0:00:00.000,0:00:02.150
PROFESSOR TODA: Any[br]questions so far?

0:00:02.150,0:00:06.450
I mean, conceptual,[br]theoretical questions first,

0:00:06.450,0:00:08.754
and then we will[br]do the second part

0:00:08.754,0:00:10.185
of [INAUDIBLE] applications.

0:00:10.185,0:00:14.478
Then you can ask[br]for more questions.

0:00:14.478,0:00:15.909
No questions so far?

0:00:15.909,0:00:18.676
I have not finished 11-4.

0:00:18.676,0:00:25.745
I still owe you a long[br]explanation about 11-4.

0:00:25.745,0:00:28.270
Hopefully it's going to[br]make more sense today

0:00:28.270,0:00:30.850
than it made last time.

0:00:30.850,0:00:34.220
I was just saying[br]that I'm doing 11-4.

0:00:34.220,0:00:36.350
This is a lot of chapter.

0:00:36.350,0:00:47.958
So second part of 11-4 today--[br]tangent plane and applications.

0:00:47.958,0:00:50.870


0:00:50.870,0:00:53.739
Now, we don't say what[br]those applications are

0:00:53.739,0:00:58.970
from the start, but these are[br]some very important concepts

0:00:58.970,0:01:00.838
called the total differential.

0:01:00.838,0:01:07.156


0:01:07.156,0:01:13.960
And the linear[br]approximation number

0:01:13.960,0:01:15.495
is going under the [INAUDIBLE].

0:01:15.495,0:01:17.180
Thank you, sir.

0:01:17.180,0:01:23.880
Linear approximation for[br]functions of the type z

0:01:23.880,0:01:29.480
equals f of xy, which means[br]graphs of two variables.

0:01:29.480,0:01:33.700
At the end of the chapter, I'll[br]take the notes copy from you.

0:01:33.700,0:01:37.130
So don't give me[br]anything until it's over.

0:01:37.130,0:01:38.910
When is that going to be over?

0:01:38.910,0:01:41.862
We have four more[br]sections to go.

0:01:41.862,0:01:47.140
So I guess right before[br]spring break you give me

0:01:47.140,0:01:50.330
the notes for chapter 11.

0:01:50.330,0:01:52.470
All right, and then[br]I'm thinking of making

0:01:52.470,0:01:54.894
copies of both chapters.

0:01:54.894,0:02:00.260
You get the-- I'm[br]distributing them to you.

0:02:00.260,0:02:02.660
I haven't started[br]and yet go ahead.

0:02:02.660,0:02:09.600
Could anybody tell[br]me what the equation

0:02:09.600,0:02:13.520
that we used last time--[br]we proved it, actually.

0:02:13.520,0:02:16.260


0:02:16.260,0:02:20.982
What is the equation[br]of the tangent plane

0:02:20.982,0:02:27.266
to a smooth surface or a patch[br]of a surface at the point

0:02:27.266,0:02:33.784
m of coordinates x0, y0,[br]z0, where the graph is

0:02:33.784,0:02:37.480
given by z equals f of x and y.

0:02:37.480,0:02:40.860
I'm going to label it on[br]the patch of a surface.

0:02:40.860,0:02:44.030
OK, imagine it[br]labeled brown there.

0:02:44.030,0:02:52.540
And can somebody tell me the[br]equation of the other plane?

0:02:52.540,0:02:54.450
But because you[br]have better memory,

0:02:54.450,0:03:00.300
being much younger, about 25[br]years younger than me or so.

0:03:00.300,0:03:05.650
So could you-- could anybody[br]tell me what the tangent

0:03:05.650,0:03:08.620
planes equation-- I'll start.

0:03:08.620,0:03:10.292
And it's going to come to you.

0:03:10.292,0:03:14.880
z minus z0 equals.

0:03:14.880,0:03:15.943
And now let's see.

0:03:15.943,0:03:18.382
I'll pick a nice color.

0:03:18.382,0:03:19.105
I'll wait.

0:03:19.105,0:03:21.845


0:03:21.845,0:03:23.650
STUDENT: fx of x.

0:03:23.650,0:03:27.220
PROFESSOR TODA: f sub x, the[br]partial derivative measured

0:03:27.220,0:03:35.780
at f0 i0 times the[br]quantity x minus x0 plus--

0:03:35.780,0:03:36.980
STUDENT: f sub y.

0:03:36.980,0:03:38.730
PROFESSOR TODA: f[br]sub y, excellent.

0:03:38.730,0:03:40.676
f sub y.

0:03:40.676,0:03:41.924
STUDENT: x0, y0.

0:03:41.924,0:03:44.930
PROFESSOR TODA: x0,[br]y0 times y minus y0.

0:03:44.930,0:03:49.259


0:03:49.259,0:03:50.221
OK.

0:03:50.221,0:03:51.630
All right.

0:03:51.630,0:03:59.110
Now thinking of what those[br]quantities mean, x minus x0, y

0:03:59.110,0:04:03.730
minus y0, z minus[br]z0, what are they?

0:04:03.730,0:04:06.830
They are small[br]displacements, aren't they?

0:04:06.830,0:04:10.380
I mean, what does it[br]mean small displacement?

0:04:10.380,0:04:20.860
Imagine that you are near[br]the point on both surfaces.

0:04:20.860,0:04:23.500
So what is a small[br]neighborhood--

0:04:23.500,0:04:27.940
what's a typical small[br]neighborhood [INAUDIBLE]?

0:04:27.940,0:04:30.280
It's a disk, right?

0:04:30.280,0:04:32.672
There are many kinds of[br]neighborhoods, but one of them,

0:04:32.672,0:04:36.956
I'd say, would be[br]this open disk, OK?

0:04:36.956,0:04:38.860
I'll draw that.

0:04:38.860,0:04:44.712
Now, if I have a[br]red point-- I don't

0:04:44.712,0:04:53.190
know how to do that pink point--[br]somewhere nearby in planes--

0:04:53.190,0:04:54.550
this is the plane.

0:04:54.550,0:04:59.356
In plane, I have this[br]point that is close.

0:04:59.356,0:05:01.030
And that point is xyz.

0:05:01.030,0:05:04.350


0:05:04.350,0:05:08.750
And you think, OK, can[br]I visualize that better?

0:05:08.750,0:05:11.790
Well, guys, it's hard to[br]visualize that better.

