0:00:00.000,0:00:01.894


0:00:01.894,0:00:03.810
MAGDALENA TODA: So what's[br]your general feeling

0:00:03.810,0:00:05.423
about Chapter 11?

0:00:05.423,0:00:06.355
STUDENT: It's OK.

0:00:06.355,0:00:07.355
MAGDALENA TODA: It's OK.

0:00:07.355,0:00:12.185
So functions of[br]two variables are

0:00:12.185,0:00:15.566
to be compared all the time with[br]the functions of one variable.

0:00:15.566,0:00:18.810
Every nothing you[br]have seen in Calc 1

0:00:18.810,0:00:23.983
has a corresponding[br]the motion in Calc 3.

0:00:23.983,0:00:27.780


0:00:27.780,0:00:33.020
So really no questions about[br]theory, concepts, Chapter 11

0:00:33.020,0:00:36.880
concepts, previous concepts?

0:00:36.880,0:00:39.760
Feel free to email[br]me this weekend.

0:00:39.760,0:00:43.370
Don't think it's the[br]weekend because we

0:00:43.370,0:00:47.140
are on a 24/7 availability.

0:00:47.140,0:00:48.970
People, we use WeBWork.

0:00:48.970,0:00:50.520
Not just me, but[br]everybody who uses

0:00:50.520,0:00:54.310
WeBWork is on a[br]24/7 availability,

0:00:54.310,0:00:59.130
answering questions[br]about WeBWork problems.

0:00:59.130,0:01:02.320
Saturday and Sunday is when[br]most of you do the homework.

0:01:02.320,0:01:05.010


0:01:05.010,0:01:09.462
It's convenient for us as well[br]because we are with the family,

0:01:09.462,0:01:11.797
but we don't have many[br]meetings to attend.

0:01:11.797,0:01:14.810
So I'll be happy to[br]answer your questions.

0:01:14.810,0:01:19.384
Last time, we discussed a[br]little bit about preparation

0:01:19.384,0:01:21.598
for The Chain Rule.

0:01:21.598,0:01:23.566
In Calc 3.

0:01:23.566,0:01:34.880
So the chain rule in Calc[br]3 was something really--

0:01:34.880,0:01:40.126
this is section 11.5.

0:01:40.126,0:01:45.549
The preparation[br]was done last time,

0:01:45.549,0:01:50.479
but I'm going to[br]review it a little bit.

0:01:50.479,0:01:54.423
Let's see what we discussed.

0:01:54.423,0:01:57.381
I'm going to split,[br]again, the board in two.

0:01:57.381,0:02:05.842
And I'll say, can we review[br]the notions of The Chain Rule.

0:02:05.842,0:02:11.360
When you start with a[br]variable-- let's say it's time.

0:02:11.360,0:02:20.795
Time going to f of t, which goes[br]into g of f of t by something

0:02:20.795,0:02:22.340
called composition.

0:02:22.340,0:02:26.284
We've done that since we[br]were kids in college algebra.

0:02:26.284,0:02:27.170
What?

0:02:27.170,0:02:28.680
You never took college algebra?

0:02:28.680,0:02:30.890
Except in high school, you[br]took high school algebra,

0:02:30.890,0:02:33.210
most of you.

0:02:33.210,0:02:35.324
So what did you do in[br]high school algebra?

0:02:35.324,0:02:38.280
We said g composed with l.

0:02:38.280,0:02:40.760
This is a composition[br]of two functions.

0:02:40.760,0:02:46.360
What I'm skipping here is the[br]theory that you learned then

0:02:46.360,0:02:56.250
that to a compose well, F of t[br]has to be in the domain of g.

0:02:56.250,0:02:59.340
So the image F of t, whatever[br]you get from this image,

0:02:59.340,0:03:01.620
has to be in the domain of g.

0:03:01.620,0:03:05.100
Otherwise, the composition[br]could not exist.

0:03:05.100,0:03:07.870
Now if you have[br]differentiability,

0:03:07.870,0:03:10.390
assuming that this[br]is g composed with F,

0:03:10.390,0:03:16.332
assuming to be c1-- c1[br]meaning differentiable

0:03:16.332,0:03:24.470
and derivatives are continuous--[br]assuming both of them are c1,

0:03:24.470,0:03:26.320
they compose well.

0:03:26.320,0:03:28.260
What am I going to do next?

0:03:28.260,0:03:35.660
I'm going to say the[br]d, dt g of F of t.

0:03:35.660,0:03:39.600
And we said last time,[br]we get The Chain Rule

0:03:39.600,0:03:44.463
from the last function[br]we applied, g prime.

0:03:44.463,0:03:50.860
And so you have dg,[br][? d2 ?] at F of t.

0:03:50.860,0:03:57.590
I'm calling this guy u variable[br]just for my own enjoyment.

0:03:57.590,0:04:01.660
And then I go du, dt.

0:04:01.660,0:04:06.020
But du, dt would be[br]nothing but a prime of t,

0:04:06.020,0:04:09.330
so remember the cowboys[br]shooting at each other?

0:04:09.330,0:04:11.060
The du and du.

0:04:11.060,0:04:16.120
I will replace the u by prime of[br]t, just like you did in Calc 1.

0:04:16.120,0:04:16.620
Why?

0:04:16.620,0:04:23.320
Because I want to a mixture of[br]notations according to Calc 1

0:04:23.320,0:04:24.871
you took here.

0:04:24.871,0:04:30.580
The idea for Calc 3 is the[br]same with [INAUDIBLE] time,

0:04:30.580,0:04:33.370
assuming everything[br]composes well,

0:04:33.370,0:04:37.910
and has differentiability, and[br]the derivatives are continuous.

0:04:37.910,0:04:39.800
Just to make your life easier.

0:04:39.800,0:04:43.700
We have x of t, y of t.

0:04:43.700,0:04:47.175
Two nice functions[br]and a function

0:04:47.175,0:04:54.690
of these variables,[br]F of x and y.

0:04:54.690,0:04:57.710
So I'm going to have[br]to say, how about x

0:04:57.710,0:05:01.330
is a function of t and[br]y is a function of t?

0:05:01.330,0:05:05.290
So I should be able to go[br]ahead and differentiate

0:05:05.290,0:05:07.620
with respect to the t.

0:05:07.620,0:05:10.850


0:05:10.850,0:05:13.354
And how did it go?

0:05:13.354,0:05:14.770
Now that I prepared[br]you last time,

0:05:14.770,0:05:21.370
a little bit, for this kind[br]of new picture, new diagram,

0:05:21.370,0:05:27.365
you should be able to tell me,[br]without looking at the notes

0:05:27.365,0:05:31.530
from last time, how this goes.

0:05:31.530,0:05:36.730
So I'll take the function[br]F of x of t, y of t.

0:05:36.730,0:05:41.721
And when I view it like that,[br]I understand it's ultimately

0:05:41.721,0:05:45.380
a big function, F of t.

0:05:45.380,0:05:49.610
It's a real valued[br]function of t,

0:05:49.610,0:05:52.780
ultimately, as the composition.

0:05:52.780,0:05:56.257
This big F.

0:05:56.257,0:05:58.370


0:05:58.370,0:06:05.290
So does anybody[br]remember how this went?

0:06:05.290,0:06:07.750
Let's see.

0:06:07.750,0:06:10.210
The derivative,[br]with respect to t,

0:06:10.210,0:06:14.965
of this whole thing,[br]F of x of t, y of t?

0:06:14.965,0:06:19.070


0:06:19.070,0:06:19.570
Thoughts?

0:06:19.570,0:06:23.160


0:06:23.160,0:06:25.270
The partial derivative[br]of F with respect

0:06:25.270,0:06:32.704
to x, evaluated at[br]x of t and y to t.

0:06:32.704,0:06:35.620
So everything has to be[br]replaced in terms of t

0:06:35.620,0:06:38.536
because it's going to be y.

0:06:38.536,0:06:43.750
We assume that this derivative[br]exists and it's continuous.

0:06:43.750,0:06:44.270
Why?

0:06:44.270,0:06:46.365
Just to make your life[br]a little bit easier.

0:06:46.365,0:06:49.550


0:06:49.550,0:06:53.990
From the beginning,[br]we had dx, dt,

0:06:53.990,0:06:57.300
which was also[br]defined everywhere

0:06:57.300,0:07:09.000
and continuous, plus df, 2y at[br]the same point times dy, dt.

0:07:09.000,0:07:12.310


0:07:12.310,0:07:20.750
Notice what happens here with[br]these guys looking diagonally,

0:07:20.750,0:07:21.745
staring at each other.

0:07:21.745,0:07:24.810


0:07:24.810,0:07:26.940
Keep in mind the plus sign.

0:07:26.940,0:07:30.670
And of course, some of you[br]told me, well, is that OK?

0:07:30.670,0:07:31.960
You know favorite, right?

0:07:31.960,0:07:35.450
F of x at x of dy of t.

0:07:35.450,0:07:37.120
That's fine.

0:07:37.120,0:07:38.500
I saw that.

0:07:38.500,0:07:39.780
In engineering you use it.

0:07:39.780,0:07:53.358
Physics majors also use[br]a lot of this notation

0:07:53.358,0:07:57.250
as sub [INAUDIBLE] Fs of t.

0:07:57.250,0:07:58.700
We've seen that.

0:07:58.700,0:07:59.600
We've seen that.

0:07:59.600,0:08:03.150
It comes as no[br]surprise to us, but we

0:08:03.150,0:08:06.879
would like to see if there[br]are any other cases we

0:08:06.879,0:08:07.670
should worry about.

0:08:07.670,0:08:18.062


0:08:18.062,0:08:22.710
Now I don't want to[br]jump to the next example

0:08:22.710,0:08:25.840
until I give you[br]something that you

0:08:25.840,0:08:32.110
know very well from Calculus 1.

0:08:32.110,0:08:38.970
It's an example that you saw[br]before that was a melting ice

0:08:38.970,0:08:41.682
sphere.

0:08:41.682,0:08:46.995
It appears a lot in problems,[br]like final exam problems

0:08:46.995,0:08:48.450
and stuff.

0:08:48.450,0:08:53.120
What is the material[br]of this ball?

0:08:53.120,0:08:54.925
It's melting ice.

0:08:54.925,0:08:59.118


0:08:59.118,0:09:08.470
And if you remember, it[br]says that at the moment t0,

0:09:08.470,0:09:13.820
assume the radius was 5 inches.

0:09:13.820,0:09:16.706


0:09:16.706,0:09:42.508
We also know that the rate of[br]change of the radius in time

0:09:42.508,0:09:49.490
will be minus 5.

0:09:49.490,0:09:54.920
But let's suppose that we say[br]that inches per-- meaning,

0:09:54.920,0:09:56.810
it's really hot in the room.

0:09:56.810,0:10:00.270
Not this room, but[br]the hypothetic room

0:10:00.270,0:10:05.760
where the ice ball is melting.

0:10:05.760,0:10:08.890
So imagine, in 1[br]minute, the radius

0:10:08.890,0:10:13.550
will go down by 5 inches.

0:10:13.550,0:10:16.786
Yes, it must be really hot.

0:10:16.786,0:10:25.980
I want to know the derivative,[br]dv, dt at the time 0.

0:10:25.980,0:10:29.600
So you go, oh my god, I don't[br]remember doing this, actually.

0:10:29.600,0:10:31.330
It is a Calc 1 type of problem.

0:10:31.330,0:10:34.580


0:10:34.580,0:10:37.620
Why am I even[br]discussing it again?

0:10:37.620,0:10:41.280
Because I want to fool you a[br]little bit into remembering

0:10:41.280,0:10:44.953
the elementary formulas for[br]the volume of a sphere, volume

0:10:44.953,0:10:47.720
of a cone, volume of a cylinder.

0:10:47.720,0:10:48.910
That was a long time ago.

0:10:48.910,0:10:53.570
When you ask you teachers in[br]K12 if you should memorize them,

0:10:53.570,0:10:55.950
they said, by all[br]means, memorize them.

0:10:55.950,0:10:59.350
That was elementary geometry,[br]but some of you know them

0:10:59.350,0:11:01.054
by heart, some of you don't.

0:11:01.054,0:11:03.180
Do you remember[br]the volume formula

0:11:03.180,0:11:05.860
for a ball with radius r?

0:11:05.860,0:11:08.161
[INTERPOSING VOICES]

0:11:08.161,0:11:09.095


0:11:09.095,0:11:10.029
What?

0:11:10.029,0:11:11.430
[? STUDENT: High RQ. ?]

0:11:11.430,0:11:12.491
STUDENT: 4/3rds.

0:11:12.491,0:11:13.366
MAGDALENA TODA: Good.

0:11:13.366,0:11:14.860
I'm proud of you guys.

0:11:14.860,0:11:18.496
I've discovered lots of people[br]who are engineering majors

0:11:18.496,0:11:19.870
and they don't[br]know this formula.

0:11:19.870,0:11:23.790
So how are we going to[br]think of this problem?

0:11:23.790,0:11:26.390
We have to think, Chain Rule.

0:11:26.390,0:11:30.570
And Chain Rule means that you[br]view this radius as a shrinking

0:11:30.570,0:11:32.280
thing because[br]that's why you have

0:11:32.280,0:11:34.740
the grade of change negative.

0:11:34.740,0:11:37.120
The radius is shrinking,[br]it's decreasing,

0:11:37.120,0:11:40.940
so you view r as[br]a function of t.

0:11:40.940,0:11:42.530
And of course, you[br]made me cube it.

0:11:42.530,0:11:44.960
I had to cube it.

0:11:44.960,0:11:48.260
And then v will be a function[br]of t ultimately, but you see,

0:11:48.260,0:11:54.400
guys, t goes to r of t,[br]r of t goes to v of t.

0:11:54.400,0:11:56.675
What's the formula[br]for this function?

0:11:56.675,0:11:59.774
v equals 4 pi i cubed over 3.

