160 --> 01:56:26.720
The idea of simplifying
these patterns

01:56:26.720 --> 01:56:31.780
by finding the sine cosine, sine
squared plus cosine squared is

01:56:31.780 --> 01:56:33.110
1, is still going to be there.

01:56:33.110 --> 01:56:35.690
But don't expect
anything that long.

01:56:35.690 --> 01:56:43.369
Also, don't expect-- once
you get to this state,

01:56:43.369 --> 01:56:44.818
I don't want an answer.

01:56:44.818 --> 01:56:46.267
This is the answer.

01:56:46.267 --> 01:56:48.199
That's the precise answer.

01:56:48.199 --> 01:56:52.560
I don't want any approximation
or anything like that.

01:56:52.560 --> 01:56:54.272
A few of you did this
with a calculator.

01:56:54.272 --> 01:56:57.655
Well, you will not have
calculators in the final.

01:56:57.655 --> 01:56:59.285
You are going to
have easy problems.

01:56:59.285 --> 01:57:03.170
If you did that
with a calculator,

01:57:03.170 --> 01:57:05.230
and you truncated
your answer later,

01:57:05.230 --> 01:57:11.270
and if you were within
0.01 of the correct answer,

01:57:11.270 --> 01:57:12.340
you were fine.

01:57:12.340 --> 01:57:14.861
But some people
approximated too much.

01:57:14.861 --> 01:57:16.785
And that's always a problem.

01:57:16.785 --> 01:57:19.490
So it's always a good
idea to enter something

01:57:19.490 --> 01:57:23.860
like that in WeBWorK.

01:57:23.860 --> 01:57:27.470
I said I wouldn't do it
except in the last 20 minutes.

01:57:27.470 --> 01:57:31.190
But I wanted to do
something like that.

01:57:31.190 --> 01:57:34.500
I want to give you another
example, because you love

01:57:34.500 --> 01:57:39.216
parametrization so much it just
occurred to me that it would

01:57:39.216 --> 01:57:41.940
be very, very helpful--
maybe, I don't

01:57:41.940 --> 01:57:47.060
know-- to give you another
problem similar to this one.

01:57:47.060 --> 01:57:50.250
It's not in the book,
but it was cooked up

01:57:50.250 --> 01:57:53.698
by one of my colleagues
for his homework.

01:57:53.698 --> 01:58:02.554
So I'd like to show it to you.

01:58:02.554 --> 01:58:06.490


01:58:06.490 --> 01:58:09.584
e to the t i is
a parametrization

01:58:09.584 --> 01:58:13.240
of a [INAUDIBLE] space.

01:58:13.240 --> 01:58:28.139
Plus e to the minus t j
plus square root of 2 tk.

01:58:28.139 --> 01:58:36.030


01:58:36.030 --> 01:58:37.470
And how do I know?

01:58:37.470 --> 01:58:41.102
Well, one of his
students came to me

01:58:41.102 --> 01:58:43.656
and asked for help
with homework.

01:58:43.656 --> 01:58:51.450
Well, we don't give help when
it comes from another colleague.

01:58:51.450 --> 01:58:55.790
So in the end, the student
went to the tutoring center.

01:58:55.790 --> 01:58:58.711
And the tutoring center
helped only in parts.

01:58:58.711 --> 01:59:00.520
She came back to me.

01:59:00.520 --> 01:59:03.860
So what was the deal here?

01:59:03.860 --> 01:59:13.662
Find f prime of t in
the most simplified form

01:59:13.662 --> 01:59:16.440
and find the absolute
value r prime of t

01:59:16.440 --> 01:59:17.830
in the most simplified form.

01:59:17.830 --> 01:59:22.830


01:59:22.830 --> 01:59:31.830
And find the length of the
arc of this curve between t

01:59:31.830 --> 01:59:33.824
equals 0 and t equals 1.

01:59:33.824 --> 01:59:36.632
If this were given
by a physicist,

01:59:36.632 --> 01:59:39.760
how would that physicist
reformulate the problem?

01:59:39.760 --> 01:59:47.895
He would say-- he or she--
what is the distance travelled

01:59:47.895 --> 01:59:54.450
by the particle between
0 seconds and 1 second?

01:59:54.450 --> 01:59:56.125
So how do you write that?