0:05:11.790,0:05:14.692
But I'll draw a triangle[br][? doing ?] a better job.

0:05:14.692,0:05:17.437


0:05:17.437,0:05:18.145
That's the frame.

0:05:18.145,0:05:22.404


0:05:22.404,0:05:24.869
This is a surface.

0:05:24.869,0:05:27.450
Imagine it's a surface, OK?

0:05:27.450,0:05:32.060
That's the point of x0, y0.

0:05:32.060,0:05:34.720
[? It's ?] the 0 and that.

0:05:34.720,0:05:36.690
Where is the point xyz again?

0:05:36.690,0:05:40.640
The point xyz is not[br]on the pink stuff.

0:05:40.640,0:05:41.790
This is a pink surface.

0:05:41.790,0:05:45.330
It looks like Pepto[br]Bismol or something.

0:05:45.330,0:05:46.310
You shaded it.

0:05:46.310,0:05:47.065
No.

0:05:47.065,0:05:48.230
That's not what I want.

0:05:48.230,0:05:55.560
I want the close enough[br]point on the blue plane.

0:05:55.560,0:06:01.180
It's actually in the blue plane[br]pie and this guy would be xyz.

0:06:01.180,0:06:05.190
So now say, OK, how[br]far I x be from x0?

0:06:05.190,0:06:06.090
Well, I don't know.

0:06:06.090,0:06:13.510
We would have to check[br]the points, the set 0,

0:06:13.510,0:06:15.948
check the blue point.

0:06:15.948,0:06:18.470
This is x.

0:06:18.470,0:06:23.940
So between x and x0, I[br]have this difference,

0:06:23.940,0:06:34.070
which is delta x displacement,[br]displacement along the x-axis,

0:06:34.070,0:06:38.915
away from the[br]point, fixed point.

0:06:38.915,0:06:41.825


0:06:41.825,0:06:44.735
This is the fixed[br]point, this point.

0:06:44.735,0:06:47.160
This point is p.

0:06:47.160,0:06:48.150
OK.

0:06:48.150,0:06:51.233
y minus y0, let's call[br]that delta y, which

0:06:51.233,0:06:53.426
is the displacement[br]along the y-axis.

0:06:53.426,0:06:56.425


0:06:56.425,0:07:02.020
And then the z minus z0 can be.

0:07:02.020,0:07:05.700
Just because I'm a mathematician[br]and I don't like writing down

0:07:05.700,0:07:11.280
a lot, I would use[br]s batch as I can,

0:07:11.280,0:07:16.780
compact symbols, to[br]speed up my computation.

0:07:16.780,0:07:19.200
So I can rewrite[br]this whole thing

0:07:19.200,0:07:27.560
as a delta z equals f sub[br]x, x0 y0, which is a number.

0:07:27.560,0:07:28.520
It's a slope.

0:07:28.520,0:07:31.570
We discussed about[br]that last time.

0:07:31.570,0:07:33.875
We even went skiing[br]last time, when

0:07:33.875,0:07:38.290
we said that's like the slope[br]in-- what's the x direction?

0:07:38.290,0:07:41.970
Slope in the x direction[br]and slope in the y direction

0:07:41.970,0:07:49.000
on the graph that was the[br]white covered with snow hill.

0:07:49.000,0:07:50.886
That was what we had last time.

0:07:50.886,0:07:54.546
Delta x plus f sub[br]0, another slope

0:07:54.546,0:07:56.742
in the y direction, delta y.

0:07:56.742,0:08:01.622


0:08:01.622,0:08:07.890
And fortunately-- OK, the book[br]is a very good book, obviously,

0:08:07.890,0:08:09.250
right?

0:08:09.250,0:08:15.630
But I wish we could've done[br]certain things better in terms

0:08:15.630,0:08:21.800
of comparisons between[br]this notion in Calc III

0:08:21.800,0:08:27.260
and some corresponding[br]notion in Calc I.

0:08:27.260,0:08:29.610
So you're probably[br]thinking, what the heck

0:08:29.610,0:08:31.240
is this witch thinking about?

0:08:31.240,0:08:34.590
Well, I'm thinking[br]of something that you

0:08:34.590,0:08:39.730
may want to remember[br]from Calc I.

0:08:39.730,0:08:42.880
And that's going to come[br]into place beautifully

0:08:42.880,0:08:47.810
right now because you have the[br]Calc I, Calc III comparison.

0:08:47.810,0:08:52.799
And that's why it would be[br]great-- the books don't even

0:08:52.799,0:08:55.270
talk about this comparison.

0:08:55.270,0:08:59.810
In Calc I, I reminded[br]you about Mr. Leibniz.

0:08:59.810,0:09:01.110
He was a very nice guy.

0:09:01.110,0:09:02.570
I have no idea, right?

0:09:02.570,0:09:04.130
Never met him.

0:09:04.130,0:09:06.660
One of the fathers of calculus.

0:09:06.660,0:09:10.630
And he introduced the[br]so-called Leibniz notation.

0:09:10.630,0:09:15.720
And one of you in office[br]hours last Wednesday

0:09:15.720,0:09:19.280
told me, so the[br]Leibnitz notation

0:09:19.280,0:09:23.455
for a function g of[br]x-- I'm intentionally

0:09:23.455,0:09:26.273
changing notation-- is what?

0:09:26.273,0:09:31.627
Well, this is just[br]the derivative

0:09:31.627,0:09:34.000
which is the limit of[br]the different quotients

0:09:34.000,0:09:38.480
of your delta g over[br]delta x-- as done by some

0:09:38.480,0:09:43.180
blutches-- 0, right, which[br]would be the same as lim

0:09:43.180,0:09:50.840
of g of x minus g of x0 over[br]x minus x0 as x approaches x0,

0:09:50.840,0:09:52.070
right?

0:09:52.070,0:09:52.570
Right.

0:09:52.570,0:09:57.720
So we've done that in Calc I.[br]But it was a long time ago.

0:09:57.720,0:10:00.630
My mission is to teach[br]you all Calc III,

0:10:00.630,0:10:03.835
but I feel that[br]my mission is also

0:10:03.835,0:10:08.635
to teach you what you may not[br]remember very well from Calc I,

0:10:08.635,0:10:11.640
because everything is related.

0:10:11.640,0:10:17.690
So what was the way we[br]could have written this,

0:10:17.690,0:10:21.260
not delta g over delta[br]x equals g prime.