0:11:59.774,0:12:02.210


0:12:02.210,0:12:04.180
So this is how the diagram goes.

0:12:04.180,0:12:10.280
You look at that composition[br]and you have dv, dt.

0:12:10.280,0:12:14.161
And I remember teaching as[br]a graduate student, that

0:12:14.161,0:12:18.320
was a long time ago,[br]in '97 or something,

0:12:18.320,0:12:23.020
with this kind of diagram with[br]compositions of functions.

0:12:23.020,0:12:25.720
And my students had told[br]me, nobody showed us

0:12:25.720,0:12:28.716
this kind of diagram before.

0:12:28.716,0:12:29.600
Well, I do.

0:12:29.600,0:12:32.120


0:12:32.120,0:12:35.290
I think they are very[br]useful for understanding

0:12:35.290,0:12:38.220
how a composition will go.

0:12:38.220,0:12:42.490
Now I would just going ahead and[br]say v prime because I'm lazy.

0:12:42.490,0:12:45.850
And I go v prime of t is 0.

0:12:45.850,0:12:51.012
Meaning, that this[br]is the dv, dt at t0.

0:12:51.012,0:12:55.440
And somebody has to help me[br]remember how we did The Chain

0:12:55.440,0:12:57.852
Rule in Calc 1.

0:12:57.852,0:12:59.470
It was ages ago.

0:12:59.470,0:13:05.490
4 pi over 3 constant times.

0:13:05.490,0:13:07.380
Who jumps down?

0:13:07.380,0:13:11.110
The 3 jumps down and he's[br]very happy to do that.

0:13:11.110,0:13:12.540
3, r squared.

0:13:12.540,0:13:15.320
But r squared is not an[br]independent variable.

0:13:15.320,0:13:18.650
He or she depends on t.

0:13:18.650,0:13:22.090
So I'll be very happy to[br]say 3 times that times.

0:13:22.090,0:13:24.020
And that's the essential part.

0:13:24.020,0:13:25.653
I'm not done.

0:13:25.653,0:13:26.872
STUDENT: It's dr over dt.

0:13:26.872,0:13:27.830
MAGDALENA TODA: dr, dt.

0:13:27.830,0:13:31.430
So I have finally[br]applied The Chain Rule.

0:13:31.430,0:13:35.440
And how do I plug[br]in the data in order

0:13:35.440,0:13:38.900
to get this as the final answer?

0:13:38.900,0:13:46.450
I just go 4 pi[br]over 3 times what?

0:13:46.450,0:13:48.980


0:13:48.980,0:13:56.420
3 times r-- who is[br]r at the time to 0,

0:13:56.420,0:14:00.112
where I want to view[br]the whole situation?

0:14:00.112,0:14:03.520
r squared at time[br]to 0 would be 25.

0:14:03.520,0:14:04.880
Are you guys with me?

0:14:04.880,0:14:09.112
dr, dt at time to[br]0 is negative 5.

0:14:09.112,0:14:10.090
All right.

0:14:10.090,0:14:12.180
I'm done.

0:14:12.180,0:14:15.195
So you are going to ask me,[br]if I'm taking the examine,

0:14:15.195,0:14:17.340
do I need this in[br]the exam like that?

0:14:17.340,0:14:18.860
Easy.

0:14:18.860,0:14:20.820
Oh, it depends on the exam.

0:14:20.820,0:14:23.330
If you have a multiple choice[br]where this is simplified,

0:14:23.330,0:14:27.440
obviously, it's not the right[br]thing to forget about it,

0:14:27.440,0:14:33.372
but I will accept[br]answers like that.

0:14:33.372,0:14:37.260
I don't care about the[br]numerical part very much.

0:14:37.260,0:14:41.335
If you want to do more, 4[br]times 25 is hundred times 5.

0:14:41.335,0:14:43.534
So I have minus what?

0:14:43.534,0:14:44.940
STUDENT: 500 pi.

0:14:44.940,0:14:47.074
MAGDALENA TODA: 500 pi.

0:14:47.074,0:14:49.050
How do we get the unit of that?

0:14:49.050,0:14:50.630
I'm wondering.

0:14:50.630,0:14:52.410
STUDENT: Cubic[br]inches per minute.

0:14:52.410,0:14:54.350
MAGDALENA TODA: Cubic[br]inches per minute.

0:14:54.350,0:14:55.320
Very good.

0:14:55.320,0:14:56.650
Cubic inches per minute.

0:14:56.650,0:14:59.020
Why don't I write it down?

0:14:59.020,0:15:01.440
Because I couldn't care less.

0:15:01.440,0:15:02.380
I'm a mathematician.

0:15:02.380,0:15:06.709
If I were a physicist, I would[br]definitely write it down.

0:15:06.709,0:15:10.100
And he was right.

0:15:10.100,0:15:15.990
Now you are going[br]to find this weird.

0:15:15.990,0:15:20.190
Why is she doing this review[br]of this kind of melting ice

0:15:20.190,0:15:22.360
problem from Calc 1?

0:15:22.360,0:15:25.790
Because today I'm[br]being sneaky and mean.

0:15:25.790,0:15:28.760
And I want to give[br]you a little challenge

0:15:28.760,0:15:30.940
for 1 point of extra credit.

0:15:30.940,0:15:33.410
You will have to compose[br]your own problem,

0:15:33.410,0:15:37.330
in Calculus 3,[br]that is like that.

0:15:37.330,0:15:49.910
So you have to compose a problem[br]about a solid cylinder made

0:15:49.910,0:15:51.840
of ice.

0:15:51.840,0:15:53.010
Say what, Magdalena?

0:15:53.010,0:15:53.730
OK.

0:15:53.730,0:15:57.450
So I'll write it down.

0:15:57.450,0:16:01.850
Solid cylinder made of ice[br]that's melting in time.

0:16:01.850,0:16:04.615


0:16:04.615,0:16:07.110
So compose your own problem.

0:16:07.110,0:16:10.020
Do you have to solve[br]your own problem?

0:16:10.020,0:16:12.530
Yes, I guess so.

0:16:12.530,0:16:14.310
Once you compose[br]your own problem,

0:16:14.310,0:16:16.545
solve your own problem[br]For extra credit, 1 point.

0:16:16.545,0:16:20.680


0:16:20.680,0:16:28.770
Compose, write, and solve--[br]you are the problem author.

0:16:28.770,0:16:36.004
Write and solve[br]your own problem,

0:16:36.004,0:16:40.396
so that the story includes--

0:16:40.396,0:16:42.836
STUDENT: A solid cylinder.

0:16:42.836,0:16:43.812
MAGDALENA TODA: Yes.

0:16:43.812,0:16:48.670
Includes-- instead of a[br]nice ball, a solid cylinder.

0:16:48.670,0:16:54.300


0:16:54.300,0:16:59.240
And necessarily, you cannot[br]write it just a story--

0:16:59.240,0:17:02.860
I once had an ice cylinder,[br]and it was melting,

0:17:02.860,0:17:05.990
and I went to watch a movie,[br]and by the time I came back,

0:17:05.990,0:17:07.098
it was all melted.

0:17:07.098,0:17:08.910
That's not what I want.

0:17:08.910,0:17:24.450
I want it so that the problem[br]is an example of applying

0:17:24.450,0:17:35.130
The Chain Rule in Calc 3.

0:17:35.130,0:17:37.730
And I won't say more.

0:17:37.730,0:17:40.688
So maybe somebody[br]can help with a hint.

0:17:40.688,0:17:43.033
Maybe I shouldn't[br]give too many hits,

0:17:43.033,0:17:46.420
but let's talk as if we[br]were chatting in a cafe,

0:17:46.420,0:17:49.290
without me writing[br]too much down.

0:17:49.290,0:17:51.380
Of course, you can take[br]notes of our discussion,

0:17:51.380,0:17:54.290
but I don't want[br]have it documented.

0:17:54.290,0:17:55.759
So we have a cylinder right.

0:17:55.759,0:17:58.753


0:17:58.753,0:18:01.747
There is the cylinder.

0:18:01.747,0:18:02.745
Forget about this.

0:18:02.745,0:18:04.250
So there's the cylinder.

0:18:04.250,0:18:09.620
It's made of ice[br]and it's melting.

0:18:09.620,0:18:14.010
And the volume should be a[br]function of two variables

0:18:14.010,0:18:16.725
because otherwise, you[br]don't have it in Calc 3.

0:18:16.725,0:18:18.426
So a function of two variables.

0:18:18.426,0:18:21.807


0:18:21.807,0:18:25.182
What other two variables[br]am I talking about?

0:18:25.182,0:18:26.640
STUDENT: The radius[br]and the height.

0:18:26.640,0:18:28.640
MAGDALENA TODA: The radius[br]would be one of them.

0:18:28.640,0:18:30.120
You don't have to say x and y.

0:18:30.120,0:18:33.840
This is r and h.

0:18:33.840,0:18:39.002
So h and r are in that formula.

0:18:39.002,0:18:40.460
I'm not going to[br]say which formula,

0:18:40.460,0:18:44.740
you guys should know of[br]the volume of the cylinder.

0:18:44.740,0:18:48.974
But both h and r, what do they[br]have in common in the story?

0:18:48.974,0:18:49.940
STUDENT: Time.

0:18:49.940,0:18:52.150
MAGDALENA TODA: They are[br]both functions of time.

0:18:52.150,0:18:53.771
They are melting in time.

0:18:53.771,0:18:55.270
STUDENT: Can I ask[br]a quick question?

0:18:55.270,0:18:55.870
MAGDALENA TODA: Yes, sir.

0:18:55.870,0:18:58.430
STUDENT: What if we solve[br]for-- what is the negative 500

0:18:58.430,0:18:59.786
[? path? ?]

0:18:59.786,0:19:04.290
MAGDALENA TODA: This is the[br]speed with which the volume is

0:19:04.290,0:19:05.638
shrinking at time to 0.

0:19:05.638,0:19:08.850


0:19:08.850,0:19:13.090
So the rate of change of[br]the volume at time to o.

0:19:13.090,0:19:15.170
And this is[br]something-- by the way,

0:19:15.170,0:19:20.260
that's how I would[br]like you to state it.

0:19:20.260,0:19:25.805
Find the rate of change[br]of the volume of the ice--

0:19:25.805,0:19:28.780
wasn't that a good cylinder?

0:19:28.780,0:19:34.850
At time to 0, if you[br]know that at time to 0

0:19:34.850,0:19:37.680
something happened.

0:19:37.680,0:19:41.120
Maybe r is given, h is given.

0:19:41.120,0:19:44.256
The derivatives are given.

0:19:44.256,0:19:47.370
You only have one[br]derivative given here,

0:19:47.370,0:19:50.460
which was our[br]prime of t minus 5.

0:19:50.460,0:19:52.160
Now I leave it to you.

0:19:52.160,0:19:57.120
I ask it to you, and I'll leave[br]it to you, and don't tell me.

0:19:57.120,0:20:01.820
When we have a[br]piece of ice-- well,

0:20:01.820,0:20:05.590
there was something in the[br]news, but I'm not going to say.

0:20:05.590,0:20:08.090
There was some nice, ice[br]sculpture in the news there.

0:20:08.090,0:20:10.880


0:20:10.880,0:20:18.980
So do the dimensions decrease[br]at the same rate, do you think?

0:20:18.980,0:20:20.560
I mean, I don't know.

0:20:20.560,0:20:22.020
It's all up to you.

0:20:22.020,0:20:24.830
Think of a case when the[br]radius and the height

0:20:24.830,0:20:27.800
would shrink at the same speed.

0:20:27.800,0:20:31.316
And think of a case when[br]the radius and the height

0:20:31.316,0:20:34.030
of the cylinder made[br]of ice would not

0:20:34.030,0:20:38.610
change at the same[br]rate for some reason.

0:20:38.610,0:20:40.700
I don't know, but[br]the simplest case

0:20:40.700,0:20:43.040
would be to assume that[br]all of the dimensions

0:20:43.040,0:20:49.520
shrink at the same speed,[br]at the same rate of change.

0:20:49.520,0:20:52.000
So you write your own problem,[br]you make up your own data.

0:20:52.000,0:20:55.335
Now you will appreciate[br]how much work people

0:20:55.335,0:20:57.370
put into that work book.

0:20:57.370,0:21:00.680
I mean, if there is a bug,[br]it's one in a thousand,

0:21:00.680,0:21:04.210
but for a programmer to be[br]able to write those problems,

0:21:04.210,0:21:08.715
he has to know calculus,[br]he has to know C++ or Java,

0:21:08.715,0:21:12.520
he has to be good-- that's[br]not a problem, right?

0:21:12.520,0:21:13.440
STUDENT: No.

0:21:13.440,0:21:14.450
That's fine.

0:21:14.450,0:21:20.360
MAGDALENA TODA: He or she has[br]to know how to write a problem,

0:21:20.360,0:21:22.600
so that you guys,[br]no matter how you

0:21:22.600,0:21:29.130
input your answer, as long as it[br]is correct, you'll get the OK.

0:21:29.130,0:21:32.990
Because you can put answers[br]in many equivalent forms

0:21:32.990,0:21:36.127
and all of them have to be--

0:21:36.127,0:21:37.210
STUDENT: The right answer.

0:21:37.210,0:21:38.043
MAGDALENA TODA: Yes.

0:21:38.043,0:21:40.780
To get the right answer.

0:21:40.780,0:21:43.950
So since I have new[br]people who just came--

0:21:43.950,0:21:47.023
And I understand you guys[br]come from different buildings

0:21:47.023,0:21:52.435
and I'm not mad for people who[br]are coming late because I know

0:21:52.435,0:21:55.280
you come from other[br]classes, I wanted

0:21:55.280,0:22:03.400
to say we started from a melting[br]ice sphere example in Calc 1

0:22:03.400,0:22:07.260
that was on many finals[br]in here, at Texas Tech.

