## TTU Math2450 Calculus3 Sec 11.2 and 11.3

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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
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review a little bit of what
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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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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
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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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Domain, range.
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After range, what did we do?
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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
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of a sum, taking a limit
of a product of functions,
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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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have to pull the truth out
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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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The class is a
construction, is working,
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is something like
a work in progress.
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We are building things together.
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This is teamwork.
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If I come up with
some slides that were
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made at home or a PDF file.
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First of all, it means I'm lazy.
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Second of all, it
means that I'm not
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willing to take it
one step at a time
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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.
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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,
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and you imagine
x, y going to x0,
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y0 by any possible paths
in any possible way,
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it's not that you have a
predetermined path to x0, y0,
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because you may be trapped.
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You may have-- as you've
seen last time, you may have,
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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,
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would be another one.
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And then you don't
have overall limits.
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So the limit-- when I call that,
that means the overall limit
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exists, exists and
equals L. It's finite.
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That's what I mean.
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And three, the value
of the function at x0,
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y0 must be equal to the limit
of the function that value
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as you approach it, x0, y0.
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And equals L, of course.
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So great.
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So it's so obvious
that we are following
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exactly the same
type of definition,
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the same type of pattern.
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to help me, to help
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me solve a harder problem
that involves continuity.
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And I'm asking you, if I
have the following function--
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I'm going to erase the
definition of continuity
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from Calc 1.
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I'm going to ask you, what if
I have this funny function?
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You've seen it before, and
I gave you a little bit
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Limit as x, y goes
to 0, 0 of x squared
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plus y squared times sine of 1
over x squared plus y squared.
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Does that exist?
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And also--
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STUDENT: It's actually--
so the limit is actually
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approaching a plane rather
than a set of [INAUDIBLE].
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MAGDALENA TODA: So
well, actually, it's
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not approaching a plane.
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Let's see what's
happening when--
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STUDENT: Sorry, sorry.
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Not a plane, a [? line. ?]
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MAGDALENA TODA: Yes.
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STUDENT: And is the z-axis--
the entire z-axis is 0, 0?
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MAGDALENA TODA: So
this is the z-axis.
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And that means exactly that
x and y-- it will be 0.
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Now I am just looking
at what happens
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in the plane, in the
floor plane x, y.
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The pairs x, y are wiggly.
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They are like
little wormy worms.
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And they float on the
water on the floor.
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And these squiggly
things approach
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x, y from any possible path.
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They go like this.
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They go like that.
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They go in every possible way.
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Let's see what happens.
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Continuity-- is this continuous?
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Well, you say,
Magdalena, come on.
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You cannot have this
continuous at 0, 0,
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because it's undefined at 0, 0.
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Yes.
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But maybe I can extend
it by continuity.
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So let me introduce-- this
is my favorite, f of x, y.
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But I'll say, f of x, y
is not defined at 0, 0.
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But how about g of x, y as
being my f of x, y for any x,
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y different from 0, 0.
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And at the origin, at the
very origin, I will say,
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I want to have--
when x, y equals 0,
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0, I want to have a value.
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Which value do you
think might extend
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this function by continuity?
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STUDENT: The limit.
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MAGDALENA TODA: The
limit if it exists
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and if-- well, you know already,
I think, what the limit is
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because some of you
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for extra credit.
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So it's not fair, right?
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No, I'm just kidding.
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So I claim that maybe--
if I put a 0 here,
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will this be continuous?
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Will g be continuous?
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So prove, prove either way,
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by a proof, a complete
proof with epsilon delta.
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Proof.
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OK.
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OK.
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So now is a worried face.
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Like, oh, my god.
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This guy is worried
because, oh, my god.
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Epsilon delta.
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Oh, my god.
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But the principle--
the intuition
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tells us that we should look
first at some sort of a graph,
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just like Ryan pointed out.
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One should close their eyes and
imagine a graph of a function
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with-- it's hard to visualize in
3D the graph of a function that
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is a surface.
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This is a surface. z
equals the whole shebang.
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But when I'm going to look
at the one dimensional case
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from last time, we
remember the sine of 1/x
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was a crazy function.
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We called it the harmonica,
well, 20-something years ago
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when I was in high school.
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calculus class.
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And our teacher was
not funny at all.
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He was also not teaching much,
gave us a lot of homework,
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very challenging.
• 14:51 - 14:54
So in order to make our
life a little bit easier,
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we always worked in
groups, which was allowed.
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So we called it a harmonica
because it was oscillating
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like that to the point
that-- you've seen
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the harmonica-- the accordion.
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When you bring it back to
the-- harmonica came to my mind
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from the harmonic function.
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So the accordion is--
when you actually
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squeeze it, all that oscillation
things, the cusps are
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closer and closer to a line.
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So what you have here is
this kind of oscillation,
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very, very rapid
oscillation for sine of 1/x.
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When we want to multiply by
an x, what's going to happen?
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Well, this has not limit at 0
because it takes all the values
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infinitesimally close to 0.
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It keeps going through all the
values between minus 1 and 1,
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closer and closer.
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So that was no good.
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But if we take this guy--
that's going to go to-- well,
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I cannot do better.
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MATLAB can do better than me.
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Mathematica can do better.
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You can do that.
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In most engineering
classes, if you are--
• 16:13 - 16:16
who is an electrical
engineering major?
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But even if you are
not, you are going
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to see this type
of function a lot.
• 16:21 - 16:25
And you're going to see it
again in differential equations.
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How can I imagine-- this
graph is hard to draw.
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Don't ask me to draw that.
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But ask me if I can use epsilon
delta to prove continuity.
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So what would it mean,
proving continuity?
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I have a feeling--
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STUDENT: Well, actually, if this
is-- going back to that graph,
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doesn't that graph look like--
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MAGDALENA TODA: This goes to 0.
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The limit exists for x
sine of 1/x, and it is 0.
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Why?
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Ryan?
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RYAN: Wouldn't the graph
with the x squared plus
• 17:01 - 17:03
y squared times that
side-- wouldn't that
• 17:03 - 17:06
just look like a ripple
in a circle going out
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from the center?
• 17:07 - 17:09
MAGDALENA TODA: Yeah,
it will be ripples.
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STUDENT: Just like a
[INAUDIBLE] from an epicenter
• 17:11 - 17:12
going outwards [INAUDIBLE].
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MAGDALENA TODA: And I think--
yes, we managed to-- you
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have a concentric image, right?
• 17:20 - 17:20
STUDENT: Yeah.
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MAGDALENA TODA: Like those
ripples, exactly like--
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STUDENT: So that's
what that looks like?
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MAGDALENA TODA: --when you
throw a stone into the water,
• 17:27 - 17:28
this kind of wave.
• 17:28 - 17:31
But it's infinitesimally close.
• 17:31 - 17:33
It's like acting weird.
• 17:33 - 17:37
But then it sort
of shrinks here.
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And that-- it
imposes the limit 0.
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How come this goes
to 0, you say?
• 17:43 - 17:46
Well, Magdalena, this
guy is crazy, right?
• 17:46 - 17:49
Sine of 1/x goes
between minus 1 and 1
• 17:49 - 17:52
infinitely many times
as I go close, close,
• 17:52 - 17:58
closer and closer, more rapidly,
more and more rapidly close
• 17:58 - 17:59
to 0.
• 17:59 - 18:01
This will oscillate
more rapidly,
• 18:01 - 18:03
more rapidly, and more rapidly.
• 18:03 - 18:05
This is crazy, right?
• 18:05 - 18:08
How does this guy, x-- how
is this guy taming this guy?
• 18:08 - 18:10
STUDENT: Because
as 0 [INAUDIBLE].
• 18:10 - 18:13
Something really small
times something [INAUDIBLE].
• 18:13 - 18:14
MAGDALENA TODA:
Something very small
• 18:14 - 18:18
that shrinks to 0 times
something bounded.
• 18:18 - 18:21
Ryan brought the main idea.
• 18:21 - 18:25
If something goes strongly to
0, and that multiplies something
• 18:25 - 18:28
that's bounded, bounded
by a finite number,
• 18:28 - 18:31
the whole problem will go to 0.
• 18:31 - 18:33
Actually, you can prove
that as a theorem.
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And some of you did.
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In most honors
classes unfortunately,
• 18:37 - 18:39
epsilon delta was not covered.
• 18:39 - 18:43
So let's see how we prove
this with epsilon delta.
• 18:43 - 18:45
And, oh, my god.
• 18:45 - 18:53
Many of you read from the book
and may be able to help me.
• 18:53 - 19:00
So what am I supposed to
show with epsilon delta?
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The limit of x squared plus
y squared sine of 1 over x
• 19:10 - 19:15
squared plus y squared is
0 as I approach the origin
• 19:15 - 19:20
with my pair, couple, x, y,
which can go any one path that
• 19:20 - 19:20
approaches 0.
• 19:20 - 19:24
• 19:24 - 19:28
So you say, oh, well, Magdalena,
the Ryan principle-- this
• 19:28 - 19:29
is the Ryan theorem.
• 19:29 - 19:32
It's the same because
this guy will be
• 19:32 - 19:34
bounded between minus 1 and 1.
• 19:34 - 19:38
I multiplied with a guy
that very determinedly goes
• 19:38 - 19:40
to 0 very strongly.
• 19:40 - 19:41
And he knows where he's going.
• 19:41 - 19:44
x squared plus y squared
says, I know what I'm doing.
• 19:44 - 19:46
I'm not going to change my mind.
• 19:46 - 19:49
This is like the guy who changes
his major too many times.
• 19:49 - 19:52
And this guy knows
what he's doing.
• 19:52 - 19:55
He's going there, and he's
a polynomial, goes to 0,
• 19:55 - 19:56
0 very rapidly.
• 19:56 - 20:01
Now it's clear what
happens intuitively.
• 20:01 - 20:03
But I'm a mathematician.
• 20:03 - 20:07
And if I don't publish
my proof, my article
• 20:07 - 20:13
will be very nicely rejected
by all the serious journals
• 20:13 - 20:14
on the market.
• 20:14 - 20:17
This is how it goes
in mathematics.
• 20:17 - 20:19
Even before journals
existed, mathematicians
• 20:19 - 20:23
proof of their work,
• 20:23 - 20:26
of their conjecture.
• 20:26 - 20:27
OK.
• 20:27 - 20:35
So I go, for every epsilon
positive, no matter how small,
• 20:35 - 20:41
there must exist a
delta positive, which
• 20:41 - 20:52
depends on epsilon-- that
depends on epsilon-- such that
• 20:52 - 20:59
as soon as-- how did
we write the distance?
• 20:59 - 21:02
I'll write the distance
again because I'm lazy.
• 21:02 - 21:06
The distance between the
point x, y and the origin
• 21:06 - 21:08
is less than delta.
• 21:08 - 21:17
It follows that the
absolute value--
• 21:17 - 21:24
these are all real numbers--
of f of x, y or g of x,
• 21:24 - 21:27
y-- g of x, y is the extension.
• 21:27 - 21:32
• 21:32 - 21:36
f of x, y minus 0, which
I claim to be the limit,
• 21:36 - 21:39
will be less than epsilon.
• 21:39 - 21:40
So you go, oh, my god.
• 21:40 - 21:43
What is this woman doing?
• 21:43 - 21:44
It's not hard.
• 21:44 - 21:46
• 21:46 - 21:49
I need your help to do that.
• 21:49 - 21:53
So it's hard to see how you
should-- you take any epsilon.
• 21:53 - 21:58
epsilon, infinitesimally small,
• 21:58 - 22:01
any small number, but
then you go, but then I
• 22:01 - 22:03
have to show this delta exists.
• 22:03 - 22:07
You have to grab that delta
and say, you are my delta.
• 22:07 - 22:09
You cannot escape me.
• 22:09 - 22:11
I tell you who you are.
• 22:11 - 22:14
And that's the
hardest part in here,
• 22:14 - 22:18
figuring out who that delta must
be as a function of epsilon.
• 22:18 - 22:19
Is that hard?
• 22:19 - 22:21
How do you build
such a construction?
• 22:21 - 22:27
First of all,
understand what proof.
• 22:27 - 22:30
"Choose any positive epsilon."
• 22:30 - 22:33
• 22:33 - 22:36
and he's going to do whatever
you want to do with him.
• 22:36 - 22:40
Delta, chasing
after delta is going
• 22:40 - 22:42
to be a little bit harder.
• 22:42 - 22:56
"Chasing after delta
with that property."
• 22:56 - 22:58
Dot, dot, dot, dot, dot.
• 22:58 - 23:00
What is this distance?
• 23:00 - 23:02
You guys have
helped me last time,
• 23:02 - 23:05
you cannot let me down now.
• 23:05 - 23:08
So as soon as this distance,
• 23:08 - 23:11
is less than delta,
you must have
• 23:11 - 23:13
that f of x, y [INAUDIBLE].
• 23:13 - 23:15
Could you tell me
what that would be?
• 23:15 - 23:16
It was Euclidean, right?
• 23:16 - 23:22
root of-- did I?
• 23:22 - 23:30
Square root of x minus 0
squared plus y minus 0 squared.
• 23:30 - 23:33
You say, but that's
silly, Magdalena.
• 23:33 - 23:38
So you have to write
it down like that?
• 23:38 - 23:39
STUDENT: It's the [INAUDIBLE].
• 23:39 - 23:40
MAGDALENA TODA: Huh?
• 23:40 - 23:42
Yeah.
• 23:42 - 23:47
So square root of this
plus square root of that
• 23:47 - 23:53
plus then delta,
that means what?
• 23:53 - 24:00
If and only if x squared plus
y squared is less than delta
• 24:00 - 24:01
squared.
• 24:01 - 24:08
• 24:08 - 24:11
And what do I want to do,
what do I want to build?
• 24:11 - 24:15
• 24:15 - 24:19
So we are thinking how
to set up all this thing.
• 24:19 - 24:21
How to choose the delta.
• 24:21 - 24:23
How to choose the delta.
• 24:23 - 24:26
• 24:26 - 24:28
OK, so what do I--
what am I after?
• 24:28 - 24:34
"I am after having" double dot.
• 24:34 - 24:40
F of x, y must be Mr. Ugly.
• 24:40 - 24:41
This one.
• 24:41 - 24:46
So absolute value of x squared
plus y squared, sine of 1
• 24:46 - 24:51
over x squared plus
y squared minus 0.
• 24:51 - 24:52
Duh.
• 24:52 - 24:55
I'm not going to write it.
• 24:55 - 24:59
We all know what that means.
• 24:59 - 25:00
Less than epsilon.
• 25:00 - 25:06
This is what must
• 25:06 - 25:12
This is what must
follow, must happen.
• 25:12 - 25:13
Must happen.
• 25:13 - 25:16
• 25:16 - 25:18
Now I'm getting excited.
• 25:18 - 25:18
Why?
• 25:18 - 25:21
Because I am thinking.
• 25:21 - 25:23
I started thinking.
• 25:23 - 25:26
Once I started thinking,
I'm dangerous, man.
• 25:26 - 25:32
So here sine of 1 over x squared
plus y squared is your friend.
• 25:32 - 25:34
• 25:34 - 25:37
Sine of 1 over x squared
plus y squared, this
• 25:37 - 25:39
is always an absolute value.
• 25:39 - 25:43
The absolute value of that
is always less than 1.
• 25:43 - 25:43
OK?
• 25:43 - 25:45
STUDENT: Can't it be 4?
• 25:45 - 25:50
MAGDALENA TODA:
So-- so-- so what
• 25:50 - 25:55
shall I take in terms of
delta-- this is my question.
• 25:55 - 25:57
What shall I take
in terms of delta?
• 25:57 - 26:04
"Delta equals 1 as a
function of epsilon
• 26:04 - 26:20
in order to have the
conclusion satisfied."
• 26:20 - 26:21
You say, OK.
• 26:21 - 26:25
It's enough to choose delta
like that function of epsilon,
• 26:25 - 26:29
and I'm done, because then
everything will be fine.
• 26:29 - 26:34
So you chose your own epsilon,
positive, small, or God
• 26:34 - 26:34
gave you an epsilon.
• 26:34 - 26:37
You don't care how
you got the epsilon.
• 26:37 - 26:38
The epsilon is arbitrary.
• 26:38 - 26:41
You pick positive and small.
• 26:41 - 26:45
Now, it's up to
you to find delta.
• 26:45 - 26:49
So what delta would
satisfy everything?
• 26:49 - 26:51
What delta would
be good enough--
• 26:51 - 26:53
you don't care
for all the good--
• 26:53 - 26:55
it's like when you get married.
• 26:55 - 26:58
Do you care for all the
people who'd match you?
• 26:58 - 27:01
Hopefully not, because
then you would probably
• 27:01 - 27:05
have too large of a pool,
and it's hard to choose.
• 27:05 - 27:13
You only need one that satisfies
that assumption, that satisfies
• 27:13 - 27:15
all the conditions you have.
• 27:15 - 27:19
So what is the delta that
satisfies all the conditions
• 27:19 - 27:20
that I have?
• 27:20 - 27:21
[INTERPOSING VOICES]
• 27:21 - 27:22
MAGDALENA TODA: [INAUDIBLE].
• 27:22 - 27:23
Who?
• 27:23 - 27:25
[INTERPOSING VOICES]
• 27:25 - 27:28
MAGDALENA TODA: For example,
delta equals epsilon.
• 27:28 - 27:29
Would that satisfy?
• 27:29 - 27:32
• 27:32 - 27:34
Well, let's see.