0:10:21.260,0:10:22.486
No.

0:10:22.486,0:10:29.040
But it's an approximation of[br]g prime around a very small

0:10:29.040,0:10:33.745
[INAUDIBLE], very close to x0.

0:10:33.745,0:10:36.550


0:10:36.550,0:10:39.676
So if you wanted to[br]rewrite this approximation,

0:10:39.676,0:10:42.091
how would you have rewritten it?

0:10:42.091,0:10:47.410


0:10:47.410,0:10:48.140
Delta g--

0:10:48.140,0:10:54.866


0:10:54.866,0:10:57.310
STUDENT: g prime sub x.

0:10:57.310,0:11:02.490
PROFESSOR TODA: g prime[br]of x0 times delta x.

0:11:02.490,0:11:03.930
OK?

0:11:03.930,0:11:08.280
Now, why this approximation?

0:11:08.280,0:11:11.710
What if I had put equal?

0:11:11.710,0:11:14.120
If I had put equal, it[br]would be all nonsense.

0:11:14.120,0:11:15.405
Why?

0:11:15.405,0:11:19.210
Well, say, Magdalena, if you[br]put equal, it's another object.

0:11:19.210,0:11:19.770
What object?

0:11:19.770,0:11:20.330
OK.

0:11:20.330,0:11:22.120
Let's look at the objects.

0:11:22.120,0:11:22.995
Let's draw a picture.

0:11:22.995,0:11:25.730


0:11:25.730,0:11:27.176
This is g.

0:11:27.176,0:11:28.622
This is x0.

0:11:28.622,0:11:30.560
This is g of x.

0:11:30.560,0:11:32.440
What's g prime?

0:11:32.440,0:11:39.420
g prime-- thank god-- is the[br]slope of g prime x0 over here.

0:11:39.420,0:11:46.610
So if I want to write the[br]line, the line is exactly this.

0:11:46.610,0:11:50.170
The red object is the line.

0:11:50.170,0:11:52.770
So what is the red object again?

0:11:52.770,0:11:58.350
It's y minus y over x[br]minus x0 equals m, which

0:11:58.350,0:12:00.290
is g prime number 0.

0:12:00.290,0:12:01.750
m is the slope.

0:12:01.750,0:12:05.210
That's the point slope[br]formula, thank you very much.

0:12:05.210,0:12:06.770
So the red object is this.

0:12:06.770,0:12:08.990
This is the line.

0:12:08.990,0:12:10.770
Attention is not the same.

0:12:10.770,0:12:15.625
The blue thing is my[br]curve, more precisely

0:12:15.625,0:12:17.600
a tiny portion of my curve.

0:12:17.600,0:12:21.610
This neighborhood around the[br]point is what I have here.

0:12:21.610,0:12:22.805
What I'm actually-- what?

0:12:22.805,0:12:26.010


0:12:26.010,0:12:29.510
I'm trying to[br]approximate my curve

0:12:29.510,0:12:32.310
function with a little line.

0:12:32.310,0:12:36.420
And I say, I would rather[br]approximate with a red line

0:12:36.420,0:12:38.582
because this is the[br]best approximation

0:12:38.582,0:12:44.200
to the blue arc of a curve[br]which is on the curve, right?

0:12:44.200,0:12:46.985
So this is what it is[br]is just an approximation

0:12:46.985,0:12:54.620
of a curve, approximation of[br]a curve of an arc of a curve.

0:12:54.620,0:12:57.590
But Magdalena's lazy[br]today-- approximation

0:12:57.590,0:13:03.550
of an arc of a curve[br]with a segment of a line,

0:13:03.550,0:13:07.102
with a segment of[br]the tangent line

0:13:07.102,0:13:10.735
of the tangent [INAUDIBLE].

0:13:10.735,0:13:13.360
How do we call[br]such a phenomenon?

0:13:13.360,0:13:17.650
An approximation of[br]an arc of a circle

0:13:17.650,0:13:23.115
with a little segment[br]of a tangent line

0:13:23.115,0:13:26.040
is like a discretization, right?

0:13:26.040,0:13:29.416
But we call it[br]linear approximation.

0:13:29.416,0:13:32.460
It's called a linear[br]approximation.

0:13:32.460,0:13:36.590


0:13:36.590,0:13:40.220
A-P-P, approx.

0:13:40.220,0:13:42.460
Have you ever seen a[br]linear approximation

0:13:42.460,0:13:46.880
before coming from Calc II?

0:13:46.880,0:13:49.700
Well, in Calc II you've[br]seen the Taylor's formula.

0:13:49.700,0:13:51.510
What is the Taylor's formula?

0:13:51.510,0:13:55.246
It's a beautiful[br]thing that said what?

0:13:55.246,0:13:55.990
I don't know.

0:13:55.990,0:13:56.990
Let's remember together.

0:13:56.990,0:14:00.206
So relationship[br]with Calc II, I'm

0:14:00.206,0:14:04.670
going to go and make an arrow--[br]relationship with Calc II,

0:14:04.670,0:14:08.160
because everything[br]is actually related.

0:14:08.160,0:14:13.750
In Calc II-- how did we[br]introduce Taylor's formula?

0:14:13.750,0:14:16.930
Well, instead of little a that[br]you're so used to in Calc II,

0:14:16.930,0:14:21.170
we are going to put x0[br]is the same thing, right?

0:14:21.170,0:14:23.550
So what was Taylor's[br]formula saying?

0:14:23.550,0:14:28.150
You have this kind of[br]smooth, beautiful curve.

0:14:28.150,0:14:30.860
But being smooth is not enough.

0:14:30.860,0:14:33.860
You have that real analytic.

0:14:33.860,0:14:36.070
Real analytic means[br]that the function can be

0:14:36.070,0:14:41.100
expanded in Taylor's formula.

0:14:41.100,0:14:42.250
So what does it mean?

0:14:42.250,0:14:53.000
It means that we have f of x[br]prime is f of x0 equals-- or g.

0:14:53.000,0:14:54.920
You want-- it doesn't matter.

0:14:54.920,0:15:01.160
f prime of x0 times[br]x minus x0 plus

0:15:01.160,0:15:06.010
dot, dot, dot, dot something[br]that I'm going to put.

0:15:06.010,0:15:09.300
This is [? O. ?] It's a small[br]quantity that's maybe not

0:15:09.300,0:15:12.898
so small, but I declare[br]it to be negligible.