0:22:07.260,0:22:14.470
And I want you to compose your[br]own problem based on that.

0:22:14.470,0:22:17.490
This time, involving[br]a cylinder made

0:22:17.490,0:22:23.650
of ice whose dimensions are[br]doing something special.

0:22:23.650,0:22:26.130
That shouldn't be hard.

0:22:26.130,0:22:29.500
I'm going to erase this[br]part because it's not

0:22:29.500,0:22:30.600
the relevant one.

0:22:30.600,0:22:32.840
I'm going to keep this[br]one a little bit more

0:22:32.840,0:22:35.960
for people who[br]want to take notes.

0:22:35.960,0:22:37.271
And I'm going to move on.

0:22:37.271,0:22:42.460


0:22:42.460,0:22:47.000
Another example we[br]give you in the book

0:22:47.000,0:22:54.700
is that one where x and y,[br]the variables the function f,

0:22:54.700,0:22:58.820
are not just[br]functions of time, t.

0:22:58.820,0:23:03.570
They, themselves, are functions[br]of other two variables.

0:23:03.570,0:23:07.990
Is that a lot more different[br]from what I gave you already?

0:23:07.990,0:23:08.490
No.

0:23:08.490,0:23:10.730
The idea is the same.

0:23:10.730,0:23:13.930
And you are imaginative.

0:23:13.930,0:23:20.500
You are able to come up[br]with your own answers.

0:23:20.500,0:23:26.950
I'm going to ask you to think[br]about what I'll have to write.

0:23:26.950,0:23:28.330
This is finished.

0:23:28.330,0:23:32.250


0:23:32.250,0:23:37.796
So assume that you have[br]function z equals F of x,y.

0:23:37.796,0:23:42.470


0:23:42.470,0:23:48.840
As we had it before,[br]this is example 2

0:23:48.840,0:23:58.330
where x is a function[br]of u and v itself.

0:23:58.330,0:24:02.510
And y is a function[br]of u and v itself.

0:24:02.510,0:24:07.201
And we assume that all[br]the partial derivatives

0:24:07.201,0:24:09.536
are defined and continuous.

0:24:09.536,0:24:12.030
And we make the[br]problem really nice.

0:24:12.030,0:24:21.490
And now we'll come[br]up with some example

0:24:21.490,0:24:38.630
you know from before where[br]x equals x of uv equals uv.

0:24:38.630,0:24:48.760
And y equals y of[br]uv equals u plus v.

0:24:48.760,0:24:51.860
So these functions are[br]the sum and the product

0:24:51.860,0:24:53.195
of other variables.

0:24:53.195,0:24:56.130


0:24:56.130,0:25:06.780
Can you tell me how I am going[br]to compute the derivative of 0,

0:25:06.780,0:25:17.320
or of f, with the script[br]of u at x of uv, y of uv?

0:25:17.320,0:25:18.756
Is this hard?

0:25:18.756,0:25:19.502
STUDENT: It is.

0:25:19.502,0:25:20.710
MAGDALENA TODA: I don't know.

0:25:20.710,0:25:27.512
You have to help me because--[br]why don't I put d here?

0:25:27.512,0:25:29.000
STUDENT: Because [INAUDIBLE].

0:25:29.000,0:25:30.630
MAGDALENA TODA:[br]Because you have 2.

0:25:30.630,0:25:32.592
So the composition[br]in itself will

0:25:32.592,0:25:35.510
be a function of two variables.

0:25:35.510,0:25:38.610
So of course, I[br]have [INAUDIBLE].

0:25:38.610,0:25:48.270
I'm going to go ahead and do[br]it as you say without rushing.

0:25:48.270,0:25:51.360
Of course, I know[br]you are watching.

0:25:51.360,0:25:53.250
What will happen?

0:25:53.250,0:25:54.159
STUDENT: 2x and 2y.

0:25:54.159,0:25:55.450
MAGDALENA TODA: No, in general.

0:25:55.450,0:25:58.470
Over here, I know you[br]want to do it right away,

0:25:58.470,0:26:01.790
but I would like you to give[br]me a general formula mimicking

0:26:01.790,0:26:06.530
the same thing you had before[br]when you had one parameter, t.

0:26:06.530,0:26:08.120
Now you have u and d separately.

0:26:08.120,0:26:10.376
You want it to do it straight.

0:26:10.376,0:26:19.088
So we have df, dx[br]at x of uv, y of uv.

0:26:19.088,0:26:20.540
Shut up, Magdalene.

0:26:20.540,0:26:24.450
Let people talk and help[br]you because you're tired.

0:26:24.450,0:26:26.880
It's a Thursday.

0:26:26.880,0:26:28.366
df, dx.

0:26:28.366,0:26:29.259
STUDENT: [INAUDIBLE].

0:26:29.259,0:26:30.050
MAGDALENA TODA: dx.

0:26:30.050,0:26:34.250
Again, [INAUDIBLE] notation,[br]partial with respect

0:26:34.250,0:26:42.890
to u, plus df, dy.

0:26:42.890,0:26:46.430
So the second argument--[br]so I prime in respect

0:26:46.430,0:26:50.610
to the second argument,[br]computing everything

0:26:50.610,0:26:55.180
in the end, which means[br]in terms of u and v times,

0:26:55.180,0:27:00.350
again, the dy with respect to u.

0:27:00.350,0:27:01.550
You are saying that.

0:27:01.550,0:27:04.367
Now I'd like you[br]to see the pattern.

0:27:04.367,0:27:06.450
Of course, you see the[br]pattern here, smart people,

0:27:06.450,0:27:11.660
but I want to[br]emphasize the cowboys.

0:27:11.660,0:27:15.021
And green for the other cowboy.

0:27:15.021,0:27:16.854
I'm trying to match the[br]college beautifully.

0:27:16.854,0:27:22.636


0:27:22.636,0:27:26.116
And the independent[br]variable, Mr. u.

0:27:26.116,0:27:27.610
Not u, but Mr. u.

0:27:27.610,0:27:28.606
Yes, ma'am?

0:27:28.606,0:27:31.096
STUDENT: Is it the[br]partial of dx, du?

0:27:31.096,0:27:37.771
Or is it-- like you did[br]the partial for the--

0:27:37.771,0:27:39.562
MAGDALENA TODA: So did[br]I do anything wrong?

0:27:39.562,0:27:41.554
I don't think I[br]did anything wrong.

0:27:41.554,0:27:44.400
STUDENT: So it is the[br]partial for dx over du?

0:27:44.400,0:27:48.663
MAGDALENA TODA: So I go du with[br]respect to the first variable,

0:27:48.663,0:27:50.865
times that variable[br]with respect to u.

0:27:50.865,0:27:52.156
STUDENT: But is it the partial?

0:27:52.156,0:27:53.660
That's my question.

0:27:53.660,0:27:57.180
MAGDALENA TODA: But it has[br]to be a partial because x is

0:27:57.180,0:28:02.995
a function of u and[br]v, so I cannot put d.

0:28:02.995,0:28:07.146
And then the same plus[br]the same idea as before.

0:28:07.146,0:28:10.110
df with respect to[br]the second argument

0:28:10.110,0:28:15.650
times that second argument[br]with respect to the u.

0:28:15.650,0:28:20.420
You see, Mr. u is[br]replacing Mr. t.

0:28:20.420,0:28:21.485
He is independent.

0:28:21.485,0:28:24.170


0:28:24.170,0:28:27.310
He's the guy who is moving.

0:28:27.310,0:28:29.010
We don't care[br]about anybody else,

0:28:29.010,0:28:32.030
but he replaces the time[br]in this kind of problem.

0:28:32.030,0:28:35.500


0:28:35.500,0:28:38.590
Now the other one.

0:28:38.590,0:28:41.064
I will let you speak.

0:28:41.064,0:28:43.926
Df, dv.

0:28:43.926,0:28:50.136
The same idea, but somebody[br]else is going to talk.

0:28:50.136,0:28:52.580
STUDENT: It would[br]be del f, del y.

0:28:52.580,0:28:54.450
MAGDALENA TODA: Del f, del x?

0:28:54.450,0:28:56.890
Well, let's try[br]to start in order.

0:28:56.890,0:29:01.940


0:29:01.940,0:29:05.200
And I tried to be[br]organized and write neatly

0:29:05.200,0:29:12.700
because I looked at-- so these[br]videos are new and in progress.

0:29:12.700,0:29:16.670
And I'm trying to see what[br]I did well and I didn't.

0:29:16.670,0:29:18.490
And at times, I wrote neatly.

0:29:18.490,0:29:20.785
At times, I wrote not so neatly.

0:29:20.785,0:29:23.260
I'm just learning about myself.

0:29:23.260,0:29:28.070
It's one thing, what you think[br]about yourself from the inside

0:29:28.070,0:29:30.610
and to you see yourself[br]the way other people

0:29:30.610,0:29:33.350
see from the outside.

0:29:33.350,0:29:34.420
It's not fun.

0:29:34.420,0:29:35.670
STUDENT: Can you say it again?

0:29:35.670,0:29:41.900
MAGDALENA TODA: This is[br]v. So I'll say it again.

0:29:41.900,0:29:46.370
We all have a certain[br]impression about ourselves,

0:29:46.370,0:29:49.100
but when you see a[br]movie of yourself,

0:29:49.100,0:29:51.740
you see the way[br]other people see you.

0:29:51.740,0:29:53.069
And it's not fun.

0:29:53.069,0:29:56.010
STUDENT: So what--

0:29:56.010,0:29:58.750
MAGDALENA TODA: So[br]let's see the cowboys.

0:29:58.750,0:30:06.360
Ryan is looking at the[br][? man. ?] He is all [? man. ?]

0:30:06.360,0:30:10.480
And y is here, right?

0:30:10.480,0:30:15.830
And who is the time[br]variable, kind of, this time?

0:30:15.830,0:30:18.205
This time, which[br]one is the time?

0:30:18.205,0:30:28.456
v. And v is the only ultimate[br]variable that we care about.

0:30:28.456,0:30:31.858
So everything you did[br]before with respect to t,

0:30:31.858,0:30:35.307
you do now with[br]respect to u, you

0:30:35.307,0:30:37.130
do now with respect[br]to v. It shouldn't

0:30:37.130,0:30:38.970
be hard to understand.

0:30:38.970,0:30:42.040
I want to work the[br]example, of course.

0:30:42.040,0:30:44.440
With your help, I will work it.

0:30:44.440,0:30:49.270
Now remember how my students[br]cheated on this one?

0:30:49.270,0:30:57.090
So I told my colleague, he did[br]not say, five or six years ago,

0:30:57.090,0:31:00.810
by first writing The Chain Rule[br]for functions of two variables,

0:31:00.810,0:31:08.430
express all the df, du, df,[br]dv, but he said by any method.

0:31:08.430,0:31:12.970
Of course, what they[br]did-- they were sneaky.

0:31:12.970,0:31:15.962
They took something[br]like x equals uv

0:31:15.962,0:31:17.926
and they plugged it in here.

0:31:17.926,0:31:19.872
They took the function[br][? u and v, ?]

0:31:19.872,0:31:20.872
they plugged it in here.

0:31:20.872,0:31:22.836
They computed everything[br]in terms of u and v

0:31:22.836,0:31:24.309
and took the partials.

0:31:24.309,0:31:26.756
STUDENT: Why don't[br]you [INAUDIBLE]?

0:31:26.756,0:31:29.130
MAGDALENA TODA: It depends[br]how the problem is formulated.

0:31:29.130,0:31:30.455
STUDENT: So if you[br]make it [INAUDIBLE],

0:31:30.455,0:31:31.626
then it's [INAUDIBLE].

0:31:31.626,0:31:35.716


0:31:35.716,0:31:39.360
MAGDALENA TODA: So when they[br]give you the precise functions,

0:31:39.360,0:31:39.919
you're right.

0:31:39.919,0:31:41.710
But if they don't give[br]you those functions,

0:31:41.710,0:31:44.320
if they keep them a[br]secret, then you still

0:31:44.320,0:31:47.230
have to write the[br]general formula.

0:31:47.230,0:31:51.450
If they don't give you[br]the functions, all of them

0:31:51.450,0:31:54.060
explicitly.

0:31:54.060,0:31:57.296
So let's see what[br]to do in this case.

0:31:57.296,0:32:09.300
df, du at x of u, vy[br]of uv will be what?

0:32:09.300,0:32:11.520
Now people, help me, please.

0:32:11.520,0:32:15.710


0:32:15.710,0:32:21.980
I want to teach you how[br]engineers and physicists very,

0:32:21.980,0:32:26.610
very often express[br]those at x and y.

0:32:26.610,0:32:29.080
And many of you know[br]because we talked

0:32:29.080,0:32:31.280
about that in office hours.

0:32:31.280,0:32:36.661
2x, I might write,[br]but evaluated at--

0:32:36.661,0:32:38.160
and this is a very[br]frequent notation

0:32:38.160,0:32:42.030
image in the engineering[br]and physicist world.

0:32:42.030,0:32:45.170
So 2x evaluated at where?

0:32:45.170,0:32:51.530
At the point where x is[br]uv and y is u plus v.

0:32:51.530,0:32:59.670
So I say x of uv, y of uv.

0:32:59.670,0:33:05.040
And I'll replace later[br]because I'm not in a hurry.

0:33:05.040,0:33:06.970
dx, du.

0:33:06.970,0:33:09.300
Who is dx, du?

0:33:09.300,0:33:11.403
The derivative of x[br]or with respect to u?

0:33:11.403,0:33:12.690
Are you guys awake?

0:33:12.690,0:33:13.370
STUDENT: Yes.

0:33:13.370,0:33:14.730
So it's v.

0:33:14.730,0:33:19.470
MAGDALENA TODA: v. Very good.[br]v plus-- the next term, who's

0:33:19.470,0:33:22.610
going to tell me what we have?