01:59:56.125 --> 02:00:03.550
Integral from 0 to 1 of
r prime of t [INAUDIBLE].

02:00:03.550 --> 02:00:05.530
And you have to do the rest.

02:00:05.530 --> 02:00:08.510


02:00:08.510 --> 02:00:13.040
So arguably, this is
the Chapter 10 review.

02:00:13.040 --> 02:00:15.070
It's very useful for
the midterm exam.

02:00:15.070 --> 02:00:17.570
So although we are
just doing this review,

02:00:17.570 --> 02:00:20.690
you should not erase
it from your memory.

02:00:20.690 --> 02:00:24.380
Because I don't like to
put surprise problems

02:00:24.380 --> 02:00:25.250
on the midterm.

02:00:25.250 --> 02:00:28.950
But if you worked a
certain type of problem,

02:00:28.950 --> 02:00:31.320
you may expect
something like that.

02:00:31.320 --> 02:00:33.720
Maybe it's different
but in the same spirit.

02:00:33.720 --> 02:00:37.690
r prime of t, who's going to
help me with r prime of t?

02:00:37.690 --> 02:00:40.720


02:00:40.720 --> 02:00:44.140
This fellow-- e to the t.

02:00:44.140 --> 02:00:46.863
And how about that?

02:00:46.863 --> 02:00:50.300
Negative e to the negative t.

02:00:50.300 --> 02:00:53.246
STUDENT: I thought the arc
length was the square root of 1

02:00:53.246 --> 02:00:56.192
plus f prime of t squared.

02:00:56.192 --> 02:00:58.730


02:00:58.730 --> 02:01:02.355
MAGDALENA TODA:
For a plane curve.

02:01:02.355 --> 02:01:04.440
OK, let me remind you.

02:01:04.440 --> 02:01:05.980
If you have a plane
curve y equals

02:01:05.980 --> 02:01:12.467
f of x, then this thing
would become integral from A

02:01:12.467 --> 02:01:17.740
to B square root of 1
plus f prime of x dx.

02:01:17.740 --> 02:01:22.010
And that, did you do that
with your Calc II instructor?

02:01:22.010 --> 02:01:25.740
How many of you
had Dr. Williams?

02:01:25.740 --> 02:01:28.000
That was a wonderful
class, wasn't it?

02:01:28.000 --> 02:01:29.380
And he taught that.

02:01:29.380 --> 02:01:31.460
And of course he
was not supposed

02:01:31.460 --> 02:01:36.120
to tell you that was the
speed of a parametric curve.

02:01:36.120 --> 02:01:39.020
If you were to
parametrize here, x of t

02:01:39.020 --> 02:01:44.000
was t and y of t
would be f of t.

02:01:44.000 --> 02:01:45.450
He could have told you.

02:01:45.450 --> 02:01:46.320
Maybe he told you.

02:01:46.320 --> 02:01:47.470
Maybe you don't remember.

02:01:47.470 --> 02:01:48.990
OK, let's forget about it.

02:01:48.990 --> 02:01:50.340
That was Calc II.

02:01:50.340 --> 02:01:54.120
Now, coming back here,
I have to list what?

02:01:54.120 --> 02:01:57.916
Square root of 2 times
t prime is one k.

02:01:57.916 --> 02:01:59.582
Who's going to help
me compute the speed

02:01:59.582 --> 02:02:02.380
and put it in a nice formula?

02:02:02.380 --> 02:02:04.163
Well, my god--

02:02:04.163 --> 02:02:04.996
STUDENT: [INAUDIBLE]

02:02:04.996 --> 02:02:08.230


02:02:08.230 --> 02:02:10.790
MAGDALENA TODA: Ahh,
you are too smart.

02:02:10.790 --> 02:02:15.152
Today you had some what is
that called with caffeine

02:02:15.152 --> 02:02:17.036
and vitamins and--

02:02:17.036 --> 02:02:18.920
STUDENT: You're
thinking of Red Bull.

02:02:18.920 --> 02:02:20.340
MAGDALENA TODA: I know.

02:02:20.340 --> 02:02:22.660
That was very nice.

02:02:22.660 --> 02:02:23.740
I try to stay away.

02:02:23.740 --> 02:02:28.223
What is that called
with the energy booster?