0:15:12.898,0:15:14.690
And so they're going[br]to be negligible.

0:15:14.690,0:15:18.920
I have to make a face,[br]a smiley face and eyes,

0:15:18.920,0:15:23.530
meaning that it's OK to[br]neglect the second order

0:15:23.530,0:15:25.420
term, the third order term.

0:15:25.420,0:15:28.370
So what happens, that[br]little h, when I square it,

0:15:28.370,0:15:29.336
say the heck with it.

0:15:29.336,0:15:30.800
It's going to be very small.

0:15:30.800,0:15:36.700
Like if h is 0.1 and then[br]h squared will be 0.0001.

0:15:36.700,0:15:40.445
And I have a certain range[br]of error that I allow,

0:15:40.445,0:15:41.540
a threshold.

0:15:41.540,0:15:43.470
I say that's negligible.

0:15:43.470,0:15:47.430
If h squared and h cubed and h[br]to the fourth are negligible,

0:15:47.430,0:15:49.930
then I'm fine.

0:15:49.930,0:15:53.440
If I take all the[br]other spot, that's

0:15:53.440,0:15:55.960
the linear approximation.

0:15:55.960,0:15:59.730
And that's exactly[br]what I wrote here

0:15:59.730,0:16:02.140
with little g instead of f.

0:16:02.140,0:16:05.120
The only difference is this is[br]little f and this is little g.

0:16:05.120,0:16:09.340
But it's the same exact[br]formula, linear approximation.

0:16:09.340,0:16:14.596
Do you guys remember then next[br]terms of the Taylor's formula?

0:16:14.596,0:16:15.310
STUDENT: fw--

0:16:15.310,0:16:16.437
PROFESSOR TODA: fw--

0:16:16.437,0:16:19.920
STUDENT: w over--

0:16:19.920,0:16:23.430
PROFESSOR TODA: So[br]fw prime at x0 over--

0:16:23.430,0:16:24.384
STUDENT: 1 factorial.

0:16:24.384,0:16:25.550
PROFESSOR TODA: 2 factorial.

0:16:25.550,0:16:26.625
This was 1 factorial.

0:16:26.625,0:16:28.950
This was over 1 factorial.

0:16:28.950,0:16:30.573
But I don't write[br]it because it's one.

0:16:30.573,0:16:31.197
STUDENT: Right.

0:16:31.197,0:16:35.823
PROFESSOR TODA: Here I would[br]have f double prime of blah,

0:16:35.823,0:16:41.100
blah, blah over-- what did[br]you say-- 2 factorial times x

0:16:41.100,0:16:44.376
minus x0 squared plus, plus,[br]plus, the cubic [INAUDIBLE]

0:16:44.376,0:16:49.730
of the-- this is the quadratic[br]term that I neglect, right?

0:16:49.730,0:16:51.180
So that was Taylor's formula.

0:16:51.180,0:16:54.790
Do I mention anything[br]about it now?

0:16:54.790,0:16:55.905
We should.

0:16:55.905,0:16:58.250
But practically, the[br]authors of the book

0:16:58.250,0:17:00.400
thought, well, everything[br]is in the book.

0:17:00.400,0:17:02.120
You can go back and forth.

0:17:02.120,0:17:05.300
It's not like that unless[br]somebody opens your eyes.

0:17:05.300,0:17:09.930
For example, I didn't[br]see that when I was 21.

0:17:09.930,0:17:13.040
I couldn't make any connection[br]between these Calc I,

0:17:13.040,0:17:14.920
Calc II, Calc III notions.

0:17:14.920,0:17:17.886
Because nobody told me, hey,[br]Magdalena, open your eyes

0:17:17.886,0:17:20.118
and look at that in[br]perspective and make

0:17:20.118,0:17:24.720
a comparison between what you[br]learned in different chapters.

0:17:24.720,0:17:26.220
I had to grow.

0:17:26.220,0:17:29.030
After 20 years, I[br]said, oh, I finally

0:17:29.030,0:17:33.680
see the picture of linearization[br]of a function of, let's say,

0:17:33.680,0:17:35.390
n variables.

0:17:35.390,0:17:38.480
So all these total[br]differentials will come in place

0:17:38.480,0:17:41.050
when time comes.

0:17:41.050,0:17:46.410
You have a so-called[br]differential in Calc I.

0:17:46.410,0:17:47.920
And that's not delta g.

0:17:47.920,0:17:49.890
Some people say, OK,[br]no, that's delta g.

0:17:49.890,0:17:52.000
No, no, no, no.

0:17:52.000,0:17:53.610
The delta x is a displacement.

0:17:53.610,0:17:57.305
The delta g is the[br]induced displacement.

0:17:57.305,0:17:59.985
If you want this to be[br]come a differential,

0:17:59.985,0:18:02.840
then you shrink[br]that displacement

0:18:02.840,0:18:05.640
to infinitesimally small.

0:18:05.640,0:18:06.230
OK?

0:18:06.230,0:18:09.684
So it's like going from[br]a molecule to an atom

0:18:09.684,0:18:13.990
to an electron to subatomic[br]particles but even more,

0:18:13.990,0:18:16.060
something infinitesimally small.

0:18:16.060,0:18:17.070
So what do we do?

0:18:17.070,0:18:22.810
We shrink delta x into dx[br]which is infinitesimally small.

0:18:22.810,0:18:26.390


0:18:26.390,0:18:28.932
It's like the notion of[br]God but microscopically

0:18:28.932,0:18:33.920
or like microbiology[br]compared to the universe, OK?

0:18:33.920,0:18:42.210
So dx is multiplied[br]by g prime of x0.

0:18:42.210,0:18:46.430
And instead of delta g, I'm[br]going to have a so-called dg,

0:18:46.430,0:18:49.060
and that's a form.

0:18:49.060,0:18:53.260
In mathematics, this is[br]called a form or a one form.

0:18:53.260,0:18:58.520
And it's a special[br]kind of object, OK?

0:18:58.520,0:19:01.550
So Mr. Leibniz was very smart.

0:19:01.550,0:19:09.720
He said, but I can rewrite this[br]form like dg dx equals g prime.