0:33:22.610,0:33:24.070
STUDENT: 2y evaluated at--

0:33:24.070,0:33:28.340
MAGDALENA TODA: 2y evaluated[br]at-- look how lazy I am.

0:33:28.340,0:33:37.540
Times the derivative[br]of y with respect to u.

0:33:37.540,0:33:39.640
So you were right because of 2y.

0:33:39.640,0:33:42.740


0:33:42.740,0:33:44.200
Attention, right?

0:33:44.200,0:33:48.140
So it's dy, du is 1.

0:33:48.140,0:33:49.910
It's very easy to[br]make a mistake.

0:33:49.910,0:33:52.430
I've had mistakes who[br]made mistakes in the final

0:33:52.430,0:33:55.740
from just miscalculating[br]because when

0:33:55.740,0:33:57.700
you are close to[br]some formula, you

0:33:57.700,0:34:00.040
don't see the whole picture.

0:34:00.040,0:34:01.038
What do you do?

0:34:01.038,0:34:05.030
At the end of your[br]exams, go back and rather

0:34:05.030,0:34:08.620
than quickly turning in[br]a paper, never do that,

0:34:08.620,0:34:11.510
go back and check[br]all your problems.

0:34:11.510,0:34:13.489
It's a good habit.

0:34:13.489,0:34:20.880
2 times x, which is uv, I plug[br]it as a function of u and v,

0:34:20.880,0:34:23.350
right?

0:34:23.350,0:34:27.370
Times a v plus-- who is 2y?

0:34:27.370,0:34:28.870
That's the last of the Mohicans.

0:34:28.870,0:34:30.310
One is out.

0:34:30.310,0:34:31.100
STUDENT: 2.

0:34:31.100,0:34:37.130
MAGDALENA TODA: 2y 2 times[br]replace y in terms of u and v.

0:34:37.130,0:34:38.050
And you're done.

0:34:38.050,0:34:40.489
So do you like it?

0:34:40.489,0:34:42.020
I don't.

0:34:42.020,0:34:44.440
And how would you write it?

0:34:44.440,0:34:48.510
Not much better than that,[br]but at least let's try.

0:34:48.510,0:34:53.380
2uv squared plus 2u plus 2v.

0:34:53.380,0:34:55.004
You can do a little[br]bit more than that,

0:34:55.004,0:35:01.130
but if you want to list it[br]in the order of the degrees

0:35:01.130,0:35:04.190
of the polynomials, that's OK.

0:35:04.190,0:35:06.220
Now next one.

0:35:06.220,0:35:10.015
df, dv, x of uv, y of uv.

0:35:10.015,0:35:12.790


0:35:12.790,0:35:15.861
Such examples are in the book.

0:35:15.861,0:35:17.860
Many things are in the[br]book and out of the book.

0:35:17.860,0:35:21.400
I mean, on the white board.

0:35:21.400,0:35:25.180
I don't know why it gives you so[br]many combinations of this type,

0:35:25.180,0:35:31.130
u plus v, u minus-- 2u[br]plus 2v, 2u you minus 2v.

0:35:31.130,0:35:31.960
Well, I know why.

0:35:31.960,0:35:35.210
Because that's a[br]rotation and rescaling.

0:35:35.210,0:35:37.150
So there is a[br]reason behind that,

0:35:37.150,0:35:42.110
but I thought of something[br]different for df, dv.

0:35:42.110,0:35:44.800
Now what do I do?

0:35:44.800,0:35:45.355
df, dx.

0:35:45.355,0:35:46.022
STUDENT: You [? have to find[br]something symmetrical to that.

0:35:46.022,0:35:46.860
?]

0:35:46.860,0:35:48.443
MAGDALENA TODA:[br]Again, the same thing.

0:35:48.443,0:35:52.781
2x evaluated at whoever times--

0:35:52.781,0:35:53.280
STUDENT: u.

0:35:53.280,0:35:55.831


0:35:55.831,0:35:58.080
MAGDALENA TODA: Because you[br]have dx with respect to v,

0:35:58.080,0:36:02.090
so you have u plus--

0:36:02.090,0:36:03.770
STUDENT: df, dy.

0:36:03.770,0:36:07.060
MAGDALENA TODA: df, dy,[br]which is 2y, evaluated

0:36:07.060,0:36:10.002
at the same kind of guy.

0:36:10.002,0:36:13.360
So all you have to do is[br]replace with respect to u and v.

0:36:13.360,0:36:16.240
And finally, multiplied by-

0:36:16.240,0:36:16.870
STUDENT: dy.

0:36:16.870,0:36:18.180
MAGDALENA TODA: dy, dv.

0:36:18.180,0:36:21.650
dy, dv is 1 again.

0:36:21.650,0:36:23.723
Just pay attention[br]when you plug in

0:36:23.723,0:36:25.722
because you realize you[br]can know these very well

0:36:25.722,0:36:28.780
and understand it as a process,[br]but if you make an algebra

0:36:28.780,0:36:31.134
and everything is out.

0:36:31.134,0:36:32.800
And then you send me[br]an email that says,

0:36:32.800,0:36:34.960
I've tried this[br]problem 15 times.

0:36:34.960,0:36:37.820
And I don't even hold[br]you responsible for that

0:36:37.820,0:36:41.770
because I can make[br]algebra mistakes anytime.

0:36:41.770,0:36:54.400
So 2uv times u plus 2 times u[br]plus v. So what did I do here?

0:36:54.400,0:37:00.906
I simply replaced the given[br]functions in terms of u and v.

0:37:00.906,0:37:03.200
And I'm done.

0:37:03.200,0:37:03.900
Do I like it?

0:37:03.900,0:37:08.532
No, but I'd like you to notice[br]something as soon as I'm done.

0:37:08.532,0:37:11.899
2u squared v plus 2u plus 2v.

0:37:11.899,0:37:17.610


0:37:17.610,0:37:20.040
Could I have expected that?

0:37:20.040,0:37:21.540
Look at the beauty[br]of the functions.

0:37:21.540,0:37:24.490


0:37:24.490,0:37:27.550
Z is a symmetric function.

0:37:27.550,0:37:31.515
x and y have some of[br]the symmetry as well.

0:37:31.515,0:37:34.550
If you swap u and v, these[br]are symmetric polynomials

0:37:34.550,0:37:38.615
of order 2 and 1.

0:37:38.615,0:37:40.070
[INAUDIBLE]

0:37:40.070,0:37:42.620
Swap the variables, you[br]still get the same thing.

0:37:42.620,0:37:45.615
Swap the variables u and[br]v, you get the same thing.

0:37:45.615,0:37:48.380
So how could I have[br]imagined that I'm

0:37:48.380,0:37:54.470
going to get-- if I were smart,[br]without doing all the work,

0:37:54.470,0:37:57.920
I could figure out[br]this by just swapping

0:37:57.920,0:38:01.760
the u and v, the rows of u[br]and v. I would have said,

0:38:01.760,0:38:07.980
2vu squared, dv plus 2u and[br]it's the same thing I got here.

0:38:07.980,0:38:13.520
But not always are you so[br]lucky to be given nice data.

0:38:13.520,0:38:15.690
Well, in real life, it's a mess.

0:38:15.690,0:38:21.940
If you are, let's say, working[br]with geophysics real data,

0:38:21.940,0:38:27.560
you two parameters and for[br]each parameter, x and y,

0:38:27.560,0:38:28.810
you have other parameters.

0:38:28.810,0:38:31.565
You will never have[br]anything that nice.

0:38:31.565,0:38:35.940
You may have nasty[br]truncations of polynomials

0:38:35.940,0:38:39.057
with many, many[br]terms that you work

0:38:39.057,0:38:41.390
with approximating polynomials[br]all the time. [INAUDIBLE]

0:38:41.390,0:38:43.350
or something like that.

0:38:43.350,0:38:46.870
So don't expect these miracles[br]to happen with real data,

0:38:46.870,0:38:49.740
but the process is the same.

0:38:49.740,0:38:52.960
And, of course,[br]there are programs

0:38:52.960,0:38:56.410
that incorporate all of[br]the Calculus 3 notions

0:38:56.410,0:38:59.380
that we went over.

0:38:59.380,0:39:03.670
There were people[br]who already wrote

0:39:03.670,0:39:08.310
lots of programs that enable[br]you to compute derivatives

0:39:08.310,0:39:11.105
of function of[br]several variables.

0:39:11.105,0:39:23.690


0:39:23.690,0:39:27.490
Now let me take your[br]temperature again.

0:39:27.490,0:39:29.520
Is this hard?

0:39:29.520,0:39:30.420
No.

0:39:30.420,0:39:33.770
It's sort of logical you just[br]have to pay attention to what?

0:39:33.770,0:39:36.450


0:39:36.450,0:39:41.523
Pay attention to not making too[br]many algebra mistakes, right?

0:39:41.523,0:39:42.522
That's kind of the idea.

0:39:42.522,0:39:45.330


0:39:45.330,0:39:48.701
More things that I[br]wanted to-- there

0:39:48.701,0:39:50.950
are many more things I wanted[br]to share with you today,

0:39:50.950,0:39:56.230
but I'm glad we reached[br]some consensus in the sense

0:39:56.230,0:40:01.718
that you feel there is logic and[br]order in this type of problem.

0:40:01.718,0:40:21.900


0:40:21.900,0:40:35.170
I tried to give you a little[br]bit of an introduction to why

0:40:35.170,0:40:39.645
the gradient is so[br]important last time.

0:40:39.645,0:40:41.220
And I'm going to[br]come back to that

0:40:41.220,0:40:45.239
again, so I'm not going[br]to leave you in the air.

0:40:45.239,0:40:49.071
But before then,[br]I would like to do

0:40:49.071,0:40:50.687
the directional[br]derivative, which

0:40:50.687,0:40:52.914
is a very important section.

0:40:52.914,0:40:57.378
So I'm going to start again.

0:40:57.378,0:41:07.890
And I'll also do, at the same[br]time, some review of 11.5.

0:41:07.890,0:41:10.060
So I will combine them.

0:41:10.060,0:41:11.844
And I want to[br]introduce the notion

0:41:11.844,0:41:14.528
of directional[br]derivatives because it's

0:41:14.528,0:41:15.992
right there for us to grab it.

0:41:15.992,0:41:26.240


0:41:26.240,0:41:28.850
And you say, well,[br]that sounds familiar.

0:41:28.850,0:41:32.290
It sounds like I dealt[br]with direction before,

0:41:32.290,0:41:35.780
but I didn't what that was.

0:41:35.780,0:41:37.500
That's exactly true.

0:41:37.500,0:41:41.360
You dealt with it before, you[br]just didn't know what it was.

0:41:41.360,0:41:44.060
And I'll give you the[br]general definition,

0:41:44.060,0:41:49.120
but then I would like you to[br]think about if you have ever

0:41:49.120,0:41:50.260
seen that before.

0:41:50.260,0:41:53.040


0:41:53.040,0:41:59.420
I'm going to say I have the[br]derivative of a function, f,

0:41:59.420,0:42:01.440
in the direction, u.

0:42:01.440,0:42:04.290
And I'm going put u bar[br]as if you were free,

0:42:04.290,0:42:05.480
not a married man.

0:42:05.480,0:42:09.180
But u as a direction as[br]always a unit vector.

0:42:09.180,0:42:10.380
STUDENT: [INAUDIBLE].

0:42:10.380,0:42:11.921
MAGDALENA TODA: I[br]told you last time,

0:42:11.921,0:42:18.188
just to prepare you, direction,[br]u, is always a unit vector.

0:42:18.188,0:42:23.544


0:42:23.544,0:42:24.044
Always.

0:42:24.044,0:42:28.450


0:42:28.450,0:42:29.390
Computed at x0y0.

0:42:29.390,0:42:35.556
But x0y0 is a given view point.

0:42:35.556,0:42:40.860


0:42:40.860,0:42:46.490
And I'm going to say[br]what that's going to be.

0:42:46.490,0:42:49.100
I have a limit.

0:42:49.100,0:42:51.300
I'm going to use the h.

0:42:51.300,0:42:53.760
And you say, why in the[br]world is she using h?

0:42:53.760,0:42:57.426
You will see in a second--[br]h goes to 0-- because we

0:42:57.426,0:42:59.130
haven't used h in awhile.

0:42:59.130,0:43:02.480
h is like a small displacement[br]that shrinks to 0.

0:43:02.480,0:43:07.480


0:43:07.480,0:43:25.818
And I put here, f of x0[br]plus hu1, y0 plus hu2,

0:43:25.818,0:43:32.300
close, minus f of x0y0.

0:43:32.300,0:43:34.912
So you say, wait a minute,[br]Magdalena, oh my god, I've

0:43:34.912,0:43:37.986
got a headache.

0:43:37.986,0:43:38.970
I'm not here.

0:43:38.970,0:43:42.530
Z0 is easy to understand[br]for everybody, right?

0:43:42.530,0:43:47.040
That's going to be[br]altitude at the point x0y0.

0:43:47.040,0:43:48.620
It shouldn't be hard.

0:43:48.620,0:43:51.560


0:43:51.560,0:43:54.530
On the other hand,[br]what am I doing?

0:43:54.530,0:44:00.090
I have to look at a real[br]graph, in the real world.

0:44:00.090,0:44:04.860
And that's going to be a[br]patch of a smooth surface.

0:44:04.860,0:44:08.640
And I say, OK, this[br]is my favorite point.

0:44:08.640,0:44:12.330
I have x0y0 on the ground.

0:44:12.330,0:44:16.030
And the corresponding[br]point in three dimensions,

0:44:16.030,0:44:21.990
would be x0y0 and z0,[br]which is the f of x0y0.

0:44:21.990,0:44:24.210
And you say, wait a[br]minute, what do you mean

0:44:24.210,0:44:25.860
I can't move in a direction?