• 27:34 - 27:37
If I take delta to
be epsilon, then x
• 27:37 - 27:40
squared plus y squared would
be less than epsilon squared.
• 27:40 - 27:47
Now the question is is epsilon
squared less than epsilon?
• 27:47 - 27:49
Not always.
• 27:49 - 27:53
If epsilon is between 0
and 1, then epsilon squared
• 27:53 - 27:54
is less then epsilon.
• 27:54 - 27:59
But if I choose epsilon
to be greater than 1,
• 27:59 - 28:00
then oh, my God.
• 28:00 - 28:03
Then if it's greater than
1, then epsilon squared
• 28:03 - 28:07
is greater than 1--
greater than it.
• 28:07 - 28:15
So what if I choose
delta to be what?
• 28:15 - 28:19
• 28:19 - 28:20
STUDENT: 0?
• 28:20 - 28:21
MAGDALENA TODA: No, no, no.
• 28:21 - 28:22
Delta cannot be 0.
• 28:22 - 28:26
So delta-- look, there exists
delta strictly bigger than 0,
• 28:26 - 28:29
that depends on epsilon.
• 28:29 - 28:34
Maybe if epsilon is very small,
in a way Alexander was right.
• 28:34 - 28:37
But the delta [INAUDIBLE],
we don't go with epsilon
• 28:37 - 28:38
greater than 1.
• 28:38 - 28:39
Come on.
• 28:39 - 28:40
Be serious.
• 28:40 - 28:42
Epsilon is always
between 0 and 1.
• 28:42 - 28:45
I mean, it's a lot
smaller than that.
• 28:45 - 28:47
It's infinitesimal small.
• 28:47 - 28:49
So in the end, yes, in
that case epsilon squared
• 28:49 - 28:53
would be less than epsilon,
which would be OK for us
• 28:53 - 28:55
and that would be fine.
• 28:55 - 28:56
OK?
• 28:56 - 28:58
So that would be a
possibility to say, hey,
• 28:58 - 29:01
since epsilon-- Alexander,
if you write that as a proof
• 29:01 - 29:02
I'll be OK.
• 29:02 - 29:05
You say, I took my epsilon
to be a very small number,
• 29:05 - 29:07
so anyway it's going
to be less than 1.
• 29:07 - 29:09
So epsilon squared
is less than epsilon.
• 29:09 - 29:14
So when I take
delta to be epsilon,
• 29:14 - 29:18
for sure this guy will be less
than epsilon squared, which
• 29:18 - 29:21
is less than epsilon,
so I'm satisfied.
• 29:21 - 29:23
I'll give you a 100%.
• 29:23 - 29:24
I'm happy.
• 29:24 - 29:25
Is that the only way?
• 29:25 - 29:27
STUDENT: But what
• 29:27 - 29:28
• 29:28 - 29:28
STUDENT: Yeah.
• 29:28 - 29:30
MAGDALENA TODA: So
this doesn't matter.
• 29:30 - 29:32
Let me write it down.
• 29:32 - 29:40
So note that x squared
plus y squared sine of 1
• 29:40 - 29:43
over x squared plus
y square would always
• 29:43 - 29:46
be less than absolute
value of x squared
• 29:46 - 29:50
plus y, which is positive.
• 29:50 - 29:52
Why is that?
• 29:52 - 29:53
Is this true?
• 29:53 - 29:54
Yeah.
• 29:54 - 29:55
Why is that?
• 29:55 - 29:58
STUDENT: Because the sine can
only be one of these negatives.
• 29:58 - 30:00
MAGDALENA TODA: So
in absolute value,
• 30:00 - 30:06
sine of 1 over x squared plus y
squared is always less than 1.
• 30:06 - 30:09
STUDENT: Can't it equal 1?
• 30:09 - 30:12
MAGDALENA TODA: Well,
when does it equal 1?
• 30:12 - 30:14
STUDENT: Wouldn't it be x
squared plus y squared equals 1
• 30:14 - 30:16
[INAUDIBLE]?
• 30:16 - 30:17
MAGDALENA TODA: Less
than or equal to.
• 30:17 - 30:18
For some values it will.
• 30:18 - 30:19
STUDENT: Yeah.
• 30:19 - 30:20
OK.
• 30:20 - 30:22
MAGDALENA TODA: Now, will
that be a problem with us?
• 30:22 - 30:22
No.
• 30:22 - 30:23
Let's put it here.
• 30:23 - 30:27
Less than or equal to x
squared plus y squared, which
• 30:27 - 30:35
has to be less than epsilon
if and only if-- well,
• 30:35 - 30:39
if delta is what?
• 30:39 - 30:41
So, again, Alexander said,
well, but if I take delta
• 30:41 - 30:43
to be epsilon, I'm done.
• 30:43 - 30:46
• 30:46 - 30:47
STUDENT: [INAUDIBLE].
• 30:47 - 30:50
MAGDALENA TODA: How
• 30:50 - 30:52
Can I take delta to be
square root of epsilon.
• 30:52 - 30:54
STUDENT: That's what I said.
• 30:54 - 30:54
MAGDALENA TODA: No.
• 30:54 - 30:56
You said epsilon.
• 30:56 - 30:57
STUDENT: I said square
root of epsilon.
• 30:57 - 30:58
MAGDALENA TODA: OK.
• 30:58 - 31:01
If delta is square
root of epsilon,
• 31:01 - 31:05
then everything will be perfect
and it will be a perfect match.
• 31:05 - 31:06
In what case?
• 31:06 - 31:08
STUDENT: If epsilon
is in between 0 and 1
• 31:08 - 31:10
and if delta is equal
to bigger than epsilon.
• 31:10 - 31:13
• 31:13 - 31:18
MAGDALENA TODA: So that's
exactly the same assumption.
• 31:18 - 31:22
Epsilon should be
• 31:22 - 31:24
STUDENT: But I thought
delta was supposed
• 31:24 - 31:26
to be less than
epsilon in every case.
• 31:26 - 31:29
So if epsilon is between 0 and
1, the square root of epsilon
• 31:29 - 31:32
is going to be [INAUDIBLE].
• 31:32 - 31:38
MAGDALENA TODA: So when
both of them are small,
• 31:38 - 31:45
delta squared will be-- if
I take delta-- so take delta
• 31:45 - 31:48
to be square root of epsilon.
• 31:48 - 31:50
STUDENT: Then anything less
than 1 and greater than 0,
• 31:50 - 31:52
epsilon would be great
than [INAUDIBLE].
• 31:52 - 31:55
MAGDALENA TODA: "Delta to
be square root of epsilon,
• 31:55 - 32:02
then x squared plus y squared
less than delta squared equals
• 32:02 - 32:04
epsilon."
• 32:04 - 32:12
Then x squared plus
y squared sine of 1
• 32:12 - 32:15
over x squared plus
y squared less than
• 32:15 - 32:17
or equal to x squared
plus y squared.
• 32:17 - 32:19
I dont' need the absolute value.
• 32:19 - 32:20
I can [INAUDIBLE].
• 32:20 - 32:23
Less than epsilon [INAUDIBLE].
• 32:23 - 32:24
Qed.
• 32:24 - 32:26
STUDENT: Well, but
you told us delta
• 32:26 - 32:28
has to be less than epsilon.
• 32:28 - 32:28
Well, if--
• 32:28 - 32:31
MAGDALENA TODA: No,
I didn't say that.
• 32:31 - 32:35
I didn't say that delta has
to be less than epsilon.
• 32:35 - 32:36
Absolutely--
• 32:36 - 32:36
STUDENT: Yeah.
• 32:36 - 32:39
You said for all the values
of epsilon greater than 0,
• 32:39 - 32:42
there's a value of delta that is
greater than 0 that [INAUDIBLE]
• 32:42 - 32:46
such that as soon as the
distance between is less than
• 32:46 - 32:47
delta-- I don't remember what--
• 32:47 - 32:48
MAGDALENA TODA: OK, so, again--
• 32:48 - 32:50
STUDENT: Such that the
distance is less than--
• 32:50 - 32:52
MAGDALENA TODA: So again,
for epsilon positive,
• 32:52 - 32:57
there is a delta
positive, very small.
• 32:57 - 32:59
Very small means very small, OK?
• 32:59 - 33:01
I'm not threatened by-- what?
• 33:01 - 33:05
For epsilon greater
than 0, very small,
• 33:05 - 33:07
there is a delta greater
than 0, very small,
• 33:07 - 33:11
which depends on epsilon-- I
didn't say it cannot be equal
• 33:11 - 33:21
to epsilon-- that depends on
epsilon such that whenever x,
• 33:21 - 33:30
y is within delta
distance from origin,
• 33:30 - 33:45
[INAUDIBLE] that f of x, y
is within epsilon of from l.
• 33:45 - 33:48
• 33:48 - 33:48
All right?
• 33:48 - 33:53
And now I will actually give
you another example where
• 33:53 - 33:56
maybe delta will be epsilon.
• 33:56 - 33:59
And let me challenge you
with another problem that's
• 33:59 - 34:01
not hard.
• 34:01 - 34:01
OK?
• 34:01 - 34:04
So let me give
you the function g
• 34:04 - 34:16
of x, y equals x sine
of 1 over y as x, y.
• 34:16 - 34:19
• 34:19 - 34:30
y is equal [? to delta 0. ?]
And let's say 0 for the rest.
• 34:30 - 34:36
• 34:36 - 34:49
Can you show-- can you check
if g is continuous at 0, 0?
• 34:49 - 34:56
• 34:56 - 34:59
This is one of the
• 34:59 - 35:02
So how do you check
that with epsilon delta?
• 35:02 - 35:04
Again, we recite the poetry.
• 35:04 - 35:06
We have to say that.
• 35:06 - 35:12
"For every epsilon
positive, small, very small,
• 35:12 - 35:16
there is a delta
positive that depends
• 35:16 - 35:34
on epsilon, such that as soon
as--" how is the distance?
• 35:34 - 35:42
Square root of x squared plus
y squared is less than delta.
• 35:42 - 35:47
This is the distance
between point and origin.
• 35:47 - 36:09
"It follows that absolute value
of x sine of 1 over y minus--"
• 36:09 - 36:12
so practically x, y no 0.
• 36:12 - 36:16
x, y different from 0.
• 36:16 - 36:18
OK?
• 36:18 - 36:22
I"m careful here, because
if y is 0, then I blow up.
• 36:22 - 36:23
And I don't want to blow up.
• 36:23 - 36:26
So x sine of 1 over y minus who?
• 36:26 - 36:31
Minus 0 is less than epsilon.
• 36:31 - 36:33
So now you're thinking,
OK, you want me
• 36:33 - 36:35
to prove there is such a delta?
• 36:35 - 36:36
Yes.
• 36:36 - 36:37
That depends on epsilon?
• 36:37 - 36:39
Yes.
• 36:39 - 36:40
And what would that delta be?
• 36:40 - 36:44
The simplest choice you
can have in this case.
• 36:44 - 36:45
So you go, oh, my God.
• 36:45 - 36:46
How do I do that?
• 36:46 - 36:48
You have to always
think backwards.
• 36:48 - 36:59
So "we need to satisfy
absolute value of x sine of 1
• 36:59 - 37:02
over y less than epsilon."
• 37:02 - 37:06
Is this hard?
• 37:06 - 37:10
• 37:10 - 37:14
• 37:14 - 37:20
Remark absolute value
of x sine of 1 over y
• 37:20 - 37:23
is smaller than who?
• 37:23 - 37:27
Smaller than the product
of absolute values.
• 37:27 - 37:28
Say it again?
• 37:28 - 37:28
Yes?
• 37:28 - 37:32
STUDENT: But, like, for
example, the only condition
• 37:32 - 37:35
for that equation is that
y must not be equal to 0.
• 37:35 - 37:39
What if you used
another point for x?
• 37:39 - 37:43
delta be different?
• 37:43 - 37:45
MAGDALENA TODA:
Well, x is-- you can
• 37:45 - 37:49
choose-- you were right here.
• 37:49 - 37:53
You can say, OK, can you be
more restrictive, Magdelena,
• 37:53 - 37:59
and say, for every point
of the type x equals 0
• 37:59 - 38:01
and y not 0, it's still OK?
• 38:01 - 38:03
Yes.
• 38:03 - 38:07
So you could be a
professional mathematician.
• 38:07 - 38:14
So practically all I care
about is x, y in the disk.
• 38:14 - 38:15
What disk?
• 38:15 - 38:17
What is this disk?
• 38:17 - 38:24
• 38:24 - 38:32
Delta-- such that your
y should not be 0.
• 38:32 - 38:36
So a more rigorous
point would be
• 38:36 - 38:39
like take all the
couples that are
• 38:39 - 38:44
in this small disk
• 38:44 - 38:46
except for those where y is 0.
• 38:46 - 38:49
So what do you actually remove?
• 38:49 - 38:55
You remove this stinking line.
• 38:55 - 39:01
But everybody else in this
disk, every couple in this disk
• 39:01 - 39:04
should be happy,
should be analyzed
• 39:04 - 39:06
• 39:06 - 39:08
Right?
• 39:08 - 39:09
OK.
• 39:09 - 39:13
x sine of 1 over y less
than-- is that true?
• 39:13 - 39:16
Is that less than the
absolute value of x?
• 39:16 - 39:17
STUDENT: Yeah.
• 39:17 - 39:18
MAGDALENA TODA: Right.
• 39:18 - 39:21
So it should be-- less
• 39:21 - 39:24
should be less than epsilon.
• 39:24 - 39:27
When is this happening
on that occasion?
• 39:27 - 39:28
If I take delta-- meh?
• 39:28 - 39:30
STUDENT: When delta's epsilon.
• 39:30 - 39:31
MAGDALENA TODA: So if
I take-- very good.
• 39:31 - 39:36
So Alex saw that, hey,
• 39:36 - 39:38
And I mean it's over.
• 39:38 - 39:43
Take delta, which is delta
of epsilon, to be epsilon.
• 39:43 - 39:44
You're done.
• 39:44 - 39:46
Why?
• 39:46 - 39:48
Let me explain what
Alex wants, because he
• 39:48 - 39:50
doesn't want to explain
much, but it's not his job.
• 39:50 - 39:51
• 39:51 - 39:52
Right?
• 39:52 - 39:54
So why is this working?
• 39:54 - 40:03
Because in this case,
note that if I take delta
• 40:03 - 40:06
to be exactly epsilon,
what's going to happen?
• 40:06 - 40:09
• 40:09 - 40:14
x, Mr. x, could be
positive or negative.
• 40:14 - 40:16
See, x could be
positive or negative.
• 40:16 - 40:19
Let's take this guy and
protect him in absolute value.
• 40:19 - 40:23
He's always less than square
root of x square plus y
• 40:23 - 40:26
squared.
• 40:26 - 40:27
Why is that, guys?
• 40:27 - 40:31
STUDENT: Because y can't be 0.
• 40:31 - 40:34
MAGDALENA TODA: So this
is-- square it in your mind.
• 40:34 - 40:37
You got x squared less than
x squared plus y squared.
• 40:37 - 40:39
So this is always true.
• 40:39 - 40:41
Always satisfied.
• 40:41 - 40:45
But we chose this to
be less than delta,
• 40:45 - 40:49
and if we choose delta to be
epsilon, that's our choice.
• 40:49 - 40:54
So God gave us the epsilon,
but delta is our choice,
• 40:54 - 40:57
because you have to prove
you can do something
• 40:57 - 40:58
• 40:58 - 40:58
Right?
• 40:58 - 41:01
So delta equals epsilon.
• 41:01 - 41:03
If you take delta
equals epsilon,
• 41:03 - 41:06
then you're done, because
in that case absolute value
• 41:06 - 41:12
of x is less than epsilon, and
• 41:12 - 41:14
was satisfied.
• 41:14 - 41:17
Now, if a student
is really smart--
• 41:17 - 41:21
one time I had a student,
I gave him this proof.
• 41:21 - 41:22
That was several
years ago in honors,
• 41:22 - 41:25
because we don't do epsilon
delta in non-honors.
• 41:25 - 41:28
And we very rarely do
it in honors as well.
• 41:28 - 41:31
His proof consisted of this.
• 41:31 - 41:34
Considering the fact that
absolute value of sine
• 41:34 - 41:38
is less than 1, if I
take delta to be epsilon,
• 41:38 - 41:40
that is sufficient.
• 41:40 - 41:42
I'm done.
• 41:42 - 41:44
And of course I gave
him 100%, because this
• 41:44 - 41:46
is the essence of the proof.
• 41:46 - 41:48
He didn't show any details.
• 41:48 - 41:52
And I thought, this is the
kind of guy who is great.
• 41:52 - 41:56
He's very smart, but he's not
going to make a good teacher.
• 41:56 - 41:59
So he's probably going to
be the next researcher,
• 41:59 - 42:05
the next astronaut, the next
something else, but not--
• 42:05 - 42:11
And then, years later, he
• 42:11 - 42:14
• 42:14 - 42:18
by the Air Force.
• 42:18 - 42:21
And he works right
now for the Air Force.
• 42:21 - 42:24
He came out dressed
as a captain.
• 42:24 - 42:29
He came and gave a talk this
year at Tech in a conference--
• 42:29 - 42:30
he was rushed.
• 42:30 - 42:32
I mean, if I talk
like that, my student
• 42:32 - 42:34
wouldn't be able to follow me.
• 42:34 - 42:38
But he was the same brilliant
student that I remember.
• 42:38 - 42:46
So he's working on some very
important top secret projects.
• 42:46 - 42:49
Very intelligent guy.