02:02:28.223 --> 02:02:29.264
STUDENT: I wouldn't know.

02:02:29.264 --> 02:02:30.491
STUDENT: 5-Hour Energy.

02:02:30.491 --> 02:02:31.719
MAGDALENA TODA: 5-Hour, OK.

02:02:31.719 --> 02:02:33.192
I used to have that.

02:02:33.192 --> 02:02:36.670
When I had that, I could
anticipate two steps computing.

02:02:36.670 --> 02:02:39.809
Just a joke, Alex,
don't take it up.

02:02:39.809 --> 02:02:40.725
Very good observation.

02:02:40.725 --> 02:02:43.460
So Alex saw.

02:02:43.460 --> 02:02:45.650
He has a premonition.

02:02:45.650 --> 02:02:48.820
He can see two steps in advance.

02:02:48.820 --> 02:02:50.915
He said, OK, square that.

02:02:50.915 --> 02:02:52.710
You have e to the 2t.

02:02:52.710 --> 02:02:53.395
Square this.

02:02:53.395 --> 02:02:55.606
The minus doesn't matter.

02:02:55.606 --> 02:03:00.330
Plus e to the minus
2t, and square that.

02:03:00.330 --> 02:03:02.560
Then he saw patterns.

02:03:02.560 --> 02:03:06.130
Because he is the
wizard 101 today.

02:03:06.130 --> 02:03:09.090
So what is the
witchcraft he performed?

02:03:09.090 --> 02:03:10.470
Do you see?

02:03:10.470 --> 02:03:13.350
Does anybody else
see the pattern?

02:03:13.350 --> 02:03:15.360
[? Nateesh ?] sees the pattern.

02:03:15.360 --> 02:03:16.719
Anybody illuminated?

02:03:16.719 --> 02:03:18.010
I didn't see it from the start.

02:03:18.010 --> 02:03:19.660
You guys saw it faster than me.

02:03:19.660 --> 02:03:23.190
It took me about a
minute and a half

02:03:23.190 --> 02:03:26.710
when I saw this
for the first time.

02:03:26.710 --> 02:03:29.930
Is this a perfect square?

02:03:29.930 --> 02:03:32.060
Of who?

02:03:32.060 --> 02:03:36.350
e to the t plus e to
the minus 2 squared

02:03:36.350 --> 02:03:40.390
is-- anybody else sees the
pattern I don't have candy.

02:03:40.390 --> 02:03:44.210
Next time-- Alex,
[INAUDIBLE], anybody else?

02:03:44.210 --> 02:03:47.000
Do you now see the
pattern, e to the 2t plus

02:03:47.000 --> 02:03:51.340
e to the minus 2t plus
twice the product?

02:03:51.340 --> 02:03:54.470
And that's where the student
was having the problem.

02:03:54.470 --> 02:03:56.550
Where do you see the product?

02:03:56.550 --> 02:03:58.474
The product is 1.

02:03:58.474 --> 02:03:59.920
The product is 1 doubled.

02:03:59.920 --> 02:04:02.100
So you get 2.

02:04:02.100 --> 02:04:06.690
So it's indeed exactly
the perfect square.

02:04:06.690 --> 02:04:09.430
So once-- it was a she.

02:04:09.430 --> 02:04:14.490
Once she saw the perfect
square, she was so happy.

02:04:14.490 --> 02:04:16.850
Because you get square
root of the square.

02:04:16.850 --> 02:04:19.560
You get e to the t
plus e to the minus t.

02:04:19.560 --> 02:04:22.694
And that's a trivial thing
to integrate that you

02:04:22.694 --> 02:04:23.860
have no problem integrating.

02:04:23.860 --> 02:04:26.980
It's a positive
function, very beautiful.

02:04:26.980 --> 02:04:31.880
The professor who gave this was
Dr. [INAUDIBLE] from Denmark.

02:04:31.880 --> 02:04:34.730
He's one of the best
teachers we have.

02:04:34.730 --> 02:04:40.690
But he makes up his
homework as far as I know.

02:04:40.690 --> 02:04:43.200
I think in the sixth
edition, this edition,

02:04:43.200 --> 02:04:48.770
we actually stole his idea,
and we made a problem like that

02:04:48.770 --> 02:04:51.490
in the book somewhere.