0:44:25.860,0:44:33.430
Is it like when took[br]a sleigh and we went

0:44:33.430,0:44:35.680
to have fun on the hill?

0:44:35.680,0:44:38.430
Yes, but I said that[br]would be the last time

0:44:38.430,0:44:42.620
we talked about the hilly[br]area with snow on it.

0:44:42.620,0:44:47.545
It was a good preparation[br]for today in the sense that--

0:44:47.545,0:44:51.522
Remember, we went somewhere when[br]I picked your direction north,

0:44:51.522,0:44:52.830
east?

0:44:52.830,0:44:54.590
i plus j?

0:44:54.590,0:44:57.150
And in the direction[br]of i plus j,

0:44:57.150,0:44:59.104
which is not quite[br]the direction and I'll

0:44:59.104,0:45:04.530
ask you why in a second, I was[br]going down along a meridian.

0:45:04.530,0:45:05.570
Remember last time?

0:45:05.570,0:45:11.820
And then that was the direction[br]of the steepest descent.

0:45:11.820,0:45:12.830
I was sliding down.

0:45:12.830,0:45:16.650
If I wanted the direction[br]of the steepest ascent,

0:45:16.650,0:45:19.680
that would have been[br]minus i minus j.

0:45:19.680,0:45:23.460
So I had plus i plus[br]j, minus i minus j.

0:45:23.460,0:45:26.087
And I told you last time, why[br]are those not quite directions?

0:45:26.087,0:45:27.670
STUDENT: Because[br]they are not unitary.

0:45:27.670,0:45:29.211
MAGDALENA TODA: They[br]are not unitary.

0:45:29.211,0:45:31.840
So to make them like[br]this u, I should

0:45:31.840,0:45:34.270
have said, in the[br]direction i plus

0:45:34.270,0:45:37.140
j, that was one minus[br]x squared minus y

0:45:37.140,0:45:41.430
squared, the parabola way,[br]that was the hill full of snow.

0:45:41.430,0:45:45.220
So in the direction[br]i plus j, I go down

0:45:45.220,0:45:47.140
the fastest possible way.

0:45:47.140,0:45:50.760
In the direction i plus[br]j over square root of 2,

0:45:50.760,0:45:53.580
I would be fine[br]with a unit vector.

0:45:53.580,0:45:57.660
In the opposite direction, I[br]go up the fastest way possible,

0:45:57.660,0:46:01.600
but you don't want to because[br]it's-- can you imagine hiking

0:46:01.600,0:46:09.021
the steepest possible[br]direction in the steepest way?

0:46:09.021,0:46:14.750


0:46:14.750,0:46:16.360
Now with my direction.

0:46:16.360,0:46:22.340
My direction in plane[br]should be the i vector.

0:46:22.340,0:46:25.600
And that magic vector should[br]have length 1 from here

0:46:25.600,0:46:26.100
to here.

0:46:26.100,0:46:29.500
And when you measure this[br]guy, he has to have length 1.

0:46:29.500,0:46:35.466
And if you decompose, you have[br]to decompose him along the--

0:46:35.466,0:46:36.940
what is this?

0:46:36.940,0:46:40.090
The x direction and[br]the y direction, right?

0:46:40.090,0:46:47.450
How do you split a vector[br]in such a decomposition?

0:46:47.450,0:46:54.110
Well, Mr. u will be u1i plus 1i.

0:46:54.110,0:46:55.710
It sounds funny.

0:46:55.710,0:46:58.550
Plus u2j.

0:46:58.550,0:47:02.220
So you have u1[br]from here to here.

0:47:02.220,0:47:04.070
I don't well you can draw.

0:47:04.070,0:47:06.790
I think some of you can[br]draw really well, especially

0:47:06.790,0:47:10.680
better than me because you[br]took technical drawing.

0:47:10.680,0:47:13.952
How many of you took technical[br]drawing in this glass?

0:47:13.952,0:47:15.860
STUDENT: Only in this class?

0:47:15.860,0:47:17.270
MAGDALENA TODA: In anything.

0:47:17.270,0:47:17.630
STUDENT: In high school.

0:47:17.630,0:47:19.254
MAGDALENA TODA: High[br]school or college.

0:47:19.254,0:47:21.700
STUDENT: I went to[br]it in middle school.

0:47:21.700,0:47:23.660
So it gives you so[br]that [INAUDIBLE]

0:47:23.660,0:47:24.740
and you'd have to[br]draw it. [INAUDIBLE].

0:47:24.740,0:47:26.364
MAGDALENA TODA: It's[br]really helping you

0:47:26.364,0:47:31.090
with the perspective view,[br]3D view, from an angle.

0:47:31.090,0:47:33.490
So now you're looking[br]at this u direction

0:47:33.490,0:47:35.930
as being u1i plus u2j.

0:47:35.930,0:47:39.810
And you say, OK, I think[br]I know what's going on.

0:47:39.810,0:47:47.085
You have a displacement[br]in the direction of the x

0:47:47.085,0:47:51.740
coordinate by 1 times h.

0:47:51.740,0:47:54.570
So it's a small displacement[br]that you're talking about.

0:47:54.570,0:47:56.890
And-- yes?

0:47:56.890,0:47:58.326
STUDENT: Why 1 [INAUDIBLE]?

0:47:58.326,0:48:01.917


0:48:01.917,0:48:03.000
MAGDALENA TODA: Which one?

0:48:03.000,0:48:04.319
STUDENT: You said 1 times H.

0:48:04.319,0:48:05.110
MAGDALENA TODA: u1.

0:48:05.110,0:48:08.719


0:48:08.719,0:48:09.760
You will see in a second.

0:48:09.760,0:48:12.416
That's the way you define it.

0:48:12.416,0:48:15.080
This is adjusted information.

0:48:15.080,0:48:18.940
I would like you to tell me[br]what the whole animal is, if I

0:48:18.940,0:48:21.240
want to represent it later.

0:48:21.240,0:48:23.980
And if you can give[br]me some examples.

0:48:23.980,0:48:29.450
And if I go in a y direction[br]with a small displacement,

0:48:29.450,0:48:33.600
from y0, I have to leave and go.

0:48:33.600,0:48:38.030
So I am here at x0y0.

0:48:38.030,0:48:41.960
And this is the x direction[br]and this is the y direction.

0:48:41.960,0:48:47.660
And when I displace a little[br]bit, I displace with the green.

0:48:47.660,0:48:49.890
I displace in this direction.

0:48:49.890,0:48:54.455
I will have to displace[br]and see what happens here.

0:48:54.455,0:48:58.100


0:48:58.100,0:49:02.182
And then in this direction--[br]I'm not going to write it yet.

0:49:02.182,0:49:04.190
So I'm displacing[br]in this direction

0:49:04.190,0:49:06.530
and in that direction.

0:49:06.530,0:49:08.490
Why am I keeping it h?

0:49:08.490,0:49:12.940
Well, because I have the[br]coordinates x0y0 plus--

0:49:12.940,0:49:21.125
how do you give me a collinear[br]vector to u, but a small one?

0:49:21.125,0:49:23.480
You say, wait a minute,[br]I know what you mean.

0:49:23.480,0:49:28.410
I start from the point x0, this[br]is p, plus a small multiple

0:49:28.410,0:49:31.990
of the direction you give me.

0:49:31.990,0:49:34.550
So here, you had it[br]before in Calc 2.

0:49:34.550,0:49:40.640
You had t times uru2,[br]which is my vector, u.

0:49:40.640,0:49:55.112
So give me a very small[br]displacement vector

0:49:55.112,0:50:04.665
in the direction u, which[br]is u1u2, u2 as a vector.

0:50:04.665,0:50:06.162
You like angular graphics.

0:50:06.162,0:50:07.660
I don't, but it doesn't matter.

0:50:07.660,0:50:09.829
STUDENT: So basically, h.

0:50:09.829,0:50:11.245
MAGDALENA TODA:[br]So basically, this

0:50:11.245,0:50:16.060
is x0 plus-- you want t or h?

0:50:16.060,0:50:17.640
t or h, it doesn't matter.

0:50:17.640,0:50:23.470
hu1, ui0 plus hu2.

0:50:23.470,0:50:24.430
Why not t?

0:50:24.430,0:50:27.310
Why did I take h?

0:50:27.310,0:50:30.470
It is like time parameter[br]that I'm doing with h,

0:50:30.470,0:50:33.990
but h is a very[br]small time parameter.

0:50:33.990,0:50:36.190
It's an infinitesimally[br]small time.

0:50:36.190,0:50:41.050
It's just a fraction of[br]a second after I start.

0:50:41.050,0:50:43.800
That's why I use little[br]h and not little t.

0:50:43.800,0:50:46.500


0:50:46.500,0:50:52.230
H, in general, indicates a[br]very small time displacement.

0:50:52.230,0:50:58.180
So tried to say,[br]where am I here?

0:50:58.180,0:51:02.480
I'm here, just one step further[br]with a small displacement.

0:51:02.480,0:51:06.150
And that's going to p[br]at this whole thing.

0:51:06.150,0:51:11.030


0:51:11.030,0:51:17.322
Let's call this F of--[br]the blue one is F of x0y0.

0:51:17.322,0:51:23.140


0:51:23.140,0:51:27.940
And the green altitude, or the[br]altitude of the green point,

0:51:27.940,0:51:29.780
will be what?

0:51:29.780,0:51:31.850
Well, this is[br]something, something,

0:51:31.850,0:51:44.002
and the altitude would be F[br]of x0 plus hu1, y0 plus hu2.

0:51:44.002,0:51:49.630
And I measure how far[br]away the altitudes are.

0:51:49.630,0:51:50.750
They are very close.

0:51:50.750,0:51:53.380
The blue altitude and[br]the green altitude

0:51:53.380,0:51:55.185
varies the displacement.

0:51:55.185,0:51:57.640
And how can I draw that?

0:51:57.640,0:51:58.630
Here.

0:51:58.630,0:52:00.134
You see this one?

0:52:00.134,0:52:02.062
This is the delta z.

0:52:02.062,0:52:05.920
So this thing is like[br]a delta z kind of guy.

0:52:05.920,0:52:06.504
Any questions?

0:52:06.504,0:52:08.711
It's a little bit hard, but[br]you will see in a second.

0:52:08.711,0:52:09.330
Yes, sir?

0:52:09.330,0:52:11.460
STUDENT: Is it like[br]a small displacement

0:52:11.460,0:52:17.290
that has to be perpendicular[br]to the [INAUDIBLE]?

0:52:17.290,0:52:18.202
MAGDALENA TODA: No.

0:52:18.202,0:52:19.785
STUDENT: It's a[br]result of [INAUDIBLE]?

0:52:19.785,0:52:21.282
MAGDALENA TODA: It[br]is in the direction.

0:52:21.282,0:52:22.365
STUDENT: In the direction?

0:52:22.365,0:52:24.290
MAGDALENA TODA: So[br]let's model it better.

0:52:24.290,0:52:27.170
I don't have a three[br]dimensional-- they sent me

0:52:27.170,0:52:29.310
an email this morning[br]from the library saying,

0:52:29.310,0:52:31.780
do you want your three[br]dimensional print--

0:52:31.780,0:52:35.400
do you want to support the idea[br]of Texas Tech having a three

0:52:35.400,0:52:39.691
dimensional printer available[br]for educational purposes?

0:52:39.691,0:52:41.232
STUDENT: Did you[br]say, of course, yes?

0:52:41.232,0:52:43.235
MAGDALENA TODA: Of[br]course, I would.

0:52:43.235,0:52:45.110
But I don't have a three[br]dimensional printer,

0:52:45.110,0:52:47.590
but you have[br]imagination and imagine

0:52:47.590,0:52:50.450
we have a surface that,[br]again, looks like a hill.

0:52:50.450,0:52:52.710
That's my hand.

0:52:52.710,0:52:58.354
And this engagement ring[br]that I have is actually p0,

0:52:58.354,0:52:59.020
which is x0y0zz.

0:52:59.020,0:53:03.590


0:53:03.590,0:53:09.246
And I'm going in a[br]direction of somebody.

0:53:09.246,0:53:10.246
It doesn't have to be u.

0:53:10.246,0:53:12.240
No, [INAUDIBLE].

0:53:12.240,0:53:14.450
So I'm going in the[br]direction of u-- yu2,

0:53:14.450,0:53:18.000
is that horizontal thing.

0:53:18.000,0:53:20.340
I'm going in that direction.

0:53:20.340,0:53:22.570
So this is the[br]direction I'm going in

0:53:22.570,0:53:25.430
and I say, OK, where do I go?

0:53:25.430,0:53:29.030
We'll do a small displacement,[br]an infinitesimally small

0:53:29.030,0:53:32.340
displacement in[br]that direction here.

0:53:32.340,0:53:37.970
So the two points are[br]related to one another.

0:53:37.970,0:53:42.480
And you say, but there's such[br]a small difference in altitudes

0:53:42.480,0:53:44.980
because you have an[br]infinitesimally small

0:53:44.980,0:53:47.080
displacement in that direction.

0:53:47.080,0:53:47.580
Yes, I know.

0:53:47.580,0:53:50.955
But when you make the ratio[br]between that small delta

0:53:50.955,0:53:57.670
z and the small h, the ratio[br]could be 65 or 120 minus 32.

0:53:57.670,0:53:59.420
You don't know what you get.

0:53:59.420,0:54:04.290
So just like in general limit[br]of the difference quotient

0:54:04.290,0:54:10.260
being the derivative, you'll get[br]the ratio between some things

0:54:10.260,0:54:12.110
that are very small.

0:54:12.110,0:54:15.050
But in the end, you can[br]get something unexpected.

0:54:15.050,0:54:16.360
Finite or anything.

0:54:16.360,0:54:21.340
Now what do you think[br]this guy-- according

0:54:21.340,0:54:27.280
to your previous Chain[br]Rule preparation.