• 42:49 - 42:53
And every now and than going
to give talks at conferences.
• 42:53 - 42:58
Like, research talks
• 42:58 - 43:02
In his class-- he took
• 43:02 - 43:04
level [INAUDIBLE]--
• 43:04 - 43:09
I explained epsilon delta, and
he had it very well understood.
• 43:09 - 43:13
And after I left the classroom
he explained it to his peers,
• 43:13 - 43:15
to his classmates.
• 43:15 - 43:17
And he explained
it better than me.
• 43:17 - 43:21
And I was there listening,
and I remember being jealous,
• 43:21 - 43:23
because although
he was very rushed,
• 43:23 - 43:27
understanding of how
• 43:27 - 43:31
you take an epsilon, no
matter how small, and then
• 43:31 - 43:34
you take a little ball
• 43:34 - 43:39
So the image of that little
ball will fit in that ball
• 43:39 - 43:40
that you take here.
• 43:40 - 43:44
So even if you
shrink on the image,
• 43:44 - 43:46
you can take this
ball even smaller
• 43:46 - 43:49
so the image will
still fit inside.
• 43:49 - 43:51
And I was going, gosh,
this is the essence,
• 43:51 - 43:55
but I wish I could convey
it, because no book
• 43:55 - 43:59
will say it just-- or show you
how to do it with your hands.
• 43:59 - 43:59
• 43:59 - 44:00
STUDENT: [INAUDIBLE]
• 44:00 - 44:01
MAGDALENA TODA: Right.
• 44:01 - 44:05
So he was rushed, but he
• 44:05 - 44:07
of what is going on.
• 44:07 - 44:08
OK.
• 44:08 - 44:11
11.3 is a completely new start.
• 44:11 - 44:14
• 44:14 - 44:16
because that's
partial derivatives.
• 44:16 - 44:20
And you say, Magdalena,
finally, this is piece of cake.
• 44:20 - 44:23
You see, I know these things.
• 44:23 - 44:26
I can do them in
my-- in my sleep.
• 44:26 - 44:30
So f of x and y
is still a graph.
• 44:30 - 44:33
And then you say,
how do we introduce
• 44:33 - 44:38
the partial derivative with
respect to one variable only.
• 44:38 - 44:40
You think, I draw the graph.
• 44:40 - 44:42
OK.
• 44:42 - 44:45
On this graph, I
pick a point x0, y0.
• 44:45 - 44:54
And if I were to take x to
be 0, what is-- what is the z
• 44:54 - 44:56
equals f of x0, y?
• 44:56 - 45:03
• 45:03 - 45:04
So I'll try to draw it.
• 45:04 - 45:05
It's not easy.
• 45:05 - 45:11
• 45:11 - 45:16
This is x and y and z, and you
want your x0 to be a constant.
• 45:16 - 45:17
STUDENT: [INAUDIBLE]
• 45:17 - 45:20
MAGDALENA TODA: It's a
so-called coordinate curve.
• 45:20 - 45:20
Very good.
• 45:20 - 45:23
It's a curve, but I want to
be good enough to draw it.
• 45:23 - 45:25
So you guys have
to wish me luck,
• 45:25 - 45:28
because I don't-- didn't have
enough coffee and I don't feel
• 45:28 - 45:30
like I can draw very well.
• 45:30 - 45:34
x0 is here.
• 45:34 - 45:40
So x is there, so you
cut with this board-- are
• 45:40 - 45:41
you guys with me?
• 45:41 - 45:44
You cut with this board
at the level x0 over here.
• 45:44 - 45:46
You cut.
• 45:46 - 45:49
When you cut with
this board-- you
• 45:49 - 45:53
with this board--
• 45:53 - 45:55
you get a curve like that.
• 45:55 - 46:00
And we call that a
curve f of x0, y.
• 46:00 - 46:06
Some people who are a little bit
in a hurry and smarter than me,
• 46:06 - 46:08
they say x equals x0.
• 46:08 - 46:10
That's called coordinate curve.
• 46:10 - 46:17
• 46:17 - 46:20
So, the thing is, this--
it's a curve in plane.
• 46:20 - 46:21
This is the blue plane.
• 46:21 - 46:22
I don't know how to call it.
• 46:22 - 46:23
Pi.
• 46:23 - 46:26
You know I love to call it pi.
• 46:26 - 46:28
Since I'm in plane with
a point in a curve--
• 46:28 - 46:34
a plane curve-- this curve
has a slope at x0, y0.
• 46:34 - 46:35
Can I draw that slope?
• 46:35 - 46:37
I'll try.
• 46:37 - 46:39
The slope of the
blue line, though.
• 46:39 - 46:40
Let me make it red.
• 46:40 - 46:44
The slope of the red line--
now, if you don't have colors
• 46:44 - 46:47
you can make it a dotted line.
• 46:47 - 46:58
The slope of the dotted line
is-- who the heck is that?
• 46:58 - 47:07
The derivative of f with respect
to y, because x0 is a constant.
• 47:07 - 47:09
So how do we write that?
• 47:09 - 47:13
Because x0 is sort of in
our way, driving us crazy.
• 47:13 - 47:15
Although he was fixed.
• 47:15 - 47:18
We keep him fixed by
keeping him in this plane.
• 47:18 - 47:20
x0 is fixed.
• 47:20 - 47:22
We have to write
another notation.
• 47:22 - 47:24
We cannot say f prime.
• 47:24 - 47:27
Because f depends
on two variables.
• 47:27 - 47:32
f prime were for when we
were babies in calculus 1.
• 47:32 - 47:33
We cannot use f prime anymore.
• 47:33 - 47:34
We have two variables.
• 47:34 - 47:36
Life became too complicated.
• 47:36 - 47:37
So we have to say--
• 47:37 - 47:38
STUDENT: Professor?
• 47:38 - 47:41
of df dy-- yes, sir.
• 47:41 - 47:42
May you use a subscript?
• 47:42 - 47:45
MAGDALENA TODA: You use--
yeah, you can do that as well.
• 47:45 - 47:47
That's what I do.
• 47:47 - 47:49
Let me do both.
• 47:49 - 47:56
f sub y at-- who
was fixed? x0 and y.
• 47:56 - 47:59
But this is my
favorite notation.
• 47:59 - 48:01
I'm going to make a
face because I love it.
• 48:01 - 48:03
This is what engineers love.
• 48:03 - 48:05
This is what we physicists love.
• 48:05 - 48:07
Mathematicians, though,
are crazy people.
• 48:07 - 48:08
They are.
• 48:08 - 48:09
All of them.
• 48:09 - 48:13
And they invented
another notation.
• 48:13 - 48:15
Do you remember
that Mr. Leibniz,
• 48:15 - 48:19
because he had nothing better to
do, when he invented calculus,
• 48:19 - 48:23
he did df dy, or df dx?
• 48:23 - 48:24
What is that?
• 48:24 - 48:27
That was the limit of
delta f, delta y, right?
• 48:27 - 48:29
That's what Leibniz did.
• 48:29 - 48:31
He introduced this
delta notation,
• 48:31 - 48:35
and then he said if you have
delta space over delta time,
• 48:35 - 48:38
then shrink both, and you
make a ratio in the limit,
• 48:38 - 48:41
should write it df dy.
• 48:41 - 48:44
And that's the so-called
Leibniz notation, right?
• 48:44 - 48:47
That was in calc 1.
• 48:47 - 48:50
But I erased it because
that was calc 1.
• 48:50 - 48:54
Now, mathematicians, to
imitate the Leibniz notation,
• 48:54 - 48:58
they said, I cannot use df dy.
• 48:58 - 49:01
So what the heck shall I use?
• 49:01 - 49:03
After they thought
• 49:03 - 49:05
• 49:05 - 49:07
they invented this,
they said, let's take
• 49:07 - 49:10
the Greek-- the Greek d.
• 49:10 - 49:12
Which is the del.
• 49:12 - 49:14
That's partial.
• 49:14 - 49:19
The del f, del y, at x0, y.
• 49:19 - 49:22
When I was 20--
no, I was 18 when
• 49:22 - 49:27
I saw this the first time--
I had the hardest time making
• 49:27 - 49:28
this sign.
• 49:28 - 49:30
It's all in the wrist.
• 49:30 - 49:32
It's very-- OK.
• 49:32 - 49:33
Now.
• 49:33 - 49:34
df dy.
• 49:34 - 49:36
If you don't like it,
then what do you do?
• 49:36 - 49:39
• 49:39 - 49:42
And what is the meaning
of this by definition?
• 49:42 - 49:45
You say, you haven't even
defined it, Magdalena.
• 49:45 - 49:48
It has to be limit of
a difference quotient,
• 49:48 - 49:49
just like here.
• 49:49 - 49:53
But we have to be happy
and think of that.
• 49:53 - 49:57
What is the delta f
versus the delta y?
• 49:57 - 49:59
It has to be like that.
• 49:59 - 50:03
f of Mr. x0 is fixed.
• 50:03 - 50:07
x0, comma, y.
• 50:07 - 50:10
We have an increment in y.
• 50:10 - 50:16
y plus delta y. y plus
delta y minus-- that's
• 50:16 - 50:18
the difference quotient.
• 50:18 - 50:23
f of what-- the original
point was, well--
• 50:23 - 50:24
STUDENT: x0, y0.
• 50:24 - 50:27
MAGDALENA TODA:
x0-- let me put y0
• 50:27 - 50:30
because our original
point was x0, y0.
• 50:30 - 50:38
x0, y0 over-- over delta y.
• 50:38 - 50:43
But if I am at x0, y0, I better
put x0, y0 fixed point here.
• 50:43 - 50:47
• 50:47 - 50:52
And I would like you to
photograph or put this thing--
• 50:52 - 50:55
STUDENT: So is that a delta
that's in front of the f?
• 50:55 - 50:56
MAGDALENA TODA: Let me
review the whole thing
• 50:56 - 50:59
because it's very important.
• 50:59 - 51:01
Where shall I start,
here, or here?
• 51:01 - 51:02
It doesn't matter.
• 51:02 - 51:03
So the limit--
• 51:03 - 51:05
STUDENT: [INAUDIBLE] start at m.
• 51:05 - 51:06
MAGDALENA TODA: At m?
• 51:06 - 51:07
At m.
• 51:07 - 51:08
OK, I'll start at m.
• 51:08 - 51:13
The slopes of this line at
x0, y0, right at my point,
• 51:13 - 51:19
will be, my favorite
notation is f sub y at x0,
• 51:19 - 51:22
y0, which means partial
derivative of f with respect
• 51:22 - 51:26
to y at the point--
fixed point x0, y0.
• 51:26 - 51:31
Or, for most mathematicians,
df-- of del-- del f,
• 51:31 - 51:34
del y at x0, y0.
• 51:34 - 51:39
Which is by definition the limit
of this difference quotient.
• 51:39 - 51:42
So x0 is held fixed
in both cases.
• 51:42 - 51:45
y0 is allowed to
deviate a little bit.
• 51:45 - 51:50
So y0 is fixed, but you
displace it by a little delta,
• 51:50 - 51:54
or by a little-- how did we
denote that in calc 1, h?
• 51:54 - 51:54
Little h?
• 51:54 - 51:55
STUDENT: Yeah.
• 51:55 - 51:57
MAGDALENA TODA: So
delta y, sometimes it
• 51:57 - 51:58
was called little h.
• 51:58 - 52:01
And this is the
same as little h.
• 52:01 - 52:04
Over that h.
• 52:04 - 52:07
Now you, without my
help, because you
• 52:07 - 52:11
have all the knowledge
and you're smart,
• 52:11 - 52:17
you should tell me how I
define f sub x at x0, y0,
• 52:17 - 52:22
and shut up, Magdalena,
let people talk.
• 52:22 - 52:24
This is hard.
• 52:24 - 52:25
[INTERPOSING VOICES]
• 52:25 - 52:26
MAGDALENA TODA: No.
• 52:26 - 52:28
I hope not.
• 52:28 - 52:32
As a limit of a
difference quotient,
• 52:32 - 52:35
so it's gonna be an
instantaneous rate of change.
• 52:35 - 52:37
That's the limit of a
difference quotient.
• 52:37 - 52:39
Limit of what?
• 52:39 - 52:39
Shut up.
• 52:39 - 52:41
I will zip my lips.
• 52:41 - 52:41
STUDENT: Delta x
• 52:41 - 52:43
MAGDALENA TODA:
Delta x, excellent.
• 52:43 - 52:44
Delta x going to 0.
• 52:44 - 52:48
So you shrink-- you displace
by a small displacement
• 52:48 - 52:50
only in the direction of x.
• 52:50 - 52:52
STUDENT: So f.
• 52:52 - 52:53
MAGDALENA TODA: f.
• 52:53 - 52:57
STUDENT: [INAUDIBLE] this
time, x is changing, so--
• 52:57 - 52:59
[INTERPOSING VOICES]
• 52:59 - 53:05
MAGDALENA TODA: X0 plus
delta x, y0 is still fixed,
• 53:05 - 53:11
minus f of x0, y0.
• 53:11 - 53:13
Thank God this is always fixed.
• 53:13 - 53:14
I love this guy.
• 53:14 - 53:16
STUDENT: Delta--
• 53:16 - 53:20
MAGDALENA TODA:
Delta x, which is
• 53:20 - 53:23
like the h we were
• 53:23 - 53:25
Now in reality,
you never do that.
• 53:25 - 53:29
You would die if for every
exercise, derivation exercise,
• 53:29 - 53:32
you would have to compute a
limit of a difference quotient.
• 53:32 - 53:33
You will go bananas.
• 53:33 - 53:35
What we do?
• 53:35 - 53:37
We do exactly the same thing.
• 53:37 - 53:39
How can I draw?
• 53:39 - 53:41
Can anybody help me draw?
• 53:41 - 53:46
For y0, I would need to take
this other plane through y0.
• 53:46 - 53:47
Where is y0?
• 53:47 - 53:49
Here.
• 53:49 - 53:50
Is my drawing good enough?
• 53:50 - 53:52
I hope so.
• 53:52 - 53:56
So it's something like
I have this plane with,
• 53:56 - 53:58
oh, do you see that, guys?
• 53:58 - 53:59
OK.
• 53:59 - 54:03
So what is that, the other
curve, coordinate curve, look
• 54:03 - 54:03
like?
• 54:03 - 54:07
• 54:07 - 54:09
Oh my God.
• 54:09 - 54:10
Looks like that.
• 54:10 - 54:13
Through the same point,
and then the slope
• 54:13 - 54:18
of the line will be a
blue slope and the slope
• 54:18 - 54:24
will be f sub-- well OK.
• 54:24 - 54:28
So here I have in the red
one, which was the blue one,
• 54:28 - 54:34
this is f sub y, and for
this one, this is f sub x.
• 54:34 - 54:35
Right?
• 54:35 - 54:42
So guys, don't look
at the picture.
• 54:42 - 54:43
The picture's confusing.
• 54:43 - 54:47
This is x coming
towards me, right?
• 54:47 - 54:50
And y going there
and z is going up.
• 54:50 - 54:52
This is the graph.
• 54:52 - 54:55
When I do the
derivative with respect
• 54:55 - 55:01
to what is this, y, the
derivative with respect to y,
• 55:01 - 55:04
with respect to y, y
is my only variable,
• 55:04 - 55:07
so the curve will be like that.
• 55:07 - 55:11
And the slope will be for a
curve that depends on y only.
• 55:11 - 55:14
When I do derivative
with respect to x,
• 55:14 - 55:19
it's like I'm on top of a hill
and I decide to go skiing.
• 55:19 - 55:22
And I'm-- and I point
my skis like that,
• 55:22 - 55:27
and the slope is going down,
and that's the x direction.
• 55:27 - 55:28
OK?
• 55:28 - 55:31
And what I'm going to
describe as a skier
• 55:31 - 55:34
will be a plane curve going
down in this direction.
• 55:34 - 55:36
Zzzzsssshh, like that.
• 55:36 - 55:41
And the slope at every
point, the slope of the line,
• 55:41 - 55:45
of y trajectory, will
be the derivative.
• 55:45 - 55:47
So I have a curve like
that, and a curve like this.
• 55:47 - 55:50
And they're called
coordinate curves.
• 55:50 - 55:51
Now this is hard.
• 55:51 - 55:53
You'll see how
beautiful and easy
• 55:53 - 55:57
it is when you actually
compute the partial derivatives
• 55:57 - 56:00
of functions by hand.
• 56:00 - 56:02
Examples?
• 56:02 - 56:08
Let's take f of x, y to be
x squared plus y squared.
• 56:08 - 56:13
is f sub x at x, y?
• 56:13 - 56:18
Who is f sub x at 1
minus 1, 1, 0, OK.
• 56:18 - 56:21
Who is f sub y at x, y?
• 56:21 - 56:26
And who is f sub y at 3 and 2.
• 56:26 - 56:28
Since I make up my
example-- I don't
• 56:28 - 56:30
want to copy the
examples from the book,
• 56:30 - 56:35
because you are supposedly
• 56:35 - 56:40
This is-- should be another
example, just for you.
• 56:40 - 56:44
• 56:44 - 56:49
So who's gonna help me-- I'm
pausing a little bit-- who's
• 56:49 - 56:51
gonna help me here?
• 56:51 - 56:54
• 56:54 - 56:56
So how do I think?
• 56:56 - 57:00
I think I got-- when I
prime with respect to x, y
• 57:00 - 57:01
is like a held constant.
• 57:01 - 57:03
He's held prisoner.