0:19:09.720,0:19:13.450
So if you ever forget[br]about this form which

0:19:13.450,0:19:18.169
is called differential,[br]differential form,

0:19:18.169,0:19:20.780
you remember Mr.[br]Leibniz, he taught you

0:19:20.780,0:19:25.322
how to write the derivative in[br]two different ways, dg dx or g

0:19:25.322,0:19:26.630
prime.

0:19:26.630,0:19:30.220
What you do is just formally[br]multiply g prime by dx

0:19:30.220,0:19:31.670
and you get dg.

0:19:31.670,0:19:34.700
Say it again, Magdalena--[br]multiply g prime by dx

0:19:34.700,0:19:35.880
and you get dg.

0:19:35.880,0:19:38.890
And that's your[br]so-called differential.

0:19:38.890,0:19:42.500
Now, why do you say total[br]differential-- total

0:19:42.500,0:19:46.870
differential, my god, like[br]complete differentiation?

0:19:46.870,0:19:52.280
In 11.4, we deal with[br]functions of two variables.

0:19:52.280,0:19:54.750
So can we say differentials?

0:19:54.750,0:19:57.290
Mmm, it's a little bit[br]like a differential

0:19:57.290,0:20:00.030
with respect to what variable?

0:20:00.030,0:20:02.590
If you say with respect[br]to all the variables,

0:20:02.590,0:20:08.960
then you have to be thinking[br]to be smart and event,

0:20:08.960,0:20:11.690
create this new object.

0:20:11.690,0:20:17.312
If one would write[br]Taylor's formula,

0:20:17.312,0:20:22.720
there is a Taylor's[br]formula that we don't give.

0:20:22.720,0:20:23.260
OK.

0:20:23.260,0:20:26.210
Now, you guys are looking[br]at me with excitement.

0:20:26.210,0:20:30.740
For one point extra[br]credit, on the internet,

0:20:30.740,0:20:35.310
find Taylor's formula for[br]n variables, functions

0:20:35.310,0:20:38.590
of n variables or at[br]least two variables,

0:20:38.590,0:20:43.720
which was going to look[br]like z minus z0 equals

0:20:43.720,0:20:49.140
f sub x at the point x0[br]at 0 times x minus x0 plus

0:20:49.140,0:21:00.200
f sub y at x0 y0 times x minus[br]x0 plus second order terms

0:21:00.200,0:21:04.010
plus third order terms[br]plus fourth order terms.

0:21:04.010,0:21:06.720
And the video cannot see me.

0:21:06.720,0:21:08.850
So what do we do?

0:21:08.850,0:21:13.830
We just truncate this[br]part of Taylor's I say,

0:21:13.830,0:21:18.170
I already take the Taylor[br]polynomial of degree one.

0:21:18.170,0:21:21.470
And the quadratic terms and[br]everything else, the heck

0:21:21.470,0:21:22.850
with that.

0:21:22.850,0:21:25.020
And I call that a[br]linear approximation,

0:21:25.020,0:21:28.330
but it's actually Taylor's[br]formula being discussed.

0:21:28.330,0:21:30.680
We don't tell you in[br]the book because we

0:21:30.680,0:21:31.740
don't want to scare you.

0:21:31.740,0:21:34.865
I think we would better[br]tell you at some point,

0:21:34.865,0:21:38.010
so I decided to tell you now.

0:21:38.010,0:21:38.850
All right.

0:21:38.850,0:21:42.440
So this is Taylor's formula[br]for functions of two variables.

0:21:42.440,0:21:45.630
We have to create[br]not out of nothing

0:21:45.630,0:21:49.810
but out of this the[br]total differential.

0:21:49.810,0:21:51.190
Who tells me?

0:21:51.190,0:21:54.033
Shrink the[br]displacement, Magdalena.

0:21:54.033,0:21:58.141
The delta x shrunk to[br]an infinitesimally small

0:21:58.141,0:21:58.640
will be dx.

0:21:58.640,0:22:01.110
Delta y will become dy.

0:22:01.110,0:22:06.390
The line is a smiley from the[br]skies, just looking at us.

0:22:06.390,0:22:08.040
He loves our notations.

0:22:08.040,0:22:10.896
And this is dz.

0:22:10.896,0:22:18.970
So I'm going to write dz or df's[br]the same thing equals f sub x.

0:22:18.970,0:22:22.420
At the point, you[br]could be at any point

0:22:22.420,0:22:29.780
you are taking in particular,[br]dx plus f sub y xy dy.

0:22:29.780,0:22:34.010
So this is at any point[br]at the arbitrary point xy

0:22:34.010,0:22:39.310
in the domain where your[br]function e is at least c1.

0:22:39.310,0:22:40.730
What does it mean, c1?

0:22:40.730,0:22:43.280
It means the function[br]is differentiable

0:22:43.280,0:22:47.400
and the partial[br]derivatives are continuous.

0:22:47.400,0:22:50.850
I said several times, I[br]want even more than that.

0:22:50.850,0:22:56.790
I want it maybe second[br]order derivatives

0:22:56.790,0:23:02.868
to exist and be continuous[br]and so on and so forth.

0:23:02.868,0:23:08.465
And I will assume[br]that the function can

0:23:08.465,0:23:11.585
be expanded [INAUDIBLE] series.

0:23:11.585,0:23:14.445


0:23:14.445,0:23:17.440
All right, now example[br]of a final problem

0:23:17.440,0:23:22.260
that was my first problem[br]on the final many times

0:23:22.260,0:23:26.310
and also on the common[br]final departmental final.

0:23:26.310,0:23:28.320
And many students[br]screwed up, and I

0:23:28.320,0:23:32.380
don't want you to ever[br]make such a mistake.

0:23:32.380,0:23:37.322
So this is a mistake not[br]to make, OK, mistake not

0:23:37.322,0:23:43.730
to make because after 20[br]something years of teaching,

0:23:43.730,0:23:46.010
I'm quite familiar with[br]the mistakes students

0:23:46.010,0:23:49.230
make in general and I don't[br]want you to make them.

0:23:49.230,0:23:50.664
You are too good to do this.

0:23:50.664,0:23:52.098
So problem 1.

0:23:52.098,0:23:56.600
On the final, I said-- we[br]said-- the only difference was

0:23:56.600,0:24:00.900
on some departmental finals,[br]we gave a more sophisticated

0:24:00.900,0:24:02.470
function.

0:24:02.470,0:24:06.580
I'm going to give only[br]some simple function

0:24:06.580,0:24:07.820
for this polynomial.