0:54:27.280,0:54:31.330
I taught you about Chain Rule.

0:54:31.330,0:54:36.440
What will this be[br]if we compute them?

0:54:36.440,0:54:37.830
There is a proof for this.

0:54:37.830,0:54:41.430
It would be like a[br]page or a 2 page proof

0:54:41.430,0:54:44.070
for what I'm claiming to have.

0:54:44.070,0:54:45.870
Or how do you think[br]I'm going to get

0:54:45.870,0:54:49.750
to this without doing the[br]limit of a difference quotient?

0:54:49.750,0:54:51.540
Because if I give[br]you functions and you

0:54:51.540,0:54:53.206
do the limit of the[br]difference quotients

0:54:53.206,0:54:56.970
for some nasty functions,[br]you'll never finish.

0:54:56.970,0:55:01.820
So what do you think[br]we ought to do?

0:55:01.820,0:55:06.152
This is going to be some[br]sort of derivative, right?

0:55:06.152,0:55:09.144
And it's going to be[br]a derivative of what?

0:55:09.144,0:55:11.040
Yes, sir.

0:55:11.040,0:55:14.440
STUDENT: Well, it's going to[br]be like a partial derivative,

0:55:14.440,0:55:20.006
except the plane you're[br]using to cut the surface

0:55:20.006,0:55:22.660
is not going to be in the x[br]direction or the y direction.

0:55:22.660,0:55:24.324
It's going to be[br]along the [? uz. ?]

0:55:24.324,0:55:25.240
MAGDALENA TODA: Right.

0:55:25.240,0:55:28.190
So that is a very[br]good observation.

0:55:28.190,0:55:31.600
And it would be like I would the[br]partial not in this direction,

0:55:31.600,0:55:34.080
not in that direction,[br]but in this direction.

0:55:34.080,0:55:35.330
Let me tell you what this is.

0:55:35.330,0:55:38.030
So according to a[br]theorem, this would

0:55:38.030,0:55:42.620
be df, dx, exactly[br]like The Chain Rule,

0:55:42.620,0:55:49.670
at my favorite point[br]here, x0y0 [INAUDIBLE]

0:55:49.670,0:55:55.070
p times-- now you say, oh,[br]Magdalena, I understand.

0:55:55.070,0:55:56.730
You're doing some[br]sort of derivation.

0:55:56.730,0:56:02.380
The derivative of that with[br]respect to h would be u1.

0:56:02.380,0:56:02.880
Yes.

0:56:02.880,0:56:04.090
It's a Chain Rule.

0:56:04.090,0:56:12.946
So then I go times u1 plus[br]df, dy at the point times u2.

0:56:12.946,0:56:14.790


0:56:14.790,0:56:18.370
And you say, OK, but[br]can I prove that?

0:56:18.370,0:56:21.130
Yes, you could, but[br]to prove that you

0:56:21.130,0:56:26.136
would need to play a game.

0:56:26.136,0:56:30.530
The proof will involve that[br]you multiply up and down

0:56:30.530,0:56:32.770
by an additional expression.

0:56:32.770,0:56:35.364
And then you take[br]limit of a product.

0:56:35.364,0:56:37.320
If you take product,[br]the product of limits,

0:56:37.320,0:56:43.090
and you study them separately[br]until you get to this Actually,

0:56:43.090,0:56:47.360
this is an application[br]of The Chain.

0:56:47.360,0:56:54.440
But I want to come back to[br]what Alexander just notice.

0:56:54.440,0:56:57.550
I can explain this[br]much better if we only

0:56:57.550,0:57:01.864
think of derivative in the[br]direction of i and derivative

0:57:01.864,0:57:02.780
in the direction of j.

0:57:02.780,0:57:04.680
What the heck are those?

0:57:04.680,0:57:07.360
What are they going to be?

0:57:07.360,0:57:13.690
The direction of deritivie--[br]if I have i instead of u, that

0:57:13.690,0:57:17.179
will make you understand the[br]whole notion much better.

0:57:17.179,0:57:18.970
So what would be the[br]directional derivative

0:57:18.970,0:57:22.650
of in the direction of i only?

0:57:22.650,0:57:23.850
Well, i for an i.

0:57:23.850,0:57:25.132
It goes this way.

0:57:25.132,0:57:27.020
This is a hard lesson.

0:57:27.020,0:57:29.852
And it's advanced calculus[br]rather than Calc 3,

0:57:29.852,0:57:32.220
but you're going to get it.

0:57:32.220,0:57:36.800
So if I go in the[br]direction of i,

0:57:36.800,0:57:40.720
I should have the df, dx, right?

0:57:40.720,0:57:41.890
That should be it.

0:57:41.890,0:57:42.837
Do I?

0:57:42.837,0:57:44.170
STUDENT: Yes, but [INAUDIBLE] 0.

0:57:44.170,0:57:45.170
MAGDALENA TODA: Exactly.

0:57:45.170,0:57:47.620
Was I able to[br]invent something so

0:57:47.620,0:57:53.210
when I come back to what I[br]already know, I recreate df, dx

0:57:53.210,0:57:56.060
and nothing else?

0:57:56.060,0:58:04.752
Precisely because for i as[br]being u, what will be u1 and u2?

0:58:04.752,0:58:06.680
STUDENT: [INAUDIBLE].

0:58:06.680,0:58:09.110
MAGDALENA TODA: u1 is 1.

0:58:09.110,0:58:11.621
u2 is 0.

0:58:11.621,0:58:12.120
Right?

0:58:12.120,0:58:16.300
Because when we write i[br]as a function of i and j,

0:58:16.300,0:58:19.070
that's 1 times i plus 0 times j.

0:58:19.070,0:58:22.946
So u1 is 1, u2 is zero.

0:58:22.946,0:58:24.110
Thank god.

0:58:24.110,0:58:26.950
According to the anything,[br]this difference quotient

0:58:26.950,0:58:32.155
or the simpler way to define[br]it from the theorem would

0:58:32.155,0:58:34.620
be simply the second goes away.

0:58:34.620,0:58:36.240
It vanishes.

0:58:36.240,0:58:41.540
u1 would be 1 and what[br]I'm left with is df, dx.

0:58:41.540,0:58:45.071
And that's exactly[br]what Alex noticed.

0:58:45.071,0:58:49.350
So the directional[br]derivative is defined,

0:58:49.350,0:58:53.770
as a combination of vectors,[br]such that you recreate

0:58:53.770,0:58:56.350
the directional derivative[br]in the direction of i

0:58:56.350,0:58:59.190
being the partial, df, dx.

0:58:59.190,0:59:02.810
Exactly like you[br]learned before in 11.3.

0:59:02.810,0:59:06.050
And what do I have[br]if I try to recreate

0:59:06.050,0:59:10.141
the directional derivative[br]in the direct of j?

0:59:10.141,0:59:10.640
x0y0.

0:59:10.640,0:59:14.500
We don't explain this[br]much in the book.

0:59:14.500,0:59:17.510
I think on this one, I'm doing[br]a better job than the book.

0:59:17.510,0:59:21.750
So what is df in[br]the direction of j?

0:59:21.750,0:59:23.890
j is this way.

0:59:23.890,0:59:27.130
Well, [INAUDIBLE][br]is that 1j-- you

0:59:27.130,0:59:32.170
let me write it[br]down-- is 0i plus 1j.

0:59:32.170,0:59:33.550
0 is u1.

0:59:33.550,0:59:36.250
1 is u2.

0:59:36.250,0:59:41.980
So by this formula,[br]I simply should

0:59:41.980,0:59:47.970
get the directional[br]deritive-- I mean,

0:59:47.970,0:59:50.975
directional derivative is the[br]partial deritive-- with respect

0:59:50.975,0:59:56.890
to y at my point times a 1[br]that I'm not going to write.

0:59:56.890,1:00:07.330
So it's a concoction, so that[br]in the directions of i and j,

1:00:07.330,1:00:10.600
you actually get the[br]partial deritives.

1:00:10.600,1:00:13.020
And everything else[br]is linear algebra.

1:00:13.020,1:00:19.550
So if you have a problem[br]understanding the composition

1:00:19.550,1:00:21.640
of vectors, the sum[br]of vectors, this

1:00:21.640,1:00:25.568
is because-- u1 and[br]u2 are [INAUDIBLE],

1:00:25.568,1:00:28.250
I'm sorry-- this is[br]because you haven't taken

1:00:28.250,1:00:33.348
the linear algebra yet, which[br]teaches you a lot about how

1:00:33.348,1:00:36.330
a vector decomposes in[br]two different directions

1:00:36.330,1:00:39.312
or along the standard[br]canonical bases.

1:00:39.312,1:00:41.860


1:00:41.860,1:00:44.940
Let's see some[br]problems of the type

1:00:44.940,1:00:49.452
that I've always put in the[br]midterm and the same kind

1:00:49.452,1:00:54.642
of problems like we[br]have seen in the final.

1:00:54.642,1:00:57.559
For example 3, is it, guys?

1:00:57.559,1:00:58.100
I don't know.

1:00:58.100,1:00:59.641
Example 3, 4, or[br]something like that?

1:00:59.641,1:01:00.460
STUDENT: 3.

1:01:00.460,1:01:03.130
MAGDALENA TODA: Given[br]z equals F of xy--

1:01:03.130,1:01:06.430
what do you like best,[br]the value or the hill?

1:01:06.430,1:01:09.250
This appeared in[br]most of my exams.

1:01:09.250,1:01:12.510
x squared plus y squared,[br]circular [INAUDIBLE]

1:01:12.510,1:01:14.470
was one of my favorite examples.

1:01:14.470,1:01:16.470
1 minus x squared[br]minus y squared

1:01:16.470,1:01:22.930
was the circular[br]parabola upside down.

1:01:22.930,1:01:24.330
Which one do you prefer?

1:01:24.330,1:01:25.470
I don't care.

1:01:25.470,1:01:26.495
Which one?

1:01:26.495,1:01:27.370
STUDENT: [INAUDIBLE].

1:01:27.370,1:01:27.680
MAGDALENA TODA: The [INAUDIBLE]?

1:01:27.680,1:01:29.240
The first one.

1:01:29.240,1:01:30.060
It's easier.

1:01:30.060,1:01:34.870


1:01:34.870,1:01:36.690
And a typical problem.

1:01:36.690,1:01:50.060
Compute the directional[br]derivative of z

1:01:50.060,1:02:00.050
equals F of x and y at the[br]point p of coordinates 1, 1, 2

1:02:00.050,1:02:14.050
in the following[br]directions-- A, i.

1:02:14.050,1:02:15.220
B, j.

1:02:15.220,1:02:18.060
C, i plus j.

1:02:18.060,1:02:24.000


1:02:24.000,1:02:29.180
D, the opposite, minus[br]i, minus j over square 2.

1:02:29.180,1:02:31.960
And E--

1:02:31.960,1:02:33.460
STUDENT: That's a square root 3.

1:02:33.460,1:02:34.460
MAGDALENA TODA: What?

1:02:34.460,1:02:36.040
STUDENT: You wrote[br]a square root 3.

1:02:36.040,1:02:37.080
MAGDALENA TODA: I[br]wrote square root of 3.

1:02:37.080,1:02:37.705
Thank you guys.

1:02:37.705,1:02:38.880
Thanks for being vigilant.

1:02:38.880,1:02:43.264
So always keep an eye on me[br]because I'm full of surprises,

1:02:43.264,1:02:43.805
good and bad.

1:02:43.805,1:02:46.210
No, just kidding.

1:02:46.210,1:02:47.807
So let's see.

1:02:47.807,1:02:48.932
What do I want to put here?

1:02:48.932,1:02:51.402
Something.

1:02:51.402,1:02:52.390
How about this?

1:02:52.390,1:03:01.282


1:03:01.282,1:03:06.716
3 over root 5, pi[br]plus [? y ?] over 5j.

1:03:06.716,1:03:10.668
Is this a unit vector or not?

1:03:10.668,1:03:12.150
STUDENT: No.

1:03:12.150,1:03:13.474
STUDENT: Yes, it is.

1:03:13.474,1:03:15.140
So you're going to[br]drag the [INAUDIBLE].

1:03:15.140,1:03:17.031
MAGDALENA TODA: Why[br]is that a unit vector?

1:03:17.031,1:03:18.761
STUDENT: It's[br]missing-- no, it's not.

1:03:18.761,1:03:20.927
MAGDALENA TODA: Then how[br]do I make it a unit vector?

1:03:20.927,1:03:22.875
STUDENT: [INAUDIBLE].

1:03:22.875,1:03:24.840
STUDENT: [INAUDIBLE].

1:03:24.840,1:03:28.362
STUDENT: I have to take down--[br]there's a 3 that has to be 1.

1:03:28.362,1:03:29.346
[INAUDIBLE]

1:03:29.346,1:03:32.298
And the second one has[br]to be 1, on the top,

1:03:32.298,1:03:34.266
to make it a unit vector.

1:03:34.266,1:03:39.200


1:03:39.200,1:03:41.560
MAGDALENA TODA: Give[br]me a unit vector.

1:03:41.560,1:03:46.668
Another one then[br]these easy ones.

1:03:46.668,1:03:48.233
STUDENT: 3 over 5 by 4 or 5.

1:03:48.233,1:03:49.108
MAGDALENA TODA: What?

1:03:49.108,1:03:52.050
STUDENT: 3 over 5 by 4 over 5j.

1:03:52.050,1:03:53.810
MAGDALENA TODA: 3[br]over-- I cannot hear.

1:03:53.810,1:03:54.080
STUDENT: 3 over 5--

1:03:54.080,1:03:55.420
MAGDALENA TODA: 3 over 5.

1:03:55.420,1:03:56.970
STUDENT: And 4 over 5j.

1:03:56.970,1:03:58.710
MAGDALENA TODA: And 4 over 5j.