• 57:03 - 57:05
Poor guy cannot leave his cell.
• 57:05 - 57:06
That's awful.
• 57:06 - 57:09
So you prime with respect to x.
• 57:09 - 57:11
Because x is the only variable.
• 57:11 - 57:12
And he is--
• 57:12 - 57:15
STUDENT: So then it's 2x plus y?
• 57:15 - 57:17
MAGDALENA TODA: 2x plus 0.
• 57:17 - 57:18
Plus 0.
• 57:18 - 57:20
Because y is a constant and
when you prime a constant,
• 57:20 - 57:22
you get 0.
• 57:22 - 57:24
STUDENT: So when you
take partial derivatives,
• 57:24 - 57:26
you-- when you're
taking it with respect
• 57:26 - 57:29
to the first derivative, the
first variable [INAUDIBLE]
• 57:29 - 57:30
MAGDALENA TODA: You
don't completely
• 57:30 - 57:32
know because it
might be multiplied.
• 57:32 - 57:33
But you view it as a constant.
• 57:33 - 57:35
So for you-- very good, Ryan.
• 57:35 - 57:38
So for you, it's like,
as if y would be 7.
• 57:38 - 57:40
Imagine that y would be 7.
• 57:40 - 57:44
And then you have x squared plus
7 squared prime is u, right?
• 57:44 - 57:48
STUDENT: So then that means
f of 1-- or f x of 1,0
• 57:48 - 57:48
is [INAUDIBLE]
• 57:48 - 57:50
MAGDALENA TODA: Very good.
• 57:50 - 57:51
STUDENT: OK.
• 57:51 - 57:55
And in this case, f sub y,
what do you think it is?
• 57:55 - 57:56
STUDENT: 2y.
• 57:56 - 57:57
MAGDALENA TODA: 2y.
• 57:57 - 58:00
And what is f y of 3, 2?
• 58:00 - 58:01
STUDENT: 4.
• 58:01 - 58:02
MAGDALENA TODA: It's 4.
• 58:02 - 58:05
And you say, OK, that
makes sense, that was easy.
• 58:05 - 58:07
Let's try something hard.
• 58:07 - 58:09
I'm going to build them
on so many examples
• 58:09 - 58:13
that you say, stop,
Magdalena, because I became
• 58:13 - 58:16
an expert in partial
differentiation
• 58:16 - 58:19
and I-- now everything is so
trivial that you have to stop.
• 58:19 - 58:38
So example A, example B. A was f
of x, y [INAUDIBLE] x, y plus y
• 58:38 - 58:40
sine x.
• 58:40 - 58:41
And you say, wait,
wait, wait, you're
• 58:41 - 58:44
giving me a little
bit of trouble.
• 58:44 - 58:46
No, I don't mean to.
• 58:46 - 58:47
It's very easy.
• 58:47 - 58:50
Believe me guys,
very, very easy.
• 58:50 - 58:55
We just have to
think how we do this.
• 58:55 - 59:02
f sub x at 1 and 2, f
sub y at x, y in general,
• 59:02 - 59:07
f sub y at 1 and
2, for God's sake.
• 59:07 - 59:08
OK.
• 59:08 - 59:10
All right.
• 59:10 - 59:19
And now, while you're
staring at that,
• 59:19 - 59:24
I take out my beautiful
colors that I paid \$6 for.
• 59:24 - 59:26
• 59:26 - 59:32
The department told me that
• 59:32 - 59:35
just two or three basic ones.
• 59:35 - 59:36
All right?
• 59:36 - 59:38
So what do we do?
• 59:38 - 59:41
STUDENT: First
one will be the y.
• 59:41 - 59:43
MAGDALENA TODA: It's like y
would be a constant 7, right,
• 59:43 - 59:47
but you have to keep in
mind it's mister called y.
• 59:47 - 59:49
Which for you is a constant.
• 59:49 - 59:53
So you go, I'm priming this
with respect to x only--
• 59:53 - 59:55
STUDENT: Then you get y.
• 59:55 - 59:56
MAGDALENA TODA: Very good.
• 59:56 - 59:57
Plus--
• 59:57 - 60:00
• 60:00 - 60:01
STUDENT: y cosine x.
• 60:01 - 60:02
MAGDALENA TODA: y cosine x.
• 60:02 - 60:02
Excellent.
• 60:02 - 60:04
And stop.
• 60:04 - 60:05
And stop.
• 60:05 - 60:06
Because that's all I have.
• 60:06 - 60:09
You see, it's not hard.
• 60:09 - 60:12
Let me put here a y.
• 60:12 - 60:13
OK.
• 60:13 - 60:19
And then, I plug
a different color.
• 60:19 - 60:22
I'm a girl, of course I
like different colors.
• 60:22 - 60:27
So 1, 2. x is 1, and y is 2.
• 60:27 - 60:30
2 plus 2 cosine 1.
• 60:30 - 60:33
And you say, oh, wait a minute,
what is that cosine of 1?
• 60:33 - 60:34
Never mind.
• 60:34 - 60:35
• 60:35 - 60:37
It's like cosine
of 1, [INAUDIBLE]
• 60:37 - 60:41
plug it in the
calculator, nobody cares.
• 60:41 - 60:45
Well, in the final, you
don't have a calculator,
• 60:45 - 60:48
so you leave it like that.
• 60:48 - 60:49
Who cares?
• 60:49 - 60:53
It's just the perfect--
I would actually hate it
• 60:53 - 60:54
that you gave me--
because all you
• 60:54 - 60:56
could give me would be an
approximation, a truncation,
• 60:56 - 60:58
with two decimals.
• 60:58 - 61:01
I prefer you give me the
• 61:01 - 61:04
is an exact answer like that.
• 61:04 - 61:05
f sub y.
• 61:05 - 61:08
Now, Mr. x is held prisoner.
• 61:08 - 61:09
He is a constant.
• 61:09 - 61:11
He cannot move.
• 61:11 - 61:12
Mr. y can move.
• 61:12 - 61:13
He has all the freedom.
• 61:13 - 61:17
So prime with respect
to y, what do you have?
• 61:17 - 61:17
STUDENT: x--
• 61:17 - 61:18
[INTERPOSING VOICES]
• 61:18 - 61:22
MAGDALENA TODA: x plus
sine x is a constant.
• 61:22 - 61:25
So for God's sake,
I'll write it.
• 61:25 - 61:31
So then I get 1,
plug in x equals 1. y
• 61:31 - 61:32
doesn't appear in the picture.
• 61:32 - 61:33
I don't care.
• 61:33 - 61:35
1 plus sine 1.
• 61:35 - 61:38
• 61:38 - 61:40
And now comes-- don't erase.
• 61:40 - 61:42
Now comes the-- I mean,
you cannot erase it.
• 61:42 - 61:45
I can erase it.
• 61:45 - 61:49
Comes this mean professor
who says, wait a minute,
• 61:49 - 61:51
I want more.
• 61:51 - 61:54
Mathematicians always want more.
• 61:54 - 61:57
He goes, I want the
second derivative.
• 61:57 - 62:01
f sub x x of x, y.
• 62:01 - 62:04
And you say, what in
the world is that?
• 62:04 - 62:06
Even some mathematicians,
they denote it
• 62:06 - 62:13
as del 2 f dx 2, which
is d of-- d with respect
• 62:13 - 62:17
to x sub d u with respect to x.
• 62:17 - 62:18
What does it mean?
• 62:18 - 62:21
You take the first derivative
and you derive it again.
• 62:21 - 62:23
And don't drink and derive
because you'll be in trouble.
• 62:23 - 62:24
Right?
• 62:24 - 62:28
So you have d of dx primed
again, with-- differentiated
• 62:28 - 62:30
again with respect to x.
• 62:30 - 62:31
Is that hard?
• 62:31 - 62:32
Uh-uh.
• 62:32 - 62:33
What you do?
• 62:33 - 62:36
In the-- don't do it here.
• 62:36 - 62:37
You do it in general, right?
• 62:37 - 62:43
With respect to x as a variable,
y is again held as a prisoner,
• 62:43 - 62:45
constant.
• 62:45 - 62:48
So when you prime
that y goes away.
• 62:48 - 62:51
You're gonna get 0.
• 62:51 - 62:55
I'll write 0 like a silly
because we are just starters.
• 62:55 - 62:56
And what else?
• 62:56 - 62:58
STUDENT: Negative y sine of x.
• 62:58 - 63:00
MAGDALENA TODA:
Minus y sine of x.
• 63:00 - 63:02
And I know you've gonna
love this process.
• 63:02 - 63:05
You are becoming
experts in that.
• 63:05 - 63:10
And in a way I'm a little
bit sorry it's so easy,
• 63:10 - 63:13
but I guess not
everybody gets it.
• 63:13 - 63:16
There are students who
don't get it the first time.
• 63:16 - 63:18
So what do we get here?
• 63:18 - 63:19
Minus--
• 63:19 - 63:22
STUDENT: 0.
• 63:22 - 63:26
tell me-- sine 1, 0.
• 63:26 - 63:26
Good.
• 63:26 - 63:30
I could do the same
thing for f y y.
• 63:30 - 63:35
I could do this thing--
what is f sub x y?
• 63:35 - 63:37
By definition f sub x y--
• 63:37 - 63:40
STUDENT: Is that taking the
derivative of the derivative
• 63:40 - 63:42
with respect-- is that
taking the second derivative
• 63:42 - 63:44
with respect to y after
you take the derivative
• 63:44 - 63:46
of the-- first derivative
with respect to x?
• 63:46 - 63:47
MAGDALENA TODA: Right.
• 63:47 - 63:50
So when I write like that,
because that's a little bit
• 63:50 - 63:54
confusing, when students
ask me, which one is first?
• 63:54 - 63:58
First you do f sub
x, and then you do y.
• 63:58 - 64:03
And then f sub y x would be the
derivative with respect to y
• 64:03 - 64:05
primed again with respect to x.
• 64:05 - 64:07
Now, let me tell
you the good news.
• 64:07 - 64:13
They-- the book doesn't call
it any name, because we don't
• 64:13 - 64:15
like to call anybody names.
• 64:15 - 64:16
I'm just kidding.
• 64:16 - 64:24
It's called the
Schwartz principle,
• 64:24 - 64:27
or the theorem of Schwartz.
• 64:27 - 64:31
When I told my co-authors,
they said, who cares?
• 64:31 - 64:35
Well I care, because I was a
student when my professors told
• 64:35 - 64:38
me that this German
• 64:38 - 64:41
this discovery, which
is so beautiful.
• 64:41 - 64:55
If f is twice differentiable
with respect to x and y,
• 64:55 - 64:59
and the partial derivatives--
the second partial
• 64:59 - 65:15
derivatives-- are continuous,
then, now in English
• 65:15 - 65:17
it would say it doesn't
matter in which order
• 65:17 - 65:19
you differentiate.
• 65:19 - 65:21
The mixed ones are
always the same.
• 65:21 - 65:22
Say what?
• 65:22 - 65:27
f sub x y equals f sub
y x for every point.
• 65:27 - 65:32
For every-- do you remember
what I taught you for every x, y
• 65:32 - 65:32
in the domain.
• 65:32 - 65:36
Or for every x, y
where this happens.
• 65:36 - 65:38
So what does this mean?
• 65:38 - 65:41
That means that whether
you differentiate
• 65:41 - 65:44
first with respect to x and then
with respect to, y, or first
• 65:44 - 65:46
with respect to y and
then with respect to x,
• 65:46 - 65:48
it doesn't matter.
• 65:48 - 65:51
The mixed partial
derivatives are the same.
• 65:51 - 65:52
Which is wonderful.
• 65:52 - 65:55
I mean, this is one
of the best things
• 65:55 - 65:58
that ever happened to us.
• 65:58 - 66:01
Let's see if this
is true in our case.
• 66:01 - 66:04
I mean, of course it's true
because it's a theorem,
• 66:04 - 66:06
if it weren't true
I wouldn't teach it,
• 66:06 - 66:11
but let's verify it on a baby.
• 66:11 - 66:14
Not on a real baby,
on a baby example.
• 66:14 - 66:15
Right?
• 66:15 - 66:21
So, f sub x is y plus y
equals sine x primed again
• 66:21 - 66:23
with respect to y.
• 66:23 - 66:28
And what do we get out of it?
• 66:28 - 66:29
Cosine of x.
• 66:29 - 66:31
Are you guys with me?
• 66:31 - 66:35
So f sub x was y
plus y equals sine x.
• 66:35 - 66:39
Take this guy
again, put it here,
• 66:39 - 66:43
squeeze them up a little
bit, divide by-- no.
• 66:43 - 66:48
Time with respect to y, x is
a constant, what do you think?
• 66:48 - 66:49
Cosine of x, am I right?
• 66:49 - 66:50
STUDENT: 1 plus [INAUDIBLE].
• 66:50 - 66:52
• 66:52 - 66:54
MAGDALENA TODA: That's
what it starts with.
• 66:54 - 66:57
Plus [INAUDIBLE].
• 66:57 - 67:02
So cosine of x, [INAUDIBLE]
a constant, plus 1.
• 67:02 - 67:05
Another way to have done
it is, like, wait a minute,
• 67:05 - 67:11
at this point I go, constant
out-- are you with me?--
• 67:11 - 67:15
constant out, prime with respect
to y, equals sine x plus 1.
• 67:15 - 67:17
Thank you.
• 67:17 - 67:17
All right.
• 67:17 - 67:21
• 67:21 - 67:26
F sub yx is going to be f sub y.
• 67:26 - 67:32
x plus sine x, but I have
to take it from here,
• 67:32 - 67:38
and I prime again with respect
to x, and I get the same thing.
• 67:38 - 67:40
I don't know,
maybe I'm dyslexic,
• 67:40 - 67:43
I go from the right to the
left, what's the matter with me.
• 67:43 - 67:47
I go cosine of x plus 1.
• 67:47 - 67:53
• 67:53 - 67:54
So it's the same thing.
• 67:54 - 67:55
Yes, sir.
• 67:55 - 67:59
STUDENT:I'm looking at
the f of xy from the--
• 67:59 - 68:00
MAGDALENA TODA: Which
one are you looking at?
• 68:00 - 68:01
Show me.
• 68:01 - 68:04
STUDENT: It's in the purple.
• 68:04 - 68:05
MAGDALENA TODA: It
is in the purple.
• 68:05 - 68:06
STUDENT: It's that
one right there.
• 68:06 - 68:06
So--
• 68:06 - 68:07
MAGDALENA TODA: This one?
• 68:07 - 68:07
STUDENT: Mmhm.
• 68:07 - 68:11
So, I'm looking at
the y plus y cosine x.
• 68:11 - 68:13
You got that from f of x.
• 68:13 - 68:14
MAGDALENA TODA: I
got this from f of x,
• 68:14 - 68:17
and I prime it again,
with respect to y.
• 68:17 - 68:19
The whole thing.
• 68:19 - 68:22
STUDENT: OK, so you're not
writing that as a derivative?
• 68:22 - 68:25
You're just substituting
that in for f of x?
• 68:25 - 68:27
MAGDALENA TODA: So,
let me write it better,
• 68:27 - 68:31
because I was a little bit
rushed, and I don't know,
• 68:31 - 68:32
silly or something.
• 68:32 - 68:35
When I prime this
with respect to y--
• 68:35 - 68:38
STUDENT: Then you get
the cosine of x plus 1.
• 68:38 - 68:39
MAGDALENA TODA: Yeah.
• 68:39 - 68:42
I could say, I can take
out all the constants.
• 68:42 - 68:43
STUDENT: OK.
• 68:43 - 68:46
MAGDALENA TODA: And that
constant is this plus 1.
• 68:46 - 68:47
And that's all I'm left with.
• 68:47 - 68:48
Right?
• 68:48 - 68:52
It's the same thing
as 1 plus cosine x,
• 68:52 - 68:54
which is a constant times y.
• 68:54 - 68:57
Prime this with respect
to y, I get the constant.
• 68:57 - 69:04
It's the same principal as when
you have bdy of 7y equals 7.
• 69:04 - 69:07
Right?
• 69:07 - 69:09
OK.
• 69:09 - 69:10
Is this too easy?
• 69:10 - 69:13
I'll give you a nicer function.
• 69:13 - 69:29
I'm imitating the one
in WeBWorK [INAUDIBLE]
• 69:29 - 69:31
To make it harder for you.
• 69:31 - 69:34
Nothing I can make at this
point is hard for you,
• 69:34 - 69:39
because you're becoming experts
in partial differentiation,
• 69:39 - 69:42
and I cannot
challenge you on that.
• 69:42 - 69:54
• 69:54 - 69:57
I'm just trying to
make it harder for you.
• 69:57 - 69:59
And I'm trying to
look up something.
• 69:59 - 70:03
• 70:03 - 70:04
• 70:04 - 70:07
• 70:07 - 70:09
This is harder than the
ones you have in WeBWorK.
• 70:09 - 70:12
But that was kind of
the idea-- that when
• 70:12 - 70:16
you go home, and open
those WeBWorK problem sets,
• 70:16 - 70:17
that's a piece of cake.
• 70:17 - 70:21
What we did in class was harder.
• 70:21 - 70:24
student, one professor said,
• 70:24 - 70:27
the easy examples are the
ones that the professor's
• 70:27 - 70:30
supposed to write in
class, on the board.
• 70:30 - 70:31
The hard examples
are the ones that
• 70:31 - 70:34
are left for the
students' homework.
• 70:34 - 70:36
I disagree.