0:24:07.820,0:24:09.770
That's beautiful.

0:24:09.770,0:24:18.930
And then I said we said[br]write the differential

0:24:18.930,0:24:28.090
of this function at an[br]arbitrary point x, y.

0:24:28.090,0:24:28.610
And done.

0:24:28.610,0:24:31.080
And [INAUDIBLE].

0:24:31.080,0:24:34.642
Well, let me tell you what[br]some of my students-- some

0:24:34.642,0:24:36.350
of my studentss-- don't do that.

0:24:36.350,0:24:38.302
I'm going to cross it with red.

0:24:38.302,0:24:41.770
And some of my students[br]wrote me very beautifully df

0:24:41.770,0:24:44.390
equals 2x plus 2y.

0:24:44.390,0:24:47.550
And that can send[br]me to the hospital.

0:24:47.550,0:24:53.320
If you want to go to the ER[br]soon, do this on the exam

0:24:53.320,0:24:55.960
because this is nonsense.

0:24:55.960,0:24:57.480
Why is this nonsense?

0:24:57.480,0:24:58.360
This is not--

0:24:58.360,0:24:59.840
STUDENT: [INAUDIBLE] dx or dy.

0:24:59.840,0:25:00.840
PROFESSOR TODA: Exactly.

0:25:00.840,0:25:06.980
So the most important thing[br]is that the df is like-- OK,

0:25:06.980,0:25:09.060
let me come back to driving.

0:25:09.060,0:25:14.480
I'm driving to Amarillo-- and I[br]give this example to my calc 1

0:25:14.480,0:25:18.201
students all the time because[br]it's a linear motion in terms

0:25:18.201,0:25:18.700
of time.

0:25:18.700,0:25:21.090
And let's say I'm on[br]cruise control or not.

0:25:21.090,0:25:22.780
It doesn't matter.

0:25:22.780,0:25:30.190
When we drive and I'm looking at[br]the speedometer and I see 60--

0:25:30.190,0:25:37.000
I didn't want to say more, but[br]let's say 80, 80 miles an hour.

0:25:37.000,0:25:38.620
That is a miles an hour.

0:25:38.620,0:25:43.344
That means the hour is a huge[br]chunk delta h or delta t.

0:25:43.344,0:25:45.010
Let's call it delta[br]t because it's time.

0:25:45.010,0:25:45.640
I'm silly.

0:25:45.640,0:25:47.660
Delta t is 1.

0:25:47.660,0:25:51.310
Delta s, the space,[br]the space, is going

0:25:51.310,0:25:54.970
to be the chunk of 60 miles.

0:25:54.970,0:26:00.360
But then that is the[br]average speed that I had.

0:26:00.360,0:26:02.130
So that's why I said 60.

0:26:02.130,0:26:04.806
That's the average[br]speed I had in my trip,

0:26:04.806,0:26:05.930
during my trip [INAUDIBLE].

0:26:05.930,0:26:10.600
There were moments when my[br]speed was 0 or close to 0.

0:26:10.600,0:26:12.390
Let's assume it was never 0.

0:26:12.390,0:26:14.931
But that means there were many[br]moments when my speed could've

0:26:14.931,0:26:18.990
been 100, and nobody knows[br]because they didn't catch me.

0:26:18.990,0:26:21.450
So I was just lucky.

0:26:21.450,0:26:26.300
So in average, if somebody is[br]asking you what is the average,

0:26:26.300,0:26:30.440
that doesn't tell them anything.

0:26:30.440,0:26:34.090
That reminds me of that[br]joke-- overall I'm good,

0:26:34.090,0:26:38.190
the statistician joke[br]who was, are you cold?

0:26:38.190,0:26:39.000
Are you warm?

0:26:39.000,0:26:44.142
And he was actually sitting[br]on with one half of him

0:26:44.142,0:26:47.090
on a block of ice and the[br]other half on the stove,

0:26:47.090,0:26:49.172
and he says, in[br]average, I'm fine.

0:26:49.172,0:26:52.400
But he was dying.

0:26:52.400,0:26:53.910
This is the same kind of thing.

0:26:53.910,0:26:58.360
My average was 60 miles[br]an hour, but I almost

0:26:58.360,0:27:02.110
got caught when I was[br]driving almost 100.

0:27:02.110,0:27:06.250
But nobody knows because I'm[br]not giving you that information.

0:27:06.250,0:27:12.440
That's the infinitesimally small[br]information that I have not

0:27:12.440,0:27:16.610
put correctly here[br]means that what is

0:27:16.610,0:27:18.990
what I see on the speedometer?

0:27:18.990,0:27:21.060
It's the instantaneous[br]rate of change

0:27:21.060,0:27:23.880
that I see that[br]fraction of second.

0:27:23.880,0:27:30.940
So that means maybe a few feet[br]per a fraction of a second.

0:27:30.940,0:27:33.920
It means how many[br]feet did I travel

0:27:33.920,0:27:36.470
in that fraction of a second?

0:27:36.470,0:27:41.240
And if that fraction of a second[br]is very tiny that I cannot even

0:27:41.240,0:27:44.000
express it properly, that's[br]what I'm going to have--

0:27:44.000,0:27:46.610
df equals f prime dx.

0:27:46.610,0:27:52.010
So df and dx have to be small[br]because their ratio will be

0:27:52.010,0:27:56.180
a good number, like 60, like[br]80, but [? them in ?] themselves

0:27:56.180,0:27:58.635
delta m delta [? srv, ?][br]very tiny things.

0:27:58.635,0:28:03.420
It's the ratio that matters[br]in the end to be 60, or 80,

0:28:03.420,0:28:04.470
or whatever.

0:28:04.470,0:28:08.520
So I have 2x dx plus 2y dy.

0:28:08.520,0:28:10.920
Never say that the[br]differential, which

0:28:10.920,0:28:13.160
is something[br]infinitesimally small,

0:28:13.160,0:28:17.376
is equal to this scalar[br]function that it doesn't even

0:28:17.376,0:28:18.160
make any sense.

0:28:18.160,0:28:20.060
Don't do that because[br]you get 0 points

0:28:20.060,0:28:21.900
and then we argue,[br]and I don't want

0:28:21.900,0:28:25.450
you to get 0 points on[br]this problem, right.

0:28:25.450,0:28:27.250
So it's a very simple problem.

0:28:27.250,0:28:31.080
All I want to test you on[br]would be this definition.