02:04:51.490 --> 02:04:55.190
We doubled the number of
problems more or less.

02:04:55.190 --> 02:05:00.900
So if you are to compute
0 to 1 of the speed,

02:05:00.900 --> 02:05:03.069
what is the speed?

02:05:03.069 --> 02:05:05.534
The speed is this
beautiful thing.

02:05:05.534 --> 02:05:09.971
Because you were able
to see the pattern.

02:05:09.971 --> 02:05:12.764
If you're not able
to see that, do you

02:05:12.764 --> 02:05:15.440
realize it's
impossible, practically,

02:05:15.440 --> 02:05:17.940
for you to integrate by hand?

02:05:17.940 --> 02:05:22.700
You have to go to a
calculator, Matlab, whatever.

02:05:22.700 --> 02:05:23.830
So this is easy.

02:05:23.830 --> 02:05:29.170
Why is that easy? e to the t
minus e to the minus t at 1

02:05:29.170 --> 02:05:32.040
and at 0-- you compare them.

02:05:32.040 --> 02:05:36.420
You get at 1 e minus
e to the minus 1

02:05:36.420 --> 02:05:41.090
minus the fundamental theorem
of calc e to the 0 minus

02:05:41.090 --> 02:05:42.500
e to the 0.

02:05:42.500 --> 02:05:43.620
Well, that's silly.

02:05:43.620 --> 02:05:45.440
Why is that silly?

02:05:45.440 --> 02:05:49.170
Because I'm going to give it up.

02:05:49.170 --> 02:05:52.110
So the answer was
e to the minus 1/e.

02:05:52.110 --> 02:05:54.570
And she knew what
the answer would be.

02:05:54.570 --> 02:05:57.030
But she didn't know why.

02:05:57.030 --> 02:05:58.434
So she came back to me.

02:05:58.434 --> 02:06:02.610
I don't know how the tutoring
center helped her figure

02:06:02.610 --> 02:06:03.470
out the answer.

02:06:03.470 --> 02:06:06.200
But she did not
understand the solution.

02:06:06.200 --> 02:06:08.946
So I said, I'm not going to
take anymore people coming

02:06:08.946 --> 02:06:11.020
from Professor [INAUDIBLE].

02:06:11.020 --> 02:06:12.830
I was also told it's not OK.

02:06:12.830 --> 02:06:16.670
So don't go to another
professor with homework coming

02:06:16.670 --> 02:06:18.410
for me or the other way around.

02:06:18.410 --> 02:06:20.600
Because it's not OK.

02:06:20.600 --> 02:06:25.310
But you can go to the tutoring
center asking them for hints.

02:06:25.310 --> 02:06:30.216
They're open starting 9:00
AM and until around when?

02:06:30.216 --> 02:06:31.560
Do you know?

02:06:31.560 --> 02:06:32.990
They used to have until 4:00.

02:06:32.990 --> 02:06:35.870
But now they're going to
work on an extended schedule

02:06:35.870 --> 02:06:37.850
until 8:00 PM.

02:06:37.850 --> 02:06:40.325
It's going to be
something crazy.

02:06:40.325 --> 02:06:43.790
Now, the thing is, we want
the students to be better,

02:06:43.790 --> 02:06:48.620
to do better, to not give
up, to be successful,

02:06:48.620 --> 02:06:51.730
top one, two, three.

02:06:51.730 --> 02:06:54.290
I'm a little bit
concerned, but maybe I

02:06:54.290 --> 02:06:56.572
shouldn't be, about those hours.

02:06:56.572 --> 02:06:59.939
So I don't know if they managed
to put a security camera

02:06:59.939 --> 02:07:00.901
or not.

02:07:00.901 --> 02:07:04.520
But having extended
hours may be a problem.

02:07:04.520 --> 02:07:09.780
Take advantage of
those afternoon hours,

02:07:09.780 --> 02:07:11.726
especially if you are busy.

02:07:11.726 --> 02:07:18.698
Those late hours will
be a big help for you.

02:07:18.698 --> 02:07:21.266
Do you know where it is?

02:07:21.266 --> 02:07:23.656
Room 106 over there.