• 70:36 - 70:38
I think it should be
the other way around.
• 70:38 - 70:40
So f sub x.
• 70:40 - 70:43
• 70:43 - 70:51
That means bfdx for
the pair xy, any xy.
• 70:51 - 70:54
I'm not specifying an x and a y.
• 70:54 - 70:56
I'm not making them a constant.
• 70:56 - 70:59
What am I going to
have in this case?
• 70:59 - 71:04
Chain -- if I catch you
not knowing the chain rule,
• 71:04 - 71:05
you fail the final.
• 71:05 - 71:13
Not really, but, OK,
you get some penalty.
• 71:13 - 71:14
You know it.
• 71:14 - 71:16
Just pay attention
to what you do.
• 71:16 - 71:18
I make my own
mistakes sometimes.
• 71:18 - 71:21
So 1 over.
• 71:21 - 71:24
What do you do here
when you differentiate
• 71:24 - 71:24
with respect to x?
• 71:24 - 71:32
You think, OK, from the outside
to the inside, one at a time.
• 71:32 - 71:36
1 over the variable
squared plus 1, right?
• 71:36 - 71:42
Whatever that variable,
it's like you call variable
• 71:42 - 71:45
of the argument xy, right?
• 71:45 - 71:47
STUDENT: [INAUDIBLE]
• 71:47 - 71:50
MAGDALENA TODA: Square plus 1.
• 71:50 - 71:57
Times-- cover it with your
hand-- prime with respect to x.
• 71:57 - 71:59
y, right?
• 71:59 - 72:00
Good!
• 72:00 - 72:01
And you're done.
• 72:01 - 72:03
You see how easy it was.
• 72:03 - 72:08
Just don't forget something
because it can cost you points.
• 72:08 - 72:10
Are you guys with me?
• 72:10 - 72:13
So, once we are done with
saying, 1 over argument
• 72:13 - 72:16
squared plus 1, I cover
this with my hand,
• 72:16 - 72:20
xy prime with
respect to 2x is y.
• 72:20 - 72:22
And I'm done.
• 72:22 - 72:23
And I'm done.
• 72:23 - 72:26
And here, pause.
• 72:26 - 72:30
What's the easiest
way to do that?
• 72:30 - 72:32
You look at it like,
she wants me to get
• 72:32 - 72:34
caught in the quotient rule.
• 72:34 - 72:37
She wants to catch me
not knowing this rule,
• 72:37 - 72:40
while I can do better.
• 72:40 - 72:43
One way to do it would
be numerator prime plus
• 72:43 - 72:48
denominator, minus
numerator [INAUDIBLE] What's
• 72:48 - 72:50
the easier way to do it?
• 72:50 - 72:53
STUDENT: x squared plus
y squared, all of it
• 72:53 - 72:54
to the negative one.
• 72:54 - 72:55
MAGDALENA TODA: Right.
• 72:55 - 72:57
So you say, hey,
you cannot catch me,
• 72:57 - 73:01
• 73:01 - 73:01
Good!
• 73:01 - 73:03
That was a good idea.
• 73:03 - 73:10
Chain rule, and
minus 1/2, times--
• 73:10 - 73:12
who tells me what's next?
• 73:12 - 73:13
I'm not going to say a word.
• 73:13 - 73:15
STUDENT: 2x plus y squared.
• 73:15 - 73:19
No, it's 2x.
• 73:19 - 73:21
x squared plus y squared.
• 73:21 - 73:23
MAGDALENA TODA: From the
outside to the inside.
• 73:23 - 73:25
From the outside-- to the what?
• 73:25 - 73:27
STUDENT: [INAUDIBLE]
• 73:27 - 73:28
MAGDALENA TODA: Good.
• 73:28 - 73:29
And now I'm done.
• 73:29 - 73:31
I don't see that anymore.
• 73:31 - 73:34
I focus to the core.
• 73:34 - 73:36
2x.
• 73:36 - 73:39
Times 2x.
• 73:39 - 73:42
And that is plenty.
• 73:42 - 73:45
OK, now, let me
• 73:45 - 73:51
What if you would
• 73:51 - 73:57
I don't know, somebody
who knows that,
• 73:57 - 74:02
can you pose the f sub y of xy
without doing the whole thing
• 74:02 - 74:03
all over again?
• 74:03 - 74:06
Can you sort of figure
out what it would be?
• 74:06 - 74:09
The beautiful
• 74:09 - 74:11
is that these are
symmetric polynomials.
• 74:11 - 74:13
What does it mean,
symmetric polynomials?
• 74:13 - 74:19
That means, if you swap x
and y, and you swap x and y,
• 74:19 - 74:21
it's the same thing.
• 74:21 - 74:23
Just think of that--
swapping x and y.
• 74:23 - 74:25
Swapping the roles of x and y.
• 74:25 - 74:28
So what do you think
you're going to get?
• 74:28 - 74:31
OK, one student said,
this is for smart people,
• 74:31 - 74:33
not for people like me.
• 74:33 - 74:35
And I said, well, OK,
what's the matter with you?
• 74:35 - 74:36
I'm a hard worker.
• 74:36 - 74:40
I'm the kind of guy who takes
the whole thing again, and does
• 74:40 - 74:42
the derivation from scratch.
• 74:42 - 74:45
And thinking back in high
school, I think, even
• 74:45 - 74:48
for symmetric polynomials,
• 74:48 - 74:50
I'm sure that being
smart and being
• 74:50 - 74:54
able to guess the
whole thing-- but I
• 74:54 - 74:56
did the computation
many times mechanically,
• 74:56 - 74:59
just in the same way,
because I was a hard worker.
• 74:59 - 75:01
So what do you
have in that case?
• 75:01 - 75:10
1/xy squared plus 1 times x
plus-- the same kind of thing.
• 75:10 - 75:14
Attention, this is the symmetric
polynomial, and I go to that.
• 75:14 - 75:17
And then times 2y.
• 75:17 - 75:21
So, see-- that kind
of easy, fast thing.
• 75:21 - 75:24
Why is this a good
observation when
• 75:24 - 75:26
you have symmetric polynomials?
• 75:26 - 75:29
If you are on the final and
you don't have that much time,
• 75:29 - 75:34
or on any kind of exam when
you are in a time-crunch.
• 75:34 - 75:36
Now, we want those
exams so you are not
• 75:36 - 75:38
going to be in a time-crunch.
• 75:38 - 75:42
If there is something I hate,
I hate a final of 2 hours
• 75:42 - 75:45
and a half with 25
serious problems,
• 75:45 - 75:48
and you know nobody can do that.
• 75:48 - 75:51
So, it happens a lot.
• 75:51 - 75:56
I see that-- one of my jobs
is also to look at the finals
• 75:56 - 75:59
after people wrote
them, and I still
• 75:59 - 76:05
do that every semester-- I see
too many people making finals.
• 76:05 - 76:07
The finals are not
supposed to be long.
• 76:07 - 76:11
The finals are supposed
to be comprehensive, cover
• 76:11 - 76:16
everything, but not extensive.
• 76:16 - 76:21
So maybe you'll have 15
problems that cover practically
• 76:21 - 76:23
the material entirely.
• 76:23 - 76:23
Why?
• 76:23 - 76:29
Because every little problem
can have two short questions.
• 76:29 - 76:31
You were done with
a section, you
• 76:31 - 76:35
shot half of a chapter
only one question.
• 76:35 - 76:40
This is one example just--
not involving [INAUDIBLE]
• 76:40 - 76:41
of an expression like that, no.
• 76:41 - 76:43
That's too time-consuming.
• 76:43 - 76:48
But maybe just tangent of
x-squared plus y-squared,
• 76:48 - 76:50
find the partial derivatives.
• 76:50 - 76:53
That's a good exam
question, and that's enough
• 76:53 - 76:56
when it comes to
testing partials.
• 76:56 - 76:58
By the way, how
much-- what is that?
• 76:58 - 77:01
And I'm going to let
you go right now.
• 77:01 - 77:02
Use the bathroom.
• 77:02 - 77:05
And when you come back from the
bathroom, we'll fill in this.
• 77:05 - 77:11
You know I am horrible in the
sense that I want-- I'm greedy.
• 77:11 - 77:12
I need extra time.
• 77:12 - 77:15
I want to use more time.
• 77:15 - 77:18
I will do your
problems from now on,
• 77:18 - 77:22
and you can use the bathroom,
• 77:22 - 77:26
• 77:26 - 77:28
I'll start in
• 77:28 - 77:29
Don't worry.
• 77:29 - 77:33
• 77:33 - 77:34
Alexander?
• 77:34 - 77:35
Are you here?
• 77:35 - 77:38
Come get this.
• 77:38 - 77:40
I apologize.
• 77:40 - 77:42
This is long due back to you.
• 77:42 - 77:43
STUDENT: Oh.
• 77:43 - 77:44
Thank you.
• 77:44 - 77:47
• 77:47 - 77:50
STUDENT: Is there an
attendance sheet today?
• 77:50 - 77:53
MAGDALENA TODA: I will--
I'm making up one.
• 77:53 - 77:57
one side attendance.
• 77:57 - 77:59
Let's use the other side.
• 77:59 - 78:02
Put today's date.
• 78:02 - 78:02
[INAUDIBLE]
• 78:02 - 78:44
• 78:44 - 78:48
[SIDE CONVERSATIONS]
• 78:48 - 78:56
• 78:56 - 78:58
MAGDALENA TODA: They
are spoiling me.
• 78:58 - 79:03
They give me new
sprays every week.
• 79:03 - 79:05
[INAUDIBLE] take care of this.
• 79:05 - 79:09
[SIDE CONVERSATIONS]
• 79:09 - 79:14
• 79:14 - 79:17
MAGDALENA TODA: So I'm
• 79:17 - 79:21
And you respond honestly.
• 79:21 - 79:25
browsed through three chapters.
• 79:25 - 79:27
I mean, Chapter 9
was vector spaces,
• 79:27 - 79:30
and it was all review
from-- from what?
• 79:30 - 79:31
From Calc 2.
• 79:31 - 79:35
Chapter 10 was curves in
[INAUDIBLE] and curves
• 79:35 - 79:37
in space, practically.
• 79:37 - 79:41
• 79:41 - 79:47
And Chapter 11 is functions
of several variables.
• 79:47 - 79:49
Now you have a flavor
of all of them,
• 79:49 - 79:50
which one was hardest for you?
• 79:50 - 79:51
STUDENT: 9 and 10, both.
• 79:51 - 79:53
MAGDALENA TODA: 9 and 10 both.
• 79:53 - 79:54
STUDENT: [INAUDIBLE].
• 79:54 - 79:56
MAGDALENA TODA: This is so
much better than the other.
• 79:56 - 80:01
No, I think you guys
actually-- it looks better,
• 80:01 - 80:07
because you've seen a lot more
vectors and vector functions.
• 80:07 - 80:09
STUDENT: I didn't
understand any of 9 or 10.
• 80:09 - 80:10
STUDENT: [INAUDIBLE].
• 80:10 - 80:10
MAGDALENA TODA: Yes, ma'am.
• 80:10 - 80:12
STUDENT: Could you go over
parametrization [INAUDIBLE]?
• 80:12 - 80:14
MAGDALENA TODA: I will
go over that again.
• 80:14 - 80:18
And I will go over some
other parametrizations today.
• 80:18 - 80:24
And I promised that at the
end, in those 20 minutes,
• 80:24 - 80:28
I will do that problem that
gave a few of you trouble.
• 80:28 - 80:29
Yes, sir?
• 80:29 - 80:31
STUDENT: Do we take
the same final exam
• 80:31 - 80:33
as all the other [INAUDIBLE]
classes? [INAUDIBLE]?
• 80:33 - 80:36
MAGDALENA TODA: Well, that's
• 80:36 - 80:43
So practically, it's at the
latitude of the instructor who
• 80:43 - 80:45
teaches honors if they
write their own final,
• 80:45 - 80:49
and in general make
it harder, or they
• 80:49 - 80:51
take the general final
like everybody else.
• 80:51 - 80:54
purposes, and as a study,
• 80:54 - 80:58
I would like you to
take the general final,
• 80:58 - 81:01
because I want to see
where you stand compared
• 81:01 - 81:03
to the rest of the population.
• 81:03 - 81:07
So you are my sample, and
they are the entire student
• 81:07 - 81:08
population of Calc
3, I want to make
• 81:08 - 81:14
the statistical analysis of your
performance compared to them.
• 81:14 - 81:16
STUDENT: So we'll
take the regular one?
• 81:16 - 81:17
MAGDALENA TODA: Yeah.
• 81:17 - 81:19
For this one, I just
have to make sure
• 81:19 - 81:22
that they also have that
• 81:22 - 81:26
Because if I have too much
extra credit in there,
• 81:26 - 81:27
well they also count that.
• 81:27 - 81:29
So that's what that means.
• 81:29 - 81:30
So we can [INAUDIBLE].
• 81:30 - 81:34
• 81:34 - 81:36
All right.
• 81:36 - 81:37
Let me finish this exercise.
• 81:37 - 81:41
And then [? stop ?]
[INAUDIBLE] and go
• 81:41 - 81:46
over some homework problems and
some parametrization problems.
• 81:46 - 81:49
And I will see what else.
• 81:49 - 81:56
So tangent of [INAUDIBLE].
• 81:56 - 81:59
• 81:59 - 82:00
Is this hard?
• 82:00 - 82:02
No, it's [INAUDIBLE].
• 82:02 - 82:06
But you have to
remind me, because I
• 82:06 - 82:09
pretend that I
forgot-- let me pretend
• 82:09 - 82:14
that I forgot what the
derivative [INAUDIBLE] notation
• 82:14 - 82:18
of tangent of t was.
• 82:18 - 82:20
STUDENT: Secant squared.
• 82:20 - 82:23
MAGDALENA TODA: You guys love
that secant squared thingy.
• 82:23 - 82:26
• 82:26 - 82:31
Why do you like secant squared?
• 82:31 - 82:34
I, as a student, I didn't
like expressing it like that.
• 82:34 - 82:36
I liked [INAUDIBLE].
• 82:36 - 82:37
Of course, it's the same thing.
• 82:37 - 82:40
But I always like it like
1 over cosine [INAUDIBLE].
• 82:40 - 82:45
• 82:45 - 82:48
And of course, I have
• 82:48 - 82:52
because I'm curious to
see what you remember.
• 82:52 - 82:55
And you say yeah,
curiosity killed the cat.
• 82:55 - 83:00
But where did the
derivative exist?
• 83:00 - 83:07
Because maybe was
that tangent of T--
• 83:07 - 83:08
STUDENT: Wasn't
it a quotient rule
• 83:08 - 83:10
of sine and [? cosine x? ?]
• 83:10 - 83:12
MAGDALENA TODA: Good.
• 83:12 - 83:15
I'm proud of you.
• 83:15 - 83:17
• 83:17 - 83:23
So [? my ?] [? have ?] this
blowing up, this blows up--
• 83:23 - 83:30
blows up where cosine
T was zero, right?
• 83:30 - 83:32
So where did that blow up?
• 83:32 - 83:37
[INAUDIBLE] blow up of
cosine and zero [INAUDIBLE].
• 83:37 - 83:41
The cosine was the
• 83:41 - 83:44
So here you blow up here, you
blow up here, you blow up here,
• 83:44 - 83:45
you blow up here.
• 83:45 - 83:49
• 83:49 - 83:52
So [? what does ?] [INAUDIBLE].
• 83:52 - 83:53
It should not be what?
• 83:53 - 83:55
STUDENT: Pi over 2.
• 83:55 - 83:56
MAGDALENA TODA: Yeah.
• 83:56 - 84:00
And can we express
that OK, among 0pi,
• 84:00 - 84:03
let's say you go in
between 0 and 2pi only.
• 84:03 - 84:08
I get rid of pi over
2 and 3pi over 2.
• 84:08 - 84:12
But if I express that in
general for [INAUDIBLE] T
• 84:12 - 84:15
not restricted to 0
to T, what do I say?
• 84:15 - 84:16
STUDENT: It's k.
• 84:16 - 84:19
STUDENT: So it can
[? never be ?] pi over 2
• 84:19 - 84:19
plus pi?
• 84:19 - 84:21
MAGDALENA TODA: 2k plus 1.
• 84:21 - 84:24
2k plus 1.
• 84:24 - 84:25
Odd number over--
• 84:25 - 84:26
STUDENT: Pi over 2.
• 84:26 - 84:27
MAGDALENA TODA: Pi over 2.
• 84:27 - 84:28
Odd number, pi over 2.
• 84:28 - 84:30
And all the odd
numbers are 2k plus 1.
• 84:30 - 84:31
Right?
• 84:31 - 84:32
All right.
• 84:32 - 84:39
So you have a not
existence and-- OK.
• 84:39 - 84:40
Coming back.
• 84:40 - 84:42
I'm just playing, because
we are still in the break.
• 84:42 - 84:44
• 84:44 - 84:50
What is dfdx, del f, del x, xy.
• 84:50 - 84:52
And what is del f, del y?
• 84:52 - 84:56
I'm not going to ask you for
the second partial derivative.
• 84:56 - 84:57
• 84:57 - 85:05
We also agreed that we have
important results in that.
• 85:05 - 85:09
What is the final answer here?
• 85:09 - 85:14
STUDENT: [INAUDIBLE] plus
x-squared [INAUDIBLE].
• 85:14 - 85:15
MAGDALENA TODA: 1
over [INAUDIBLE].
• 85:15 - 85:18
I love this one, OK?