1:03:58.710,1:04:00.560
And why is that a unit vector?

1:04:00.560,1:04:05.160
STUDENT: Because 3[br]squared is [INAUDIBLE].

1:04:05.160,1:04:07.284
MAGDALENA TODA: And what[br]do we call these numbers?

1:04:07.284,1:04:08.200
You say, what is that?

1:04:08.200,1:04:10.711
And interview?

1:04:10.711,1:04:12.920
Yes, it is an interview.

1:04:12.920,1:04:13.890
Pythagorean numbers.

1:04:13.890,1:04:16.162
3, 4, and 5 are[br]Pythagorean numbers.

1:04:16.162,1:04:19.180


1:04:19.180,1:04:23.840
So let me think a little[br]bit where I should write.

1:04:23.840,1:04:26.200
Is this seen by[br]the-- yes, it's seen

1:04:26.200,1:04:34.438
by the-- I'll just leave[br]what's important for me

1:04:34.438,1:04:35.875
to solve this problem.

1:04:35.875,1:04:44.990


1:04:44.990,1:04:48.160
A. So what do we do?

1:04:48.160,1:04:55.580
The same thing. i is 1.i plus[br]u, or 1 times i plus u times j.

1:04:55.580,1:04:58.675
So simply, you can write[br]the formula or you can say,

1:04:58.675,1:05:01.430
the heck with the formula.

1:05:01.430,1:05:03.920
You know that df is df, dx.

1:05:03.920,1:05:07.810
The derivative of[br]this at the point p.

1:05:07.810,1:05:14.022
So what you want to do is say,[br]2x-- are you guys with me?

1:05:14.022,1:05:15.160
STUDENT: Yes.

1:05:15.160,1:05:23.090
MAGDALENA TODA: At the[br]value 1, 1, 2, which is 2.

1:05:23.090,1:05:24.940
And at the end of[br]this exercise, I'm

1:05:24.940,1:05:28.430
going to ask you if there's[br]any connection between--

1:05:28.430,1:05:30.610
or maybe I will[br]ask you next time.

1:05:30.610,1:05:34.590
Oh, we have time.

1:05:34.590,1:05:37.540
What is d in the direction of j?

1:05:37.540,1:05:41.070
The partial derivative[br]with respect to y.

1:05:41.070,1:05:43.600
Nothing else, but[br]our old friend.

1:05:43.600,1:05:47.680
And our old friend[br]says, I have 2y

1:05:47.680,1:05:51.780
computed for the[br]point p, 1, 1, 2.

1:05:51.780,1:05:52.980
What does it mean?

1:05:52.980,1:05:58.794
Y is 1, so just plug[br]this 1 into the thingy.

1:05:58.794,1:05:59.738
It's 2.

1:05:59.738,1:06:03.990


1:06:03.990,1:06:07.110
Now do I see some--[br]I'm a scientist.

1:06:07.110,1:06:09.200
I have to find[br]interpretations when

1:06:09.200,1:06:11.210
I get results that coincide.

1:06:11.210,1:06:12.645
It's a pattern.

1:06:12.645,1:06:14.014
Why do I get the same answer?

1:06:14.014,1:06:15.930
STUDENT: Because your[br]functions are symmetric.

1:06:15.930,1:06:16.846
MAGDALENA TODA: Right.

1:06:16.846,1:06:20.070
And more than that, because[br]the function is symmetric,

1:06:20.070,1:06:24.621
it's a quadric that I love,[br]it's just a circular problem.

1:06:24.621,1:06:27.850
It's rotation is symmetric.

1:06:27.850,1:06:33.500
So I just take one parabola,[br]one branch of a parabola,

1:06:33.500,1:06:38.400
and I rotate it by 360 degrees.

1:06:38.400,1:06:45.760
So the slope will be the same[br]in both directions, i and j,

1:06:45.760,1:06:47.255
at the point that I have.

1:06:47.255,1:06:49.860


1:06:49.860,1:06:52.750
Well, it depends on the point.

1:06:52.750,1:06:55.145
If the point is,[br]itself, symmetric

1:06:55.145,1:06:58.490
like that, x and y are[br]the same, one in one,

1:06:58.490,1:07:03.750
I did it on purpose-- if[br]you didn't have one and one,

1:07:03.750,1:07:07.520
you had an x variable and[br]y variable to plug in.

1:07:07.520,1:07:10.960
But your magic point is where?

1:07:10.960,1:07:11.780
Oh my god.

1:07:11.780,1:07:15.370
I don't know how to[br]explain with my hands.

1:07:15.370,1:07:16.650
Here I am, the frame.

1:07:16.650,1:07:19.910
I am the frame. x, y, and z.

1:07:19.910,1:07:21.820
1, 1.

1:07:21.820,1:07:23.220
Go up.

1:07:23.220,1:07:25.060
Where do you meet the vase?

1:07:25.060,1:07:26.900
At c equals 2.

1:07:26.900,1:07:30.356
So it's really symmetric[br]and really beautiful.

1:07:30.356,1:07:34.140


1:07:34.140,1:07:37.800
Next I say, oh, in[br]the direction i plus

1:07:37.800,1:07:43.590
j, which is exactly the[br]direction of this meridian

1:07:43.590,1:07:47.782
that I was talking about, i[br]plus j over square root 2.

1:07:47.782,1:07:50.780
Now I've had students-- that's[br]where I was broken hearted.

1:07:50.780,1:07:53.330
Really, I didn't[br]know what to do,

1:07:53.330,1:07:56.500
how much partial credit to give.

1:07:56.500,1:08:00.680
The definition of direction[br]derivative is very strict.

1:08:00.680,1:08:04.750
It says you cannot take[br]whatever 1 and 2 that you want.

1:08:04.750,1:08:09.490
You cannot multiply[br]them by proportionality.

1:08:09.490,1:08:14.410
You have to have u[br]to be a unit vector.

1:08:14.410,1:08:18.279
And then the directional[br]derivative will be unique.

1:08:18.279,1:08:24.319
If I take 1 and 1 for u1 and[br]u2, then I can take 2 and 2,

1:08:24.319,1:08:26.189
and 7 and 7, and 9 and 9.

1:08:26.189,1:08:28.399
And that's going to[br]be a mess because

1:08:28.399,1:08:32.040
the directional derivative[br]wouldn't be unique anymore.

1:08:32.040,1:08:36.020
And that's why whoever[br]gave this definition,

1:08:36.020,1:08:39.104
I think Euler-- I tried[br]to see in the history who

1:08:39.104,1:08:42.779
was the first[br]mathematician who gave

1:08:42.779,1:08:46.950
the definition of the[br]directional derivative.

1:08:46.950,1:08:49.710
And some people[br]said it was Gateaux

1:08:49.710,1:08:53.196
because that's a french[br]mathematician who first talked

1:08:53.196,1:08:55.231
about the Gateaux[br]derivative, which

1:08:55.231,1:08:56.689
is like the[br]directional derivative,

1:08:56.689,1:08:58.859
but other people said,[br]no, look at Euler's work.

1:08:58.859,1:09:00.290
He was a genius.

1:09:00.290,1:09:04.710
He's the guy who discovered[br]the transcendental number

1:09:04.710,1:09:06.740
e and many other things.

1:09:06.740,1:09:09.080
And the exponential[br]e to the x is also

1:09:09.080,1:09:10.510
from Euler and everything.

1:09:10.510,1:09:12.569
He was one of the[br]fathers of calculus.

1:09:12.569,1:09:19.060
Apparently, he knew the first[br]32 decimals of the number e.

1:09:19.060,1:09:22.910
And how he got to[br]them is by hand.

1:09:22.910,1:09:24.090
Do you guys know of them?

1:09:24.090,1:09:29.620
2.71828-- and that's all I know.

1:09:29.620,1:09:32.000
The first five decimals.

1:09:32.000,1:09:35.729
Well, he knew 32 of them[br]and he got to them by hand.

1:09:35.729,1:09:39.200
And they are non-repeating,[br]infinitely remaining decimals.

1:09:39.200,1:09:40.460
It's a transcendental number.

1:09:40.460,1:09:41.858
STUDENT: And his 32 are correct?

1:09:41.858,1:09:42.733
MAGDALENA TODA: What?

1:09:42.733,1:09:44.180
STUDENT: His 32 are correct?

1:09:44.180,1:09:46.960
MAGDALENA TODA: His first[br]32 decimals were correct.

1:09:46.960,1:09:49.790
I don't know what--[br]I mean, the guy

1:09:49.790,1:09:53.260
was something like-- he[br]was working at night.

1:09:53.260,1:09:56.690
And he would fill out,[br]in one night, hundreds

1:09:56.690,1:10:04.270
of pages, computations, both[br]by hand formulas and numerical.

1:10:04.270,1:10:07.170
So imagine-- of course, he would[br]never make a WeBWork mistake.

1:10:07.170,1:10:10.866
I mean, if we built[br]a time machine,

1:10:10.866,1:10:13.240
and we bring Euler back,[br]and he's at Texas Tech,

1:10:13.240,1:10:16.160
and we make him solve[br]our WeBWork problems,

1:10:16.160,1:10:17.910
I think he would take[br]a thousand problems

1:10:17.910,1:10:19.880
and solve them in one night.

1:10:19.880,1:10:21.850
He need to know[br]how to type, so we

1:10:21.850,1:10:24.250
have to teach him how to type.

1:10:24.250,1:10:27.650
But he would be able to[br]compute what you guys have,

1:10:27.650,1:10:31.790
all those numerical[br]answers, in his head.

1:10:31.790,1:10:35.250
He was a scary fellow.

1:10:35.250,1:10:41.380
So u has to be [INAUDIBLE][br]in some way, made unique.

1:10:41.380,1:10:43.350
u1 and u2.

1:10:43.350,1:10:45.920
I have students-- that's[br]where the story started--

1:10:45.920,1:10:49.990
who were very good, very smart,[br]both honors and non-honors, who

1:10:49.990,1:10:54.350
took u1 to be 1, u2 to be 2[br]because they thought direction

1:10:54.350,1:11:00.640
1 and 1, which is not made[br]unique as a direction, unitary.

1:11:00.640,1:11:03.420
And they plugged in here[br]1, they plugged in here 1,

1:11:03.420,1:11:08.210
they got these correctly, what[br]was I supposed to give them, as

1:11:08.210,1:11:09.066
a [? friend? ?]

1:11:09.066,1:11:09.941
STUDENT: [INAUDIBLE].

1:11:09.941,1:11:10.430
MAGDALENA TODA: What?

1:11:10.430,1:11:11.530
STUDENT: [INAUDIBLE].

1:11:11.530,1:11:12.696
MAGDALENA TODA: I gave them.

1:11:12.696,1:11:14.413
How much do you think?

1:11:14.413,1:11:15.204
You should know me.

1:11:15.204,1:11:16.198
STUDENT: [INAUDIBLE].

1:11:16.198,1:11:17.192
STUDENT: Full.

1:11:17.192,1:11:18.186
MAGDALENA TODA: 60%.

1:11:18.186,1:11:19.504
No.

1:11:19.504,1:11:20.920
Some people don't[br]give any credit,

1:11:20.920,1:11:22.720
so pay attention to this.

1:11:22.720,1:11:31.498
In this case, this has[br]to be 1 over square root

1:11:31.498,1:11:41.602
of 2 times the derivative of f[br]at x, which is computed before

1:11:41.602,1:11:50.570
at the point, plus 1 over square[br]root of 2 times the derivative

1:11:50.570,1:11:52.025
of the function.

1:11:52.025,1:11:54.450
Again, compute it[br]at the same place.

1:11:54.450,1:12:02.514
Which is, oh my god, square[br]root of 2 plus square root of 2,

1:12:02.514,1:12:04.470
which is 2 square root of 2.

1:12:04.470,1:12:20.630


1:12:20.630,1:12:29.746
And finally, the derivative[br]of F at the same point-- I

1:12:29.746,1:12:31.222
should have put at the point.

1:12:31.222,1:12:35.160
Like a physicist[br]would say, at p.

1:12:35.160,1:12:38.290
That would make you[br]familiar with this notation.

1:12:38.290,1:12:40.330
And then measured at what?

1:12:40.330,1:12:43.540
The opposite direction,[br]minus i minus j.

1:12:43.540,1:12:46.060
And now I'm getting lazy[br]and I'm going to ask you

1:12:46.060,1:12:48.624
what the answer will be.

1:12:48.624,1:12:50.040
STUDENT: 2 minus[br]square root of 2.

1:12:50.040,1:12:53.045
MAGDALENA TODA: So you see,[br]there is another pattern.

1:12:53.045,1:12:55.190
In the opposite[br]direction, the direction

1:12:55.190,1:12:59.500
of the derivative in this case[br]would just be the negative one.

1:12:59.500,1:13:03.190
What if we took this directional[br]derivative in absolute value?

1:13:03.190,1:13:05.374
Because you see,[br]in this direction,

1:13:05.374,1:13:07.610
there's a positive[br]directional derivaty.

1:13:07.610,1:13:11.930
In the other direction, it's[br]like it's because-- I know why.

1:13:11.930,1:13:13.600
I'm a vase.

1:13:13.600,1:13:18.132
So in the direction i plus[br]j over square root of 2,

1:13:18.132,1:13:20.305
the directional derivative[br]will be positive.

1:13:20.305,1:13:21.780
It goes up.

1:13:21.780,1:13:24.260
But in the direction[br]minus i minus

1:13:24.260,1:13:28.290
j, which is the opposite, over[br]square root of 2, it goes down.

1:13:28.290,1:13:30.620
So the slope is negative.

1:13:30.620,1:13:32.200
So that's why we have negative.

1:13:32.200,1:13:34.770
Everything you get[br]in life or in math,

1:13:34.770,1:13:36.410
you have to find[br]an interpretation.

1:13:36.410,1:13:40.354


1:13:40.354,1:13:44.460
Sometimes in life and[br]mathematics, things are subtle.