0:28:31.080,0:28:36.000
Remember, you're going to[br]see that again on the midterm

0:28:36.000,0:28:39.020
and on the final, or[br]just on the final.

0:28:39.020,0:28:41.650
Any questions about that?

0:28:41.650,0:28:42.250
All right.

0:28:42.250,0:28:53.978
So I want to give you the[br]following homework out

0:28:53.978,0:29:00.680
of section 11.4 on[br]top of the web work.

0:29:00.680,0:29:07.250


0:29:07.250,0:29:16.640
Read all the solved[br]examples of the section.

0:29:16.640,0:29:23.530


0:29:23.530,0:29:24.030
OK.

0:29:24.030,0:29:30.470
So for example,[br]somebody tells you

0:29:30.470,0:29:40.110
I have to apply this[br]knowing that I have

0:29:40.110,0:29:44.610
an error of measurement of[br]some sort in the s direction

0:29:44.610,0:29:48.210
and an error of measurement of[br]some sort in the y direction.

0:29:48.210,0:29:51.010
There are two or three[br]examples like that.

0:29:51.010,0:29:54.910
They will give you all this[br]data, including the error

0:29:54.910,0:29:55.640
measurement.

0:29:55.640,0:29:58.490
For delta, it should be 0.1.

0:29:58.490,0:30:04.240
Don't confuse the 0.1 with[br]dx. dx is not a quantity.

0:30:04.240,0:30:08.608
dx is something like[br]micro cosmic thing.

0:30:08.608,0:30:14.134
It's like infinitely[br][? small ?].

0:30:14.134,0:30:15.050
Infinitesimally small.

0:30:15.050,0:30:19.560
So saying that dx should be[br]0.1 doesn't make any sense,

0:30:19.560,0:30:22.880
but delta x being[br]0.1 make sense.

0:30:22.880,0:30:26.350
Delta y being 0.3 makes sense.

0:30:26.350,0:30:29.560
And they ask you to[br]plug it in and find

0:30:29.560,0:30:32.130
the general difference.

0:30:32.130,0:30:33.730
For example, where[br]could that happen?

0:30:33.730,0:30:35.760
And you see examples[br]in the book.

0:30:35.760,0:30:40.910
Somebody measures something--[br]an area of a rectangle

0:30:40.910,0:30:42.970
or a volume of a cube.

0:30:42.970,0:30:46.110
But when you measure,[br]you make mistakes.

0:30:46.110,0:30:48.270
You have measurement errors.

0:30:48.270,0:30:53.250
In the delta x, you have[br]an error of plus minus 0.1.

0:30:53.250,0:31:00.870
In the y direction, you have[br]displacement error 0.2 or 0.3,

0:31:00.870,0:31:02.220
something like that.

0:31:02.220,0:31:05.090
What is the overall[br]error you are

0:31:05.090,0:31:08.100
going to make when you measure[br]that function of two variables?

0:31:08.100,0:31:09.730
That's what you have.

0:31:09.730,0:31:12.140
So you plug in all[br]those displacements

0:31:12.140,0:31:14.790
and you come up with the[br]computational problem.

0:31:14.790,0:31:20.200
Several of you Wednesday we[br]discussed in my office already

0:31:20.200,0:31:24.700
solved those problems through[br]web work and came to me,

0:31:24.700,0:31:27.510
and I said, how did you know[br]to plug in those [? numbers ?]?

0:31:27.510,0:31:28.900
Well, it's not so hard.

0:31:28.900,0:31:30.120
It's sort of common sense.

0:31:30.120,0:31:32.990
Plus, I looked in the book[br]and that gave me the idea

0:31:32.990,0:31:34.517
to remind you to[br]look in the book

0:31:34.517,0:31:37.250
for those numerical examples.

0:31:37.250,0:31:40.370
You will have to[br]use your calculator.

0:31:40.370,0:31:42.990
So you don't have it with[br]you, you generally, we

0:31:42.990,0:31:45.000
don't use in the classroom,[br]but it's very easy.

0:31:45.000,0:31:48.392
All you have to do is use the[br]calculator and [INAUDIBLE]

0:31:48.392,0:31:51.310
examples and see how it goes.

0:31:51.310,0:31:57.430
I wanted to show you[br]something more interesting

0:31:57.430,0:32:09.410
even, more beautiful[br]regarding something

0:32:09.410,0:32:12.930
we don't show in the[br]book until later on,

0:32:12.930,0:32:18.240
and I'm uncomfortable with the[br]idea of not showing this to you

0:32:18.240,0:32:19.610
now.

0:32:19.610,0:32:26.560
An alternate way, or[br]more advanced way,

0:32:26.560,0:32:38.390
more advanced way, to[br]define the tangent plane--

0:32:38.390,0:32:49.190
the tangent plane-- to a[br]surface S at the point p.

0:32:49.190,0:32:51.690
And I'll draw again.

0:32:51.690,0:32:56.470
Half of my job is drawing[br]in this class, which I like.

0:32:56.470,0:32:59.910
I mean, I was having an argument[br]with one of my colleagues who

0:32:59.910,0:33:03.480
said, I hate when they are[br]giving me to teach calculus 3

0:33:03.480,0:33:07.660
because I cannot draw.

0:33:07.660,0:33:09.910
I think that the[br]most beautiful part

0:33:09.910,0:33:15.450
is that we can represent[br]things visually,

0:33:15.450,0:33:20.262
and this is just pi, the[br]tangent plane I'm after,

0:33:20.262,0:33:24.880
and p will be a[br]coordinate 0 by 0, z0.

0:33:24.880,0:33:26.900
And what was the label?

0:33:26.900,0:33:27.790
Oh, the label.

0:33:27.790,0:33:28.365
The label.

0:33:28.365,0:33:34.330
The label was internal[br]where z equals f of xy.

0:33:34.330,0:33:40.160
But more generally, I'll say[br]this time plus more generally,

0:33:40.160,0:33:58.970
what if you have f of xyz[br]equals c for that surface.

0:33:58.970,0:34:00.560
Let's call it [INAUDIBLE].

0:34:00.560,0:34:04.800
F of xy is [INAUDIBLE].

0:34:04.800,0:34:08.210
And somebody even said, can[br]you have a parametrization?

0:34:08.210,0:34:10.440
And this is where[br]I wanted to go.