• 85:18 - 85:20
Don't tell me what I
want to [INAUDIBLE].
• 85:20 - 85:22
I'm just kidding.
• 85:22 - 85:24
[INAUDIBLE] squared times--
• 85:24 - 85:25
STUDENT: 2x.
• 85:25 - 85:26
MAGDALENA TODA: 2x, good.
• 85:26 - 85:28
• 85:28 - 85:28
The same thing.
• 85:28 - 85:34
• 85:34 - 85:37
Times 2y.
• 85:37 - 85:42
• 85:42 - 85:43
OK.
• 85:43 - 85:46
I want to tell you something
that I will repeat.
• 85:46 - 85:49
But you will see it
all through the course.
• 85:49 - 85:52
There is a certain
notion that Alexander,
• 85:52 - 85:54
who is not talking--
I'm just kidding,
• 85:54 - 85:58
you can talk-- he
• 85:58 - 86:03
until a few sections from now.
• 86:03 - 86:05
But I'd like to
anticipate a little bit.
• 86:05 - 86:08
a function, wherever
• 86:08 - 86:15
the partial derivatives exist,
with the partial derivative--
• 86:15 - 86:21
that is, f sub x
and f sub y exist--
• 86:21 - 86:27
I'm going to have that
delta f-- nabla f.
• 86:27 - 86:29
nabla is a [INAUDIBLE].
• 86:29 - 86:34
Nable f at xy represents what?
• 86:34 - 86:35
The vector.
• 86:35 - 86:37
• 86:37 - 86:39
And I know you love vectors.
• 86:39 - 86:46
And that's why I'm going back
to the vector notation f sub x
• 86:46 - 86:52
at xy times i, i being
the standard vector i
• 86:52 - 86:59
unit along the x axis,
f sub y at xy times j.
• 86:59 - 87:03
STUDENT: So it's just like
the notation of [INAUDIBLE]?
• 87:03 - 87:05
MAGDALENA TODA: Just
the vector notation.
• 87:05 - 87:08
How else could I write it?
• 87:08 - 87:13
Angular bracket, f sub x x
at xy, comma, f sub y at xy.
• 87:13 - 87:17
And you know-- people who
saw my videos, colleagues
• 87:17 - 87:20
who teach Calc 3
at the same time
• 87:20 - 87:25
said I have a tendency of not
going by the book notations
• 87:25 - 87:28
all the time, and just give you
the [? round ?] parentheses.
• 87:28 - 87:29
It's OK.
• 87:29 - 87:31
I mean, different books,
different notations.
• 87:31 - 87:35
But what I mean is to represent
the vector in the standard way
• 87:35 - 87:37
[INAUDIBLE].
• 87:37 - 87:38
All right.
• 87:38 - 87:39
OK.
• 87:39 - 87:42
Can you have this
notion for something
• 87:42 - 87:45
like a function of
three variables?
• 87:45 - 87:46
Absolutely.
• 87:46 - 87:48
Now I'll give you an easy one.
• 87:48 - 87:51
Suppose that you have
x-squared plus y-squared
• 87:51 - 87:54
plus z-squared equals 1.
• 87:54 - 88:00
And that is called-- let's
call it names-- f of x, y, z.
• 88:00 - 88:17
at any point x, y, z for f.
• 88:17 - 88:21
Find the meaning of that
• 88:21 - 88:29
the geometric meaning of it.
• 88:29 - 88:34
For this case, not in
general, for this case.
• 88:34 - 88:36
So you say, wait,
wait, Magdalena.
• 88:36 - 88:39
A-dah-dah, you're confusing me.
• 88:39 - 88:40
• 88:40 - 88:40
Hmm.
• 88:40 - 88:43
Depends on how many
variables you have.
• 88:43 - 88:48
So you have to show a vector
whose coordinates represent
• 88:48 - 88:51
the partial derivatives with
respect to all the variables.
• 88:51 - 88:56
If I have n variables, I have
f sub x1 comma f sub x2 comma
• 88:56 - 88:59
f sub x3 comma f
sub xn, and stop.
• 88:59 - 89:00
Yes, sir.
• 89:00 - 89:04
STUDENT: If the formula
was just f of xy,
• 89:04 - 89:05
wouldn't that be implicit?
• 89:05 - 89:07
MAGDALENA TODA:
That is implicit.
• 89:07 - 89:09
That's exactly what I meant.
• 89:09 - 89:12
What's the geometric
meaning of this animal?
• 89:12 - 89:14
Forget about the left hand side.
• 89:14 - 89:16
I'm going to clean it quickly.
• 89:16 - 89:17
What is that animal?
• 89:17 - 89:20
That is a hippopotamus.
• 89:20 - 89:20
What is that?
• 89:20 - 89:22
STUDENT: It's a sphere.
• 89:22 - 89:23
MAGDALENA TODA: It's a sphere.
• 89:23 - 89:25
But what kind of sphere?
• 89:25 - 89:28
Center 0, 0, 0 with radius 1.
• 89:28 - 89:30
What do we call that?
• 89:30 - 89:31
Unit sphere.
• 89:31 - 89:34
Do you know what notation
that mathematicians
• 89:34 - 89:37
use for that object?
• 89:37 - 89:40
You don't know but I'll
tell you. s1 is the sphere.
• 89:40 - 89:42
We have s2, I'm
sorry, the sphere
• 89:42 - 89:45
of dimension 2, which
means the surface.
• 89:45 - 89:47
s1 is the circle.
• 89:47 - 89:49
s1 is a circle.
• 89:49 - 89:52
s2 is a sphere.
• 89:52 - 89:55
So what is this number
here for a mathematician?
• 89:55 - 89:59
That's the dimension of
that kind of manifold.
• 89:59 - 90:02
So if I have just a
circle, we call it s1
• 90:02 - 90:06
because there is only a one
independent variable, which
• 90:06 - 90:08
is time, and we parameterize.
• 90:08 - 90:09
Why go clockwise?
• 90:09 - 90:10
Shame on me.
• 90:10 - 90:12
Go counterclockwise.
• 90:12 - 90:13
All right.
• 90:13 - 90:14
That's s1.
• 90:14 - 90:16
For s2, I have two
degrees of freedom.
• 90:16 - 90:19
It's a surface.
• 90:19 - 90:23
On earth, what are those
two degrees of freedom?
• 90:23 - 90:26
It's a riddle.
• 90:26 - 90:27
No extra credit.
• 90:27 - 90:30
STUDENT: The latitude
and longitude?
• 90:30 - 90:32
MAGDALENA TODA: Who said it?
• 90:32 - 90:34
Who said it first?
• 90:34 - 90:35
STUDENT: [INAUDIBLE].
• 90:35 - 90:40
MAGDALENA TODA: How many of
you said it at the same time?
• 90:40 - 90:41
Alexander said it.
• 90:41 - 90:43
STUDENT: I know there
was one other person.
• 90:43 - 90:44
I wasn't the only one.
• 90:44 - 90:45
STUDENT: I didn't.
• 90:45 - 90:47
• 90:47 - 90:49
STUDENT: [INAUDIBLE], sorry.
• 90:49 - 90:50
[INTERPOSING VOICES]
• 90:50 - 90:52
MAGDALENA TODA: I
don't have enough.
• 90:52 - 90:54
STUDENT: I'll take
the credit for it.
• 90:54 - 90:56
MAGDALENA TODA:
[INAUDIBLE] extra credit.
• 90:56 - 90:59
OK, you choose.
• 90:59 - 91:00
These are good.
• 91:00 - 91:02
They are Valentine's hearts,
chocolate [INAUDIBLE].
• 91:02 - 91:05
• 91:05 - 91:05
Wilson.
• 91:05 - 91:09
• 91:09 - 91:12
I heard you saying Wilson.
• 91:12 - 91:13
I have more.
• 91:13 - 91:14
I have more.
• 91:14 - 91:17
These are cough drops,
so I'm [INAUDIBLE].
• 91:17 - 91:20
You set it right
next time, Alexander.
• 91:20 - 91:22
STUDENT: [INAUDIBLE].
• 91:22 - 91:23
MAGDALENA TODA: OK.
• 91:23 - 91:24
Anybody else?
• 91:24 - 91:26
Anybody needing cough drops?
• 91:26 - 91:27
OK.
• 91:27 - 91:28
I'll leave them here.
• 91:28 - 91:29
Just let me see.
• 91:29 - 91:32
Do I have more chocolate?
• 91:32 - 91:33
Eh, next time.
• 91:33 - 91:35
I'm going to get some
before-- we have-- we
• 91:35 - 91:37
need before Valentine's, right?
• 91:37 - 91:38
So it's Thursday.
• 91:38 - 91:41
I'm going to bring
you a lot more.
• 91:41 - 91:47
So in that case, what
• 91:47 - 91:48
An x, y, z.
• 91:48 - 91:48
Aha.
• 91:48 - 91:50
I have three variables.
• 91:50 - 91:53
• 91:53 - 91:56
I can write it as a
bracket, angular notation.
• 91:56 - 91:58
Am I right?
• 91:58 - 92:03
Or I can write it 2xi
plus 2ij plus 2zk.
• 92:03 - 92:07
Can anybody tell me why?
• 92:07 - 92:10
What in the world are
these, 2x, 2y, 2z?
• 92:10 - 92:12
STUDENT: Those are the
partial derivatives.
• 92:12 - 92:14
MAGDALENA TODA: They are
exactly the partial derivatives
• 92:14 - 92:18
with respect to x, with respect
to y, with respect to z.
• 92:18 - 92:19
Does this have a
geometric meaning?
• 92:19 - 92:21
I don't know.
• 92:21 - 92:22
I have to draw.
• 92:22 - 92:24
And maybe when I
draw, I get an idea.
• 92:24 - 92:29
• 92:29 - 92:32
Is this a unit vector?
• 92:32 - 92:32
Uh-uh.
• 92:32 - 92:34
It's not.
• 92:34 - 92:36
Nabla s, right.
• 92:36 - 92:36
In a way it is.
• 92:36 - 92:38
It's not a unit vector.
• 92:38 - 92:41
But if I were to
[? uniterize ?] it--
• 92:41 - 92:44
and you know very well what it
means to [? uniterize it ?].
• 92:44 - 92:45
It means to--
• 92:45 - 92:46
STUDENT: Divide it by--
• 92:46 - 92:47
MAGDALENA TODA: Divide
it by its magnitude
• 92:47 - 92:51
and make it a unit vector
that would have a meaning.
• 92:51 - 92:52
This is the sphere.
• 92:52 - 92:56
• 92:56 - 92:58
What if I make like this?
• 92:58 - 93:04
n equals nabla f over
a magnitude of f.
• 93:04 - 93:10
And what is the meaning
of that going to be?
• 93:10 - 93:12
Can you tell me what
I'm going to get here?
• 93:12 - 93:19
• 93:19 - 93:25
compute the magnitude
• 93:25 - 93:29
and divide by the magnitude,
and you have exactly 15 seconds
• 93:29 - 93:31
to tell me what it is.
• 93:31 - 93:33
STUDENT: [INAUDIBLE].
• 93:33 - 93:34
MAGDALENA TODA:
[? Ryan, ?] [? Ryan, ?]
• 93:34 - 93:36
you are in a Twilight Zone.
• 93:36 - 93:40
But I'm sure once I tell you,
once I tell you, [INAUDIBLE].
• 93:40 - 93:42
STUDENT: 1 divided by
the square root of 2
• 93:42 - 93:43
for the [? i controller. ?]
• 93:43 - 93:44
STUDENT: [INAUDIBLE].
• 93:44 - 93:48
• 93:48 - 93:50
MAGDALENA TODA: Well, OK.
• 93:50 - 93:51
Say it again, somebody.
• 93:51 - 93:53
STUDENT: x plus y plus z.
• 93:53 - 93:58
MAGDALENA TODA: xi plus yj
plus zk, not x plus x, y,
• 93:58 - 94:00
z because that
would be a mistake.
• 94:00 - 94:03
It would be a scalar function.
[INAUDIBLE] has to be a vector.
• 94:03 - 94:07
If I am to draw this vector,
how am I going to draw it?
• 94:07 - 94:10
Well, this is the
position vector.
• 94:10 - 94:11
Say it again.
• 94:11 - 94:13
This is the position vector.
• 94:13 - 94:16
When I have a point on this
stinking earth, whatever
• 94:16 - 94:21
it is, x, y, z, the
position vector is x, y, z.
• 94:21 - 94:26
It's xi plus yj plus zk.
• 94:26 - 94:29
I have this identification
between the point
• 94:29 - 94:30
and the vector.
• 94:30 - 94:30
This is our vector.
• 94:30 - 94:33
So I'm going to draw these
needles, all these needles,
• 94:33 - 94:42
all these vectors whose tips
are exactly on the sphere.
• 94:42 - 94:43
So why?
• 94:43 - 94:44
You say, OK.
• 94:44 - 94:46
I understand that is
the position vector,
• 94:46 - 94:49
but why did you put an n here?
• 94:49 - 94:53
that gets a cough drops.
• 94:53 - 94:55
STUDENT: [INAUDIBLE].
• 94:55 - 94:56
MAGDALENA TODA: Because that is?
• 94:56 - 94:58
STUDENT: The normal
to the surface.
• 94:58 - 94:59
MAGDALENA TODA: You get a--
• 94:59 - 95:01
STUDENT: Yeah, cough drop.
• 95:01 - 95:02
MAGDALENA TODA: Two of them.
• 95:02 - 95:03
STUDENT: Aw, yeah.
• 95:03 - 95:04
MAGDALENA TODA: All right.
• 95:04 - 95:08
So that's the normal
to the surface, which
• 95:08 - 95:11
would be a continuation
of the position vector.
• 95:11 - 95:12
You see, guys?
• 95:12 - 95:14
So imagine you take
• 95:14 - 95:16
This is the sphere.
• 95:16 - 95:18
It's like an egg.
• 95:18 - 95:21
And these tips
are on the sphere.
• 95:21 - 95:25
If you continue from
sitting on the sphere,
• 95:25 - 95:29
colinear to that,
• 95:29 - 95:31
that would be the
normal to the sphere.
• 95:31 - 95:36
So in topology, we
have a name for that.
• 95:36 - 95:39
We call that the hairy ball.
• 95:39 - 95:42
The hairy ball in
mathematics, I'm not kidding,
• 95:42 - 95:44
it's a concentrated notations.
• 95:44 - 95:48
courses, if you're
• 95:48 - 95:50
student in mathematics,
• 95:50 - 95:52
or you want to do a
dual degree or whatever,
• 95:52 - 95:56
you're going to see the hairy
ball, all those normal vectors
• 95:56 - 95:59
of length 1.
• 95:59 - 96:02
It's also called
the normal field.
• 96:02 - 96:05
Ibragimov, because he
• 96:05 - 96:09
is in this kind of field
theory, [INAUDIBLE] normal field
• 96:09 - 96:10
to a surface.
• 96:10 - 96:13
But for the topologists
or geometers,
• 96:13 - 96:15
they say, oh, that's
the hairy ball.
• 96:15 - 96:19
So if you ask him what the
hairy ball is, he will say,
• 96:19 - 96:22
why are you talking
nonsense to me?
• 96:22 - 96:23
Right.
• 96:23 - 96:24
Exactly.
• 96:24 - 96:31
So here's where we stopped
our intrusion in chapter 11.
• 96:31 - 96:33
It's going to be as
fun as it was today
• 96:33 - 96:34
with these partial derivatives.
• 96:34 - 96:36
You're going to love them.
• 96:36 - 96:40
You have a lot of computations
like the ones we did today.
• 96:40 - 96:43
Let's go back to
something you hated,
• 96:43 - 96:46
which is the parameterizations.
• 96:46 - 96:49
So one of you--
no, three of you--
• 96:49 - 96:52
problem like the one
• 96:52 - 96:54
with the parameterization
of a circle.
• 96:54 - 96:58
But now I have to pay
attention to the data
• 96:58 - 97:00
that I come up with.
• 97:00 - 97:14
So write the parameterization
• 97:14 - 97:17
• 97:17 - 97:21
Do you want specific
data or you want letters?
• 97:21 - 97:22
STUDENT: [INAUDIBLE].
• 97:22 - 97:26
• 97:26 - 97:26
MAGDALENA TODA: OK.
• 97:26 - 97:30
Let's do it [INAUDIBLE] r,
and then I'll give an example.
• 97:30 - 97:43
And center x0, y0 in plane
where-- what is the point?
• 97:43 - 97:57
Where is the particle
moving for time t equals 0?
• 97:57 - 97:59
Where is it located?
• 97:59 - 98:00
All right.
• 98:00 - 98:03
So review.
• 98:03 - 98:16
We had frame that we always
picked at the origin.
• 98:16 - 98:23
That was bad because we could
pick x0, y0 as a center,
• 98:23 - 98:25
and that has a separate radius.
• 98:25 - 98:32
• 98:32 - 98:39
And now, they want me to write
a parameterization of a circle.
• 98:39 - 98:41
How do you achieve it?
• 98:41 - 98:49
You say the circle is x minus x0
squared plus y minus y0 squared
• 98:49 - 98:51
equals r squared.
• 98:51 - 98:54
And one of you
• 98:54 - 98:57
and that was a good
question-- you said, come on.
• 98:57 - 98:59
Look, it was [INAUDIBLE].
• 98:59 - 99:03
So you said, I was
quite good in math.
• 99:03 - 99:04
I was smart.
• 99:04 - 99:10
Why didn't I know the equations,
the parametric equations,
• 99:10 - 99:11
or even this?