02:07:23.656 --> 02:07:26.530


02:07:26.530 --> 02:07:29.800
Any other questions related
to this type of problem

02:07:29.800 --> 02:07:35.240
or related to anything
else in the material

02:07:35.240 --> 02:07:38.870
that maybe I can
give you hints on,

02:07:38.870 --> 02:07:40.970
at least the hint I'm
going to give you?

02:07:40.970 --> 02:07:44.860
Sometimes I cannot stop, and
I just give the problem away.

02:07:44.860 --> 02:07:46.330
I'm not supposed to do that.

02:07:46.330 --> 02:07:50.750


02:07:50.750 --> 02:07:54.303
Look at your WeBWorK, see what
kind of help I can give you.

02:07:54.303 --> 02:07:56.428
You still have a
little bit of time.

02:07:56.428 --> 02:07:57.261
STUDENT: [INAUDIBLE]

02:07:57.261 --> 02:08:00.712


02:08:00.712 --> 02:08:05.140
MAGDALENA TODA: That's
the maximum of what?

02:08:05.140 --> 02:08:06.677
It was--

02:08:06.677 --> 02:08:07.510
STUDENT: [INAUDIBLE]

02:08:07.510 --> 02:08:11.110


02:08:11.110 --> 02:08:12.680
MAGDALENA TODA: Was
this the problem?

02:08:12.680 --> 02:08:14.600
STUDENT: e to the 2x
or something like that.

02:08:14.600 --> 02:08:15.560
MAGDALENA TODA:
Something like that?

02:08:15.560 --> 02:08:16.060
I erased it.

02:08:16.060 --> 02:08:19.400
STUDENT: You erased
that? [INAUDIBLE].

02:08:19.400 --> 02:08:21.330
I found an answer.

02:08:21.330 --> 02:08:23.395
MAGDALENA TODA: It's
very computational I saw.

02:08:23.395 --> 02:08:26.750
But before that, I
saw that seven of you

02:08:26.750 --> 02:08:28.990
guys-- you two also did it.

02:08:28.990 --> 02:08:33.714
So I wrote-- you have a
brownie waiting for that.

02:08:33.714 --> 02:08:35.163
But then I erased it.

02:08:35.163 --> 02:08:39.510
STUDENT: You erased the previous
one too in the homework one.

02:08:39.510 --> 02:08:42.040
MAGDALENA TODA: Because
that had a bug in it.

02:08:42.040 --> 02:08:45.400
That one, the one in the
homework one, had a bug in it.

02:08:45.400 --> 02:08:46.965
It only worked for some data.

02:08:46.965 --> 02:08:50.090
And for other data
it didn't work.

02:08:50.090 --> 02:08:53.580
So every time you find
a bug, you tell me,

02:08:53.580 --> 02:08:56.200
and I will tell the programmer
of those problems, who's

02:08:56.200 --> 02:08:57.010
really careful.

02:08:57.010 --> 02:09:02.423
But one in 1,000 you
are bound to find a bug.

02:09:02.423 --> 02:09:06.207
And I'm going to
give you a chocolate

02:09:06.207 --> 02:09:08.092
or something for every bug.

02:09:08.092 --> 02:09:09.820
And any other questions?

02:09:09.820 --> 02:09:14.695


02:09:14.695 --> 02:09:17.665
STUDENT: So are you
saying this is too long?

02:09:17.665 --> 02:09:20.140
MAGDALENA TODA: Actually,
it's very beautiful.

02:09:20.140 --> 02:09:23.605
If you have a calculator,
it's easier to solve it.

02:09:23.605 --> 02:09:25.585
You can do it by hand,
write it by hand, also.

02:09:25.585 --> 02:09:27.257
But it's a long--

02:09:27.257 --> 02:09:28.090
STUDENT: [INAUDIBLE]

02:09:28.090 --> 02:09:30.776


02:09:30.776 --> 02:09:34.262
MAGDALENA TODA: Right,
so let's do it now

02:09:34.262 --> 02:09:36.752
for anybody who wants to stay.

02:09:36.752 --> 02:09:37.748
You don't have to stay.

02:09:37.748 --> 02:09:39.740
So practicing what you do--

02:09:39.740 --> 02:09:44.974
[SIDE CONVERSATIONS]

02:09:44.974 --> 02:11:55.449