1:13:44.460,1:13:46.850
People will say one thing[br]and they mean another thing.

1:13:46.850,1:13:49.824
You have to try to see[br]beyond their words.

1:13:49.824,1:13:50.700
That's sad.

1:13:50.700,1:13:53.760
And in mathematics, you have to[br]try to see beyond the numbers.

1:13:53.760,1:13:55.420
You see a pattern.

1:13:55.420,1:13:58.050
So being in opposite[br]directions, I

1:13:58.050,1:14:02.239
got opposite signs of the[br]directional derivative

1:14:02.239,1:14:03.530
because I have opposite slopes.

1:14:03.530,1:14:07.750


1:14:07.750,1:14:10.992
What else do I want to[br]learn in this example?

1:14:10.992,1:14:12.120
One last thing.

1:14:12.120,1:14:13.286
STUDENT: E.

1:14:13.286,1:14:22.670
MAGDALENA TODA: E. So[br]I have the same thing.

1:14:22.670,1:14:25.242
So it's not going to[br]matter, the direction

1:14:25.242,1:14:26.920
is the only thing that changes.

1:14:26.920,1:14:28.960
These guys are the same.

1:14:28.960,1:14:33.630
The partials are the[br]same at the same point.

1:14:33.630,1:14:35.130
I'm not going to[br]worry about them.

1:14:35.130,1:14:39.200
So I get 2 or both.

1:14:39.200,1:14:41.260
What changes is the blue guys.

1:14:41.260,1:14:47.847
They are going to be[br]3 over 5 and 4 over 5.

1:14:47.847,1:14:53.620


1:14:53.620,1:14:56.270
And what do I get?

1:14:56.270,1:15:04.885
I get-- right?

1:15:04.885,1:15:09.110


1:15:09.110,1:15:12.890
Now I want to tell[br]you something--

1:15:12.890,1:15:16.100
I already anticipated[br]something last time.

1:15:16.100,1:15:21.280
And let me tell you[br]what I said last time.

1:15:21.280,1:15:25.970


1:15:25.970,1:15:27.930
Maybe I should not[br]erase-- well, I

1:15:27.930,1:15:30.240
have to erase this[br]whether I like it or not.

1:15:30.240,1:15:33.800


1:15:33.800,1:15:35.775
And now I'll review[br]what this was.

1:15:35.775,1:15:38.180
What was this? d equals[br]x squared plus y squared?

1:15:38.180,1:15:38.960
Yes or no?

1:15:38.960,1:15:41.410
STUDENT: Yes.

1:15:41.410,1:15:46.035
MAGDALENA TODA: So what[br]did I say last time?

1:15:46.035,1:15:52.730
We have no result. We[br]noticed it last time.

1:15:52.730,1:15:55.060
We did not prove it.

1:15:55.060,1:16:08.570
We did not prove it, only[br]found it experimentally

1:16:08.570,1:16:11.810
using our physical common sense.

1:16:11.810,1:16:16.990
When you have a function[br]z equals F of xy,

1:16:16.990,1:16:30.878
we studied the[br]maximum rate of change

1:16:30.878,1:16:39.560
at the point x0y0 in the domain,[br]assuming this is a c1 function.

1:16:39.560,1:16:40.890
I don't know.

1:16:40.890,1:16:44.330
Maximum rate of change[br]was a magic thing.

1:16:44.330,1:16:48.130
And you probably thought,[br]what in the world is that?

1:16:48.130,1:17:01.120
And we also said, this[br]maximum for the rate of change

1:17:01.120,1:17:23.849
is always attained in the[br]direction of the gradient.

1:17:23.849,1:17:31.310


1:17:31.310,1:17:38.050
So you realize that it's[br]the steepest ascent,

1:17:38.050,1:17:40.920
the way it's called in[br]many, many other fields,

1:17:40.920,1:17:42.832
but mathematics.

1:17:42.832,1:17:45.420
Or the steepest descent.

1:17:45.420,1:17:51.530


1:17:51.530,1:17:58.240
Now if it's an ascent, then it's[br]in the direction gradient of F.

1:17:58.240,1:18:00.210
But if it's a[br]descent, it's going

1:18:00.210,1:18:04.630
to be in the opposite[br]direction, minus gradient of F.

1:18:04.630,1:18:07.580
But then I [INAUDIBLE][br]first of all,

1:18:07.580,1:18:11.890
it's not the same direction,[br]if you have opposites.

1:18:11.890,1:18:14.750
Well, direction is sort[br]of given by one line.

1:18:14.750,1:18:18.840
Whether you take this or the[br]opposite, it's the same thing.

1:18:18.840,1:18:21.280
What this means is[br]that we say direction

1:18:21.280,1:18:25.680
and we didn't[br][? unitarize ?] it.

1:18:25.680,1:18:31.050
So we could say,[br]or gradient of F

1:18:31.050,1:18:35.980
over length of gradient of[br]F. Or minus gradient of F

1:18:35.980,1:18:39.750
over length of gradient of F.[br]Can this theorem be proved?

1:18:39.750,1:18:41.330
Yes, it can be proved.

1:18:41.330,1:18:45.370
We are going to discuss a little[br]bit more next time about it,

1:18:45.370,1:18:49.360
but I want to tell you[br]a big disclosure today.

1:18:49.360,1:18:55.020
This maximum rate of change[br]is the directional derivative.

1:18:55.020,1:19:07.808
This maximum rate[br]of change is exactly

1:19:07.808,1:19:15.630
the directional derivative[br]in the direction

1:19:15.630,1:19:35.068
of the gradient, which is also[br]the magnitude of the gradient.

1:19:35.068,1:19:43.380


1:19:43.380,1:19:47.230
And you'll say,[br]wait a minute, what?

1:19:47.230,1:19:48.360
What did you say?

1:19:48.360,1:19:51.082
Let's first verify my claim.

1:19:51.082,1:19:53.350
I'm not even sure[br]my claim is true.

1:19:53.350,1:19:55.480
We will see next time.

1:19:55.480,1:19:59.830
Can I verify my[br]claim on one example?

1:19:59.830,1:20:01.690
Well, OK.

1:20:01.690,1:20:04.870
Maximum rate of change[br]would be exactly

1:20:04.870,1:20:07.964
as the directional[br]derivative and the direction

1:20:07.964,1:20:08.630
of the gradient?

1:20:08.630,1:20:10.070
I don't know about that.

1:20:10.070,1:20:11.382
That all sounds crazy.

1:20:11.382,1:20:12.825
So what do I have to compute?

1:20:12.825,1:20:16.673
I have to compute that[br]directional derivative

1:20:16.673,1:20:21.640
of, let's say, my function F in[br]the direction of the gradient--

1:20:21.640,1:20:22.956
what is the gradient?

1:20:22.956,1:20:26.270


1:20:26.270,1:20:28.815
We have to figure it out.

1:20:28.815,1:20:30.640
We did it last time,[br]but you forgot.

1:20:30.640,1:20:37.360
So for this guy, nabla F,[br]what will be the gradient?

1:20:37.360,1:20:39.740
Where is my function?

1:20:39.740,1:20:47.620
Nabla F will be 2x, 2y, right?

1:20:47.620,1:20:51.750
Which means 2xi plus 2yj, right?

1:20:51.750,1:20:54.676
But if I'm at the point[br]p, what does it mean?

1:20:54.676,1:20:59.020
At the point p, it means that I[br]have 2 times i plus 2 times j,

1:20:59.020,1:21:00.342
right?

1:21:00.342,1:21:06.030
And what is the magnitude[br]of the gradient?

1:21:06.030,1:21:08.140
Yes.

1:21:08.140,1:21:13.292
The magnitude of the gradient is[br]somebody I know, which is what?

1:21:13.292,1:21:18.583
Which is square root of[br]2 squared plus 2 squared.

1:21:18.583,1:21:20.784
I cannot do that now.

1:21:20.784,1:21:21.950
What's the square root of 8?

1:21:21.950,1:21:22.839
STUDENT: 2 root 2.

1:21:22.839,1:21:23.880
MAGDALENA TODA: 2 root 2.

1:21:23.880,1:21:24.629
This is a pattern.

1:21:24.629,1:21:25.230
2 root 2.

1:21:25.230,1:21:27.240
I've seen this 2 root[br]2 again somewhere.

1:21:27.240,1:21:28.880
Where the heck have I seen it?

1:21:28.880,1:21:29.922
STUDENT: That was the[br]directional derivative.

1:21:29.922,1:21:31.713
MAGDALENA TODA: The[br]directional derivative.

1:21:31.713,1:21:33.320
So the claim may be right.

1:21:33.320,1:21:36.452
It says it is the directional[br]derivative in the direction

1:21:36.452,1:21:37.810
of the gradient.

1:21:37.810,1:21:40.920
But is this really the[br]direction of the gradient?

1:21:40.920,1:21:42.770
Yes.

1:21:42.770,1:21:45.910
Because when you compote the[br]direction for the gradient, 2y

1:21:45.910,1:21:52.190
plus 2j, you don't mean 2i[br]plus 2j as a twice i plus j,

1:21:52.190,1:21:55.647
you mean the unit vector[br]correspondent to that.

1:21:55.647,1:21:57.230
So what is the[br]direction corresponding

1:21:57.230,1:22:00.550
to the gradient 2i plus 2j?

1:22:00.550,1:22:01.850
STUDENT: i plus j [? over 2. ?]

1:22:01.850,1:22:02.850
MAGDALENA TODA: Exactly.

1:22:02.850,1:22:06.140
U equals i plus j[br]divided by square 2.

1:22:06.140,1:22:09.310
So this is the[br]directional derivative

1:22:09.310,1:22:13.120
in the direction of the gradient[br]at the point p, which is 2 root

1:22:13.120,1:22:13.620
2.

1:22:13.620,1:22:18.250
And it's the same thing-- for[br]some reason that's mysterious

1:22:18.250,1:22:19.860
and we will see next time.

1:22:19.860,1:22:23.340
For some mysterious reason[br]you get exactly the same

1:22:23.340,1:22:27.780
as the length of[br]the gradient vector.

1:22:27.780,1:22:30.460
We will see about this[br]mystery next time.

1:22:30.460,1:22:35.220
I have you enough to[br]torment you until Tuesday.

1:22:35.220,1:22:38.280
What have you promised me[br]besides doing the homework?

1:22:38.280,1:22:39.756
STUDENT: To read the book.

1:22:39.756,1:22:41.130
MAGDALENA TODA:[br]To read the book.

1:22:41.130,1:22:41.950
You're very smart.

1:22:41.950,1:22:43.620
Please, read the book.

1:22:43.620,1:22:45.078
All the examples in the book.

1:22:45.078,1:22:47.070
They are short.

1:22:47.070,1:22:48.066
Thank you so much.

1:22:48.066,1:22:50.556
Have a wonderful[br]weekend and I'll

1:22:50.556,1:22:54.540
talk to you on Tuesday about[br]anything you have trouble with.

1:22:54.540,1:22:57.030
When is the homework due?

1:22:57.030,1:22:59.022
STUDENT: Saturday.

1:22:59.022,1:23:00.514
MAGDALENA TODA: On Saturday.

1:23:00.514,1:23:01.014
I was mean.

1:23:01.014,1:23:04.500
I should have given it you[br]until Sunday night, but--

1:23:04.500,1:23:05.943
STUDENT: Yes.

1:23:05.943,1:23:08.484
MAGDALENA TODA: Do you want me[br]to make it until Sunday night?

1:23:08.484,1:23:08.982
STUDENT: Yes.

1:23:08.982,1:23:10.148
MAGDALENA TODA: At midnight?

1:23:10.148,1:23:10.974
STUDENT: Yes.

1:23:10.974,1:23:12.966
MAGDALENA TODA: I'll do that.

1:23:12.966,1:23:14.958
I will extend it.

1:23:14.958,1:23:19.440


1:23:19.440,1:23:22.428
STUDENT: She asked, I said yes.

1:23:22.428,1:23:23.922
STUDENT: Why did[br]you do that, dude?

1:23:23.922,1:23:28.238
Come on, my life is ruined[br]now because I have more time

1:23:28.238,1:23:29.987
to work on my homework.

1:23:29.987,1:23:31.820
MAGDALENA TODA: And[br]I've ruined your Sunday.

1:23:31.820,1:23:32.361
STUDENT: Yes.

1:23:32.361,1:23:33.020
No.

1:23:33.020,1:23:33.920
MAGDALENA TODA: No.

1:23:33.920,1:23:36.362
Actually, I know why I did that.

1:23:36.362,1:23:37.820
I thought that the[br]28th of February

1:23:37.820,1:23:42.620
is the last day of the month,[br]but it's a short month.

1:23:42.620,1:23:45.020
So if we [? try it, ?] we[br]have to extend the months

1:23:45.020,1:23:48.620
a little bit by pulling[br]it by one more day.

1:23:48.620,1:23:49.754
STUDENT: We did?

1:23:49.754,1:23:51.920
MAGDALENA TODA: The first[br]of March is Sunday, right?

1:23:51.920,1:23:53.720
STUDENT: Yes.

1:23:53.720,1:23:55.520
[INTERPOSING VOICES]

1:23:55.520,1:24:05.812


1:24:05.812,1:24:07.520
STUDENT: You're going[br]to miss the speech.

1:24:07.520,1:24:09.320
STUDENT: Oh, we're doing that?

1:24:09.320,1:24:10.520
STUDENT: You're in English?

1:24:10.520,1:24:11.395
STUDENT: [INAUDIBLE].

1:24:11.395,1:24:13.987


1:24:13.987,1:24:15.320
STUDENT: You don't know English?

1:24:15.320,1:24:15.920
Why are you talking English?

1:24:15.920,1:24:17.720
That's what my[br]father used to say.

1:24:17.720,1:24:19.570
You don't know your own tongue?