• 99:11 - 99:14
I'll tell you why.
• 99:14 - 99:16
This used to be
covered in high school.
• 99:16 - 99:18
It's something called
college algebra.
• 99:18 - 99:21
either trigonometry
• 99:21 - 99:22
or college algebra.
• 99:22 - 99:25
analytic geometry.
• 99:25 - 99:26
This is analytic geometry.
• 99:26 - 99:29
It's the same chapter
in which you guys
• 99:29 - 99:34
covered conics, [INAUDIBLE],
ellipse, [INAUDIBLE], parabola.
• 99:34 - 99:36
It's no longer covered
in most high schools.
• 99:36 - 99:37
• 99:37 - 99:40
The teachers told
me that we reduced
• 99:40 - 99:42
the geometric
applications a lot,
• 99:42 - 99:48
according to the general
standards that are imposed.
• 99:48 - 99:52
That's a pity, because you
really need this in college.
• 99:52 - 99:53
All right.
• 99:53 - 99:56
So how do you come up
with a parameterization?
• 99:56 - 100:01
You say, I would like to
parameterize in such way
• 100:01 - 100:03
that this would be
easy to understand
• 100:03 - 100:06
this for Pythagorean theorem.
• 100:06 - 100:07
Oh, OK.
• 100:07 - 100:10
So what is the Pythagorean
theorem telling me?
• 100:10 - 100:14
It's telling you that if you are
in a unit circle practically,
• 100:14 - 100:19
then this is cosine and
theta and this is sine theta,
• 100:19 - 100:22
and the sum of
cosine theta squared
• 100:22 - 100:24
plus sine theta squared is 1.
• 100:24 - 100:27
This is 1, so that is
the Pythagorean theorem
• 100:27 - 100:29
[INAUDIBLE].
• 100:29 - 100:38
So xy plus x0 should be cosine
of theta times an R. Why an R?
• 100:38 - 100:42
Because I want, when I square,
I want the R squared up.
• 100:42 - 100:46
And here, this guy inside
will be our sine [? thing. ?]
• 100:46 - 100:48
Am I going to be in good shape?
• 100:48 - 100:51
Yes, because when I
square this fellow squared
• 100:51 - 100:55
plus this fellow squared will
give me exactly R squared.
• 100:55 - 100:58
And here is my
[INAUDIBLE] smiley face.
• 100:58 - 101:01
So I want to understand
what I'm doing.
• 101:01 - 101:05
x minus x0 must
be R cosine theta.
• 101:05 - 101:09
y minus y0 is R sine theta.
• 101:09 - 101:14
Theta in general is an
angular velocity, [INAUDIBLE].
• 101:14 - 101:17
But it's also time, right?
• 101:17 - 101:19
It has the meaning
of time parameter.
• 101:19 - 101:23
So when we wrote those--
and some of you are bored,
• 101:23 - 101:26
but I think it's not
going to harm anybody
• 101:26 - 101:27
that I do this again.
• 101:27 - 101:36
R cosine of t plus x0 y is R
sine t plus x0, or plus y0.
• 101:36 - 101:41
Now note, all those
examples in web work,
• 101:41 - 101:44
they were not very imaginative.
• 101:44 - 101:48
They didn't mean for
you to try other things.
• 101:48 - 101:54
Like if one would put here
cosine of 5t or sine of 5t,
• 101:54 - 101:57
that person would move five
times faster on the circle.
• 101:57 - 102:00
at 2 pi, in time 2 pi,
• 102:00 - 102:03
they would be there
in time 2 pi over 5.
• 102:03 - 102:07
All the examples-- and each of
you, it was randomized somehow.
• 102:07 - 102:10
Each of you has a
different data set.
• 102:10 - 102:12
Different R,
different x0 with 0,
• 102:12 - 102:16
and a different place where
the particle is moving.
• 102:16 - 102:19
But no matter what
they gave you,
• 102:19 - 102:22
it's a response to
the same problem.
• 102:22 - 102:27
And at time t equals
0, you have M. Do
• 102:27 - 102:29
you want me to call it M0?
• 102:29 - 102:33
Yes, from my initial-- M0.
• 102:33 - 102:41
For t equals 0, you're
going to have R plus x0.
• 102:41 - 102:45
And for t equals 0, you have y0.
• 102:45 - 102:50
So for example, Ryan had-- Ryan,
I don't remember what you had.
• 102:50 - 102:54
You had some where theta R was--
• 102:54 - 102:55
STUDENT: 4 and 8.
• 102:55 - 102:57
MAGDALENA TODA: 7.
• 102:57 - 102:58
You, what did you have?
• 102:58 - 103:00
STUDENT: No, R was 7
and x was 3, y was 1.
• 103:00 - 103:04
MAGDALENA TODA: R
was 7 and x0 was--
• 103:04 - 103:06
STUDENT: 3, 1.
• 103:06 - 103:12
MAGDALENA TODA: 3, 1 was x0, y0
so in that case, the point they
• 103:12 - 103:16
gave here was 7 plus 3.
• 103:16 - 103:17
Am I right, Ryan?
• 103:17 - 103:18
You can always check.
• 103:18 - 103:18
I remember.
• 103:18 - 103:22
It was 10 and God
knows, and 10 and 1.
• 103:22 - 103:26
So all of the data that
• 103:26 - 103:30
was created so that you
have these equations.
• 103:30 - 103:36
And at time 0, you were exactly
at the time t equals 0 replaced
• 103:36 - 103:37
the t.
• 103:37 - 103:38
All right.
• 103:38 - 103:39
OK.
• 103:39 - 103:40
STUDENT: What's the M0?
• 103:40 - 103:42
What is--
• 103:42 - 103:45
MAGDALENA TODA: M0
is Magdalena times 0.
• 103:45 - 103:47
I don't know.
• 103:47 - 103:51
I mean, it's the
point where you are.
• 103:51 - 103:55
I couldn't come up
with a better name.
• 103:55 - 104:02
So I'm going to
erase here and I'll
• 104:02 - 104:08
get to another problem, which
gave you guys a big headache.
• 104:08 - 104:17
And it's not so hard, but this
is the computational problem,
• 104:17 - 104:18
very pretty in itself.
• 104:18 - 104:25
• 104:25 - 104:36
[INAUDIBLE] cosine t i plus
e to the 3t sine t j plus e
• 104:36 - 104:37
to the 3tk.
• 104:37 - 104:40
• 104:40 - 104:44
And I think this was more or
less in everybody's homework
• 104:44 - 104:45
the same.
• 104:45 - 104:52
There's a position vector
given as parameterized form.
• 104:52 - 104:54
So since you love
parameterization so much,
• 104:54 - 105:01
I'm going to remind you what
that means for x and y and zr.
• 105:01 - 105:03
And what did they want from you?
• 105:03 - 105:08
I forget what number of
the problem that was.
• 105:08 - 105:16
They wanted the length of
the arc of a curve from t
• 105:16 - 105:19
equals-- I don't know.
• 105:19 - 105:20
STUDENT: 2 to 5.
• 105:20 - 105:21
MAGDALENA TODA: 2 to 5.
• 105:21 - 105:22
Thank you.
• 105:22 - 105:24
[INAUDIBLE] t equals 5.
• 105:24 - 105:30
So this is the beginning and the
end of the curve, the beginning
• 105:30 - 105:32
and the end of a curve.
• 105:32 - 105:36
So what is that going
to be [INAUDIBLE]?
• 105:36 - 105:40
How does [INAUDIBLE],
which we have
• 105:40 - 105:47
to write down 2 to 5
magnitude of r prime at t, dt.
• 105:47 - 105:50
• 105:50 - 105:53
And I don't know.
• 105:53 - 105:57
But I want to review this
because-- so what in the world?
• 105:57 - 106:00
Maybe I put this
on the midterm or I
• 106:00 - 106:04
make it a little bit easier,
but the same what I don't like,
• 106:04 - 106:05
it's time consuming.
• 106:05 - 106:08
But I can give you
something a lot easier
• 106:08 - 106:11
that tests the
concept, the idea, not
• 106:11 - 106:13
the computational power.
• 106:13 - 106:20
So r prime of t here with
a little bit of attention,
• 106:20 - 106:25
of course, most of you
computing this correctly.
• 106:25 - 106:28
You are just a little bit scared
of what happened after that,
• 106:28 - 106:30
and you should not be scared
because now I'll tell you
• 106:30 - 106:33
why you shouldn't be scared.
• 106:33 - 106:35
Chain rule, product rule.
• 106:35 - 106:38
So I have first prime--
• 106:38 - 106:39
STUDENT: 3.
• 106:39 - 106:42
MAGDALENA TODA: 3 into the
3e second and [? time ?]
• 106:42 - 106:47
cosine t plus-- I'm
going to do that later.
• 106:47 - 106:48
I know what you're thinking.
• 106:48 - 106:50
STUDENT: e 3t.
• 106:50 - 106:53
MAGDALENA TODA: e to
the 3t minus sine.
• 106:53 - 106:56
I'm not worried
• 106:56 - 106:57
I'll take care of that later.
• 106:57 - 106:58
Times i.
• 106:58 - 107:01
• 107:01 - 107:03
Now with your
permission-- when you
• 107:03 - 107:08
say, why is she not writing the
whole thing in continuation?
• 107:08 - 107:09
Because I don't want to.
• 107:09 - 107:10
No.
• 107:10 - 107:13
Because I want to help
you see what's going on.
• 107:13 - 107:16
You do the same kind of stuff
for this individual one.
• 107:16 - 107:18
I want to put it
right underneath.
• 107:18 - 107:21
If I put it right underneath,
it's going to [? agree ?].
• 107:21 - 107:24
Otherwise it's not
going to [? agree ?].
• 107:24 - 107:32
E to the 3t times sine t
plus e to the 3t cosine t.
• 107:32 - 107:34
You didn't have a
problem because you
• 107:34 - 107:36
know how to differentiate.
• 107:36 - 107:41
You started having the
problem from this point on.
• 107:41 - 107:44
3 into the 3tk.
• 107:44 - 107:47
The problem came when
you were supposed
• 107:47 - 107:56
to identify the coordinates and
square them and squeeze them
• 107:56 - 107:57
under the same square root.
• 107:57 - 108:01
And that drove you crazy
when you have enough.
• 108:01 - 108:04
Let me put the minus here to
make it more obvious what's
• 108:04 - 108:06
going to happen.
• 108:06 - 108:08
When you're going
to have problems
• 108:08 - 108:10
like that in
differential equations,
• 108:10 - 108:15
you better have the eye
for it, [INAUDIBLE].
• 108:15 - 108:19
You should be able to recognize
this is like a pattern.
• 108:19 - 108:27
Have you seen the
movie A Beautiful Mind?
• 108:27 - 108:27
STUDENT: Yeah.
• 108:27 - 108:29
MAGDALENA TODA:
OK, so Nash, when
• 108:29 - 108:34
he was writing with the finger
on everything, on the walls
• 108:34 - 108:40
at Princeton, on the window,
he was thinking of patterns.
• 108:40 - 108:42
He's actually
trying to-- and it's
• 108:42 - 108:44
hard to visualize
without drawing,
• 108:44 - 108:48
but this is what most of us
recognize all the time when
• 108:48 - 108:51
a mathematician writes
down some computations
• 108:51 - 108:53
in a different way.
• 108:53 - 108:58
All we hope for is to get a
few steps behind that board
• 108:58 - 109:00
and see a pattern.
• 109:00 - 109:02
And when you do that,
you see the pattern.
• 109:02 - 109:06
This is an a minus b
and that's an a plus b.
• 109:06 - 109:09
And then you say, OK, if
I'm going to square them,
• 109:09 - 109:11
what's going to happen?
• 109:11 - 109:15
When you square an a minus
b and you square an a plus b
• 109:15 - 109:19
and you have this giggly
guy there-- leave him there.
• 109:19 - 109:22
He's having too much fun.
• 109:22 - 109:28
You actually develop these
guys and you put them one
• 109:28 - 109:32
under the other
and say wow, what
• 109:32 - 109:34
a beautiful simplification.
• 109:34 - 109:37
When I'm going to
• 109:37 - 109:40
this thing in the middle
will simply will cancel out,
• 109:40 - 109:45
but the a squared will double
and the b squared will double.
• 109:45 - 109:47
And that's the beauty
of seeing pattern.
• 109:47 - 109:51
You see how there is
something symmetric and magic
• 109:51 - 109:56
in mathematics that make
• 109:56 - 110:01
And that allows you to compress
• 110:01 - 110:06
seemed to be a mess
into something that's
• 110:06 - 110:09
more easily expressed.
• 110:09 - 110:11
So when you're going
to compute this r
• 110:11 - 110:18
prime of t magic absolute
value of the magnitude, that's
• 110:18 - 110:22
going to be square root of--
• 110:22 - 110:25
[INAUDIBLE], I hate writing
and rewriting the whole thing
• 110:25 - 110:29
squared plus the whole thing
squared plus this squared.
• 110:29 - 110:33
If I love to write so much,
I'd be in humanities and not
• 110:33 - 110:35
in mathematics.
• 110:35 - 110:41
So as a mathematician, how
am I going to write that?
• 110:41 - 110:44
As a mathematician, I'm going
to use some sort of-- like the U
• 110:44 - 110:45
substitution.
• 110:45 - 110:49
So I say, I call this Mr.
A, and I call this Mr. B.
• 110:49 - 110:51
And that's A minus B,
and that's A plus B.
• 110:51 - 110:54
And that's somebody else.
• 110:54 - 110:57
So when I square
the first guy, and I
• 110:57 - 111:01
square the second component, and
I square the third component,
• 111:01 - 111:10
I'm going to get what?
• 111:10 - 111:16
Square root of 2A
squared plus 2B squared.
• 111:16 - 111:19
Because I know that
these are the first two.
• 111:19 - 111:21
This guy squared
plus this guy squared
• 111:21 - 111:23
is going to be
exactly 2A squared
• 111:23 - 111:26
plus 2B squared,
nothing in the middle.
• 111:26 - 111:29
These guys cancel out.
• 111:29 - 111:31
STUDENT: A and B
are not the same.
• 111:31 - 111:34
• 111:34 - 111:42
MAGDALENA TODA: Well,
yeah, you're right.
• 111:42 - 111:46
Let me call-- you're
right, this is the same,
• 111:46 - 111:48
but these are different.
• 111:48 - 111:52
So let me call them
A prime plus B prime.
• 111:52 - 111:54
No, that's derivative.
• 111:54 - 111:56
Let me call them C
and D-- very good,
• 111:56 - 112:04
thank you-- C squared
plus 2CD plus D squared.
• 112:04 - 112:06
• 112:06 - 112:08
But the principle is the same.
• 112:08 - 112:11
So I'm going to have A
squared plus C squared.
• 112:11 - 112:13
This goes away.
• 112:13 - 112:15
Why?
• 112:15 - 112:19
Because this times that is
the same as this times that.
• 112:19 - 112:20
Say it again.
• 112:20 - 112:23
If we look in the
middle, the middle term
• 112:23 - 112:28
will have 3e to the 3t cosine
t times e to the 3t sine t.
• 112:28 - 112:33
Middle term here is 3e to the
3t e to the 3t sine and cosine.
• 112:33 - 112:36
So they will cancel
out, this and that.
• 112:36 - 112:40
So here I have the
sum of the square of A
• 112:40 - 112:46
plus the square
of C. And here I'm
• 112:46 - 112:51
going to have the square
of B plus the square of D.
• 112:51 - 112:54
OK, now when I square this
and that, what do I get?
• 112:54 - 112:57
• 112:57 - 113:01
The beauty of that-- let me
write it down then explicitly.
• 113:01 - 113:07
9e to the 3t cosine squared
t remains from this guy.
• 113:07 - 113:09
Plus from the square
of that, we'll
• 113:09 - 113:20
have 9e to the 3t-- no, just 3,
9 to the 6t, 9 to the 6t sine
• 113:20 - 113:22
squared.
• 113:22 - 113:23
So I take this guy.
• 113:23 - 113:24
I square it.
• 113:24 - 113:25
I take this guy.
• 113:25 - 113:27
I square it.
• 113:27 - 113:30
The middle terms will
disappear, thank god.
• 113:30 - 113:33
Then I have this guy, I square
it, that guy, I square it,
• 113:33 - 113:35
good.
• 113:35 - 113:41
Plus another parenthesis-- e
to the 6t sine squared t plus e
• 113:41 - 113:44
to the 6t cosine squared t.
• 113:44 - 113:47
• 113:47 - 113:50
So even if they don't
double because they're not
• 113:50 - 113:53
the same thing, what
is the principle
• 113:53 - 113:54
that will make my life easier?
• 113:54 - 113:59
The same pattern
of simplification.
• 113:59 - 114:01
What is that same pattern
of simplification?
• 114:01 - 114:04
Look at the beauty
of this guy and look
• 114:04 - 114:05
at the beauty of this guy.
• 114:05 - 114:07
And then there is
something missing,
• 114:07 - 114:13
the happy guy that was quiet
because I told him to be quiet.
• 114:13 - 114:17
That's 9e to the 6t.
• 114:17 - 114:18
He was there in the corner.
• 114:18 - 114:22
And you had to square this
guy and square this guy
• 114:22 - 114:26
and square this guy and
• 114:26 - 114:28
Now what is the pattern?
• 114:28 - 114:35
The pattern is 9e to the 6t
with 9e to the 6t, same guy.
• 114:35 - 114:38
The orange guys-- that's
why I love the colors.
• 114:38 - 114:41
Cosine squared cosine
squared will be 1.
• 114:41 - 114:47
Another pattern like that, I
have e to the 6t, to the 6t,
• 114:47 - 114:52
and the same happy guys sine
squared t, sine squared t,
• 114:52 - 114:55
• 114:55 - 115:01
So all in all, this mess
is not a mess anymore.
• 115:01 - 115:11
So it becomes 9e to the 6t plus
e to the 6t plus 9e to the 6t.
• 115:11 - 115:13
Are you guys with me?
• 115:13 - 115:18
All right, now how many
e to the 6t's do we have?
• 115:18 - 115:26
9 plus 9 plus 1, 19, square
root of 19 e to the 6t.
• 115:26 - 115:30
So when we integrate,
we go integral
• 115:30 - 115:33
from 2 to 5 square root of 19.
• 115:33 - 115:35
Kick him out of your life.
• 115:35 - 115:37
He's just making
• 115:37 - 115:40
And then you have square root
of e to the 6t e to the 3t.
• 115:40 - 115:43
• 115:43 - 115:48
So after you kick
the guy out, you
• 115:48 - 115:55
have e to the 3t divided
by 3 between t equals 2
• 115:55 - 115:58
and t equals 5.
• 115:58 - 116:03
Actually, I took it right off
• 116:03 - 116:06
So if you type this
• 116:06 - 116:12
• 116:12 - 116:13
as being correct.
• 116:13 - 116:18
• 116:18 - 116:24
On the exam, do not
expect anything that long.
• 116:24 - 116:27
The idea of simplifying
these patterns
• 116:27 - 116:32
by finding the sine cosine, sine
squared plus cosine squared is
• 116:32 - 116:33
1, is still going to be there.
• 116:33 - 116:36
But don't expect
anything that long.
• 116:36 - 116:43
Also, don't expect-- once
you get to this state,
• 116:43 - 116:45
• 116:45 - 116:46
• 116:46 - 116:48
• 116:48 - 116:53
I don't want any approximation
or anything like that.
• 116:53 - 116:54
A few of you did this
with a calculator.
• 116:54 - 116:58
Well, you will not have
calculators in the final.
• 116:58 - 116:59
You are going to
have easy problems.
• 116:59 - 117:03
If you did that
with a calculator,
• 117:03 - 117:05
and you truncated
• 117:05 - 117:11
and if you were within
• 117:11 - 117:12
you were fine.
• 117:12 - 117:15
But some people
approximated too much.
• 117:15 - 117:17
And that's always a problem.
• 117:17 - 117:19
So it's always a good
idea to enter something
• 117:19 - 117:24
like that in WeBWorK.
• 117:24 - 117:27
I said I wouldn't do it
except in the last 20 minutes.
• 117:27 - 117:31
But I wanted to do
something like that.
• 117:31 - 117:34
I want to give you another
example, because you love
• 117:34 - 117:39
parametrization so much it just
occurred to me that it would
• 117:39 - 117:42
maybe, I don't
• 117:42 - 117:47
know-- to give you another
problem similar to this one.
• 117:47 - 117:50
It's not in the book,
but it was cooked up
• 117:50 - 117:54
by one of my colleagues
for his homework.
• 117:54 - 118:03
So I'd like to show it to you.
• 118:03 - 118:06
• 118:06 - 118:10
e to the t i is
a parametrization
• 118:10 - 118:13
of a [INAUDIBLE] space.
• 118:13 - 118:28
Plus e to the minus t j
plus square root of 2 tk.
• 118:28 - 118:36
• 118:36 - 118:37
And how do I know?
• 118:37 - 118:41
Well, one of his
students came to me
• 118:41 - 118:44
with homework.
• 118:44 - 118:51
Well, we don't give help when
it comes from another colleague.
• 118:51 - 118:56
So in the end, the student
went to the tutoring center.
• 118:56 - 118:59
And the tutoring center
helped only in parts.
• 118:59 - 119:01
She came back to me.
• 119:01 - 119:04
So what was the deal here?
• 119:04 - 119:14
Find f prime of t in
the most simplified form
• 119:14 - 119:16
and find the absolute
value r prime of t
• 119:16 - 119:18
in the most simplified form.
• 119:18 - 119:23
• 119:23 - 119:32
And find the length of the
arc of this curve between t
• 119:32 - 119:34
equals 0 and t equals 1.
• 119:34 - 119:37
If this were given
by a physicist,
• 119:37 - 119:40
how would that physicist
reformulate the problem?
• 119:40 - 119:48
He would say-- he or she--
what is the distance travelled
• 119:48 - 119:54
by the particle between
0 seconds and 1 second?
• 119:54 - 119:56
So how do you write that?
• 119:56 - 120:04
Integral from 0 to 1 of
r prime of t [INAUDIBLE].
• 120:04 - 120:06
And you have to do the rest.
• 120:06 - 120:09
• 120:09 - 120:13
So arguably, this is
the Chapter 10 review.
• 120:13 - 120:15
It's very useful for
the midterm exam.
• 120:15 - 120:18
So although we are
just doing this review,
• 120:18 - 120:21
you should not erase
• 120:21 - 120:24
Because I don't like to
put surprise problems
• 120:24 - 120:25
on the midterm.
• 120:25 - 120:29
But if you worked a
certain type of problem,
• 120:29 - 120:31
you may expect
something like that.
• 120:31 - 120:34
Maybe it's different
but in the same spirit.
• 120:34 - 120:38
r prime of t, who's going to
help me with r prime of t?
• 120:38 - 120:41
• 120:41 - 120:44
This fellow-- e to the t.
• 120:44 - 120:47
• 120:47 - 120:50
Negative e to the negative t.
• 120:50 - 120:53
STUDENT: I thought the arc
length was the square root of 1
• 120:53 - 120:56
plus f prime of t squared.
• 120:56 - 120:59
• 120:59 - 121:02
MAGDALENA TODA:
For a plane curve.
• 121:02 - 121:04
OK, let me remind you.
• 121:04 - 121:06
If you have a plane
curve y equals
• 121:06 - 121:12
f of x, then this thing
would become integral from A
• 121:12 - 121:18
to B square root of 1
plus f prime of x dx.
• 121:18 - 121:22
And that, did you do that
• 121:22 - 121:26
How many of you
• 121:26 - 121:28
That was a wonderful
class, wasn't it?
• 121:28 - 121:29
And he taught that.
• 121:29 - 121:31
And of course he
was not supposed
• 121:31 - 121:36
to tell you that was the
speed of a parametric curve.
• 121:36 - 121:39
If you were to
parametrize here, x of t
• 121:39 - 121:44
was t and y of t
would be f of t.
• 121:44 - 121:45
He could have told you.
• 121:45 - 121:46
Maybe he told you.
• 121:46 - 121:47
Maybe you don't remember.
• 121:47 - 121:49
• 121:49 - 121:50
That was Calc II.
• 121:50 - 121:54
Now, coming back here,
I have to list what?
• 121:54 - 121:58
Square root of 2 times
t prime is one k.
• 121:58 - 122:00
Who's going to help
me compute the speed
• 122:00 - 122:02
and put it in a nice formula?
• 122:02 - 122:04
Well, my god--
• 122:04 - 122:05
STUDENT: [INAUDIBLE]
• 122:05 - 122:08
• 122:08 - 122:11
MAGDALENA TODA: Ahh,
you are too smart.
• 122:11 - 122:15
Today you had some what is
that called with caffeine
• 122:15 - 122:17
and vitamins and--
• 122:17 - 122:19
STUDENT: You're
thinking of Red Bull.
• 122:19 - 122:20
MAGDALENA TODA: I know.
• 122:20 - 122:23
That was very nice.
• 122:23 - 122:24
I try to stay away.
• 122:24 - 122:28
What is that called
with the energy booster?
• 122:28 - 122:29
STUDENT: I wouldn't know.
• 122:29 - 122:30
STUDENT: 5-Hour Energy.
• 122:30 - 122:32
MAGDALENA TODA: 5-Hour, OK.
• 122:32 - 122:33
I used to have that.
• 122:33 - 122:37
When I had that, I could
anticipate two steps computing.
• 122:37 - 122:40
Just a joke, Alex,
don't take it up.
• 122:40 - 122:41
Very good observation.
• 122:41 - 122:43
So Alex saw.
• 122:43 - 122:46
He has a premonition.
• 122:46 - 122:49
He can see two steps in advance.
• 122:49 - 122:51
He said, OK, square that.
• 122:51 - 122:53
You have e to the 2t.
• 122:53 - 122:53
Square this.
• 122:53 - 122:56
The minus doesn't matter.
• 122:56 - 123:00
Plus e to the minus
2t, and square that.
• 123:00 - 123:03
Then he saw patterns.
• 123:03 - 123:06
Because he is the
wizard 101 today.
• 123:06 - 123:09
So what is the
witchcraft he performed?
• 123:09 - 123:10
Do you see?
• 123:10 - 123:13
Does anybody else
see the pattern?
• 123:13 - 123:15
[? Nateesh ?] sees the pattern.
• 123:15 - 123:17
Anybody illuminated?
• 123:17 - 123:18
I didn't see it from the start.
• 123:18 - 123:20
You guys saw it faster than me.
• 123:20 - 123:23
minute and a half
• 123:23 - 123:27
when I saw this
for the first time.
• 123:27 - 123:30
Is this a perfect square?
• 123:30 - 123:32
Of who?
• 123:32 - 123:36
e to the t plus e to
the minus 2 squared
• 123:36 - 123:40
is-- anybody else sees the
pattern I don't have candy.
• 123:40 - 123:44
Next time-- Alex,
[INAUDIBLE], anybody else?
• 123:44 - 123:47
Do you now see the
pattern, e to the 2t plus
• 123:47 - 123:51
e to the minus 2t plus
twice the product?
• 123:51 - 123:54
And that's where the student
was having the problem.
• 123:54 - 123:57
Where do you see the product?
• 123:57 - 123:58
The product is 1.
• 123:58 - 124:00
The product is 1 doubled.
• 124:00 - 124:02
So you get 2.
• 124:02 - 124:07
So it's indeed exactly
the perfect square.
• 124:07 - 124:09
So once-- it was a she.
• 124:09 - 124:14
Once she saw the perfect
square, she was so happy.
• 124:14 - 124:17
Because you get square
root of the square.
• 124:17 - 124:20
You get e to the t
plus e to the minus t.
• 124:20 - 124:23
And that's a trivial thing
to integrate that you
• 124:23 - 124:24
have no problem integrating.
• 124:24 - 124:27
It's a positive
function, very beautiful.
• 124:27 - 124:32
The professor who gave this was
Dr. [INAUDIBLE] from Denmark.
• 124:32 - 124:35
He's one of the best
teachers we have.
• 124:35 - 124:41
But he makes up his
homework as far as I know.
• 124:41 - 124:43
I think in the sixth
edition, this edition,
• 124:43 - 124:49
we actually stole his idea,
and we made a problem like that
• 124:49 - 124:51
in the book somewhere.
• 124:51 - 124:55
We doubled the number of
problems more or less.
• 124:55 - 125:01
So if you are to compute
0 to 1 of the speed,
• 125:01 - 125:03
what is the speed?
• 125:03 - 125:06
The speed is this
beautiful thing.
• 125:06 - 125:10
Because you were able
to see the pattern.
• 125:10 - 125:13
If you're not able
to see that, do you
• 125:13 - 125:15
realize it's
impossible, practically,
• 125:15 - 125:18
for you to integrate by hand?
• 125:18 - 125:23
You have to go to a
calculator, Matlab, whatever.
• 125:23 - 125:24
So this is easy.
• 125:24 - 125:29
Why is that easy? e to the t
minus e to the minus t at 1
• 125:29 - 125:32
and at 0-- you compare them.
• 125:32 - 125:36
You get at 1 e minus
e to the minus 1
• 125:36 - 125:41
minus the fundamental theorem
of calc e to the 0 minus
• 125:41 - 125:42
e to the 0.
• 125:42 - 125:44
Well, that's silly.
• 125:44 - 125:45
Why is that silly?
• 125:45 - 125:49
Because I'm going to give it up.
• 125:49 - 125:52
e to the minus 1/e.
• 125:52 - 125:55
And she knew what
• 125:55 - 125:57
But she didn't know why.
• 125:57 - 125:58
So she came back to me.
• 125:58 - 126:03
I don't know how the tutoring
center helped her figure
• 126:03 - 126:03
• 126:03 - 126:06
But she did not
understand the solution.
• 126:06 - 126:09
So I said, I'm not going to
take anymore people coming
• 126:09 - 126:11
from Professor [INAUDIBLE].
• 126:11 - 126:13
I was also told it's not OK.
• 126:13 - 126:17
So don't go to another
professor with homework coming
• 126:17 - 126:18
for me or the other way around.
• 126:18 - 126:21
Because it's not OK.
• 126:21 - 126:25
But you can go to the tutoring
• 126:25 - 126:30
They're open starting 9:00
AM and until around when?
• 126:30 - 126:32
Do you know?
• 126:32 - 126:33
They used to have until 4:00.
• 126:33 - 126:36
But now they're going to
work on an extended schedule
• 126:36 - 126:38
until 8:00 PM.
• 126:38 - 126:40
It's going to be
something crazy.
• 126:40 - 126:44
Now, the thing is, we want
the students to be better,
• 126:44 - 126:49
to do better, to not give
up, to be successful,
• 126:49 - 126:52
top one, two, three.
• 126:52 - 126:54
I'm a little bit
concerned, but maybe I
• 126:54 - 126:57
• 126:57 - 127:00
So I don't know if they managed
to put a security camera
• 127:00 - 127:01
or not.
• 127:01 - 127:05
But having extended
hours may be a problem.
• 127:05 - 127:10
those afternoon hours,
• 127:10 - 127:12
especially if you are busy.
• 127:12 - 127:19
Those late hours will
be a big help for you.
• 127:19 - 127:21
Do you know where it is?
• 127:21 - 127:24
Room 106 over there.
• 127:24 - 127:27
• 127:27 - 127:30
Any other questions related
to this type of problem
• 127:30 - 127:35
or related to anything
else in the material
• 127:35 - 127:39
that maybe I can
give you hints on,
• 127:39 - 127:41
at least the hint I'm
going to give you?
• 127:41 - 127:45
Sometimes I cannot stop, and
I just give the problem away.
• 127:45 - 127:46
I'm not supposed to do that.
• 127:46 - 127:51
• 127:51 - 127:54
Look at your WeBWorK, see what
kind of help I can give you.
• 127:54 - 127:56
You still have a
little bit of time.
• 127:56 - 127:57
STUDENT: [INAUDIBLE]
• 127:57 - 128:01
• 128:01 - 128:05
MAGDALENA TODA: That's
the maximum of what?
• 128:05 - 128:07
It was--
• 128:07 - 128:08
STUDENT: [INAUDIBLE]
• 128:08 - 128:11
• 128:11 - 128:13
MAGDALENA TODA: Was
this the problem?
• 128:13 - 128:15
STUDENT: e to the 2x
or something like that.
• 128:15 - 128:16
MAGDALENA TODA:
Something like that?
• 128:16 - 128:16
I erased it.
• 128:16 - 128:19
STUDENT: You erased
that? [INAUDIBLE].
• 128:19 - 128:21
• 128:21 - 128:23
MAGDALENA TODA: It's
very computational I saw.
• 128:23 - 128:27
But before that, I
saw that seven of you
• 128:27 - 128:29
guys-- you two also did it.
• 128:29 - 128:34
So I wrote-- you have a
brownie waiting for that.
• 128:34 - 128:35
But then I erased it.
• 128:35 - 128:40
STUDENT: You erased the previous
one too in the homework one.
• 128:40 - 128:42
MAGDALENA TODA: Because
that had a bug in it.
• 128:42 - 128:45
That one, the one in the
homework one, had a bug in it.
• 128:45 - 128:47
It only worked for some data.
• 128:47 - 128:50
And for other data
it didn't work.
• 128:50 - 128:54
So every time you find
a bug, you tell me,
• 128:54 - 128:56
and I will tell the programmer
of those problems, who's
• 128:56 - 128:57
really careful.
• 128:57 - 129:02
But one in 1,000 you
are bound to find a bug.
• 129:02 - 129:06
And I'm going to
give you a chocolate
• 129:06 - 129:08
or something for every bug.
• 129:08 - 129:10
And any other questions?
• 129:10 - 129:15
• 129:15 - 129:18
STUDENT: So are you
saying this is too long?
• 129:18 - 129:20
MAGDALENA TODA: Actually,
it's very beautiful.
• 129:20 - 129:24
If you have a calculator,
it's easier to solve it.
• 129:24 - 129:26
You can do it by hand,
write it by hand, also.
• 129:26 - 129:27
But it's a long--
• 129:27 - 129:28
STUDENT: [INAUDIBLE]
• 129:28 - 129:31
• 129:31 - 129:34
MAGDALENA TODA: Right,
so let's do it now
• 129:34 - 129:37
for anybody who wants to stay.
• 129:37 - 129:38
You don't have to stay.
• 129:38 - 129:40
So practicing what you do--
• 129:40 - 129:45
[SIDE CONVERSATIONS]
• 129:45 - 131:55
Title:
TTU Math2450 Calculus3 Sec 11.2 and 11.3
Description:

Functions of several variables: continuity and partial derivatives

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Video Language:
English
 jackie.luft edited English subtitles for TTU Math2450 Calculus3 Sec 11.2 and 11.3