0:34:10.440,0:34:14.469


0:34:14.469,0:34:16.230
Ryan was the first[br]one who asked me,

0:34:16.230,0:34:18.870
but then there were[br]three more of you

0:34:18.870,0:34:21.159
who have restless[br]minds plus you--

0:34:21.159,0:34:25.670
because that's the essence[br]of being active here.

0:34:25.670,0:34:29.840
We don't lose our connections.

0:34:29.840,0:34:34.300
We lose neurons anyway, but[br]we don't lose our connections

0:34:34.300,0:34:37.949
if we think, and[br]anticipate things,

0:34:37.949,0:34:40.080
and try to relate concepts.

0:34:40.080,0:34:42.590
So if you don't want to[br]get Alzheimer's, just

0:34:42.590,0:34:45.730
think about the parametrization.

0:34:45.730,0:34:49.699
So can I have a[br]parametrization for a surface?

0:34:49.699,0:34:52.179
All righty, what do you mean?

0:34:52.179,0:34:58.240
What if somebody says for a[br]curve, we have r of t, right,

0:34:58.240,0:34:59.075
which was what?

0:34:59.075,0:35:06.500
It was x of ti plus y of tj plus[br]z of tk, and we were so happy

0:35:06.500,0:35:09.825
and we were happy[br]because we were traveling

0:35:09.825,0:35:12.320
in time with respect[br]to the origin,

0:35:12.320,0:35:15.640
and this was r of t at time t.

0:35:15.640,0:35:18.330
[INAUDIBLE]

0:35:18.330,0:35:20.210
But somebody asked[br]me, [INAUDIBLE],

0:35:20.210,0:35:27.010
can you have such a position[br]vector moving on a surface?

0:35:27.010,0:35:30.240
Like look, it's a rigid motion.

0:35:30.240,0:35:32.770
If you went to the[br]robotics science

0:35:32.770,0:35:36.340
fair, Texas Tech, or something[br]like that, you know about that.

0:35:36.340,0:35:37.180
Yeah, cities.

0:35:37.180,0:35:39.977
So how do we introduce[br]such a parametrization?

0:35:39.977,0:35:44.470
We have an origin of course.

0:35:44.470,0:35:46.390
An origin is always important.

0:35:46.390,0:35:48.326
Everybody has an origin.

0:35:48.326,0:35:53.170


0:35:53.170,0:35:57.610
And I take that position[br]vector, and where does it start?

0:35:57.610,0:36:02.120
It starts at the origin, and[br]the tip of it is on the surface,

0:36:02.120,0:36:05.382
And it's gliding on the[br]surface, the tip of it.

0:36:05.382,0:36:10.500
And that's going to be r, but[br]it's not going to be r of t.

0:36:10.500,0:36:12.930
It's going to be r of[br]longitude and latitude.

0:36:12.930,0:36:16.110
Like imagine, that would[br]be the radius coming

0:36:16.110,0:36:18.360
from the center of the earth.

0:36:18.360,0:36:20.980
And it depends on[br]two parameters.

0:36:20.980,0:36:24.780
One of them would be latitude.

0:36:24.780,0:36:26.140
Am I drawing this right?

0:36:26.140,0:36:26.640
Latitude--

0:36:26.640,0:36:28.730
STUDENT: [INAUDIBLE] longitude.

0:36:28.730,0:36:30.870
PROFESSOR TODA:[br]--from a latitude 0.

0:36:30.870,0:36:32.010
I'm at the equator.

0:36:32.010,0:36:33.760
Then latitude 90 degrees.

0:36:33.760,0:36:35.970
I'm at the North Pole.

0:36:35.970,0:36:37.760
In mathematics, we are funny.

0:36:37.760,0:36:40.880
We say latitude 0,[br]latitude 90 North Pole,

0:36:40.880,0:36:45.165
latitude negative 90,[br]which is South Pole.

0:36:45.165,0:36:49.290
And longitude from 0 to 2 pi.

0:36:49.290,0:36:53.740
Meridian 0 to all around.

0:36:53.740,0:36:58.200
So r will be not a function of[br]t but a function of u and b,

0:36:58.200,0:37:02.240
thank god, because u and b[br]are the latitude and longitude

0:37:02.240,0:37:03.320
sort of.

0:37:03.320,0:37:12.324
So we have x of uv i plus[br]y of uv j plus z of uv k.

0:37:12.324,0:37:20.620


0:37:20.620,0:37:23.030
You can do that.

0:37:23.030,0:37:26.010
And you say, but can you give[br]us an example, because this

0:37:26.010,0:37:28.210
looks so abstract for god sake.

0:37:28.210,0:37:31.830
If you give me the graph[br]the way you gave it to me

0:37:31.830,0:37:37.307
before z equals f of xy,[br]please parametrize this for me.

0:37:37.307,0:37:41.880


0:37:41.880,0:37:44.640
Parametrize it for[br]me because I'm lost.

0:37:44.640,0:37:45.610
You are not lost.

0:37:45.610,0:37:47.530
We can do this together.

0:37:47.530,0:37:51.480
Now what's the simplest[br]way to parametrize

0:37:51.480,0:37:57.260
a graph of the type[br]z equals f of xy?

0:37:57.260,0:38:01.970
Take the xy to be[br]u and v. Take x

0:38:01.970,0:38:05.360
and y to be your[br]independent variables

0:38:05.360,0:38:07.850
and take z to be the[br]dependent variable.

0:38:07.850,0:38:12.700


0:38:12.700,0:38:16.930
I'm again expressing these[br]things in terms of variables

0:38:16.930,0:38:18.340
like I did last time.

0:38:18.340,0:38:23.370
Then I say, let's take this kind[br]of parametrization. [INAUDIBLE]

0:38:23.370,0:38:24.380
vu, right.

0:38:24.380,0:38:33.080
y would be v. Then I'm[br]going to write r of x and y

0:38:33.080,0:38:36.710
just like that guy will[br]be [INAUDIBLE] of xn.

0:38:36.710,0:38:38.770
[? y ?] will say, wait a minute.

0:38:38.770,0:38:42.884
I will have to re-denote[br]everybody with capitals.

0:38:42.884,0:38:46.300
Then my life will become[br]better because you

0:38:46.300,0:38:47.300
don't have to erase.

0:38:47.300,0:38:50.670
You just make little[br]x big, little y bigs,

0:38:50.670,0:38:53.890
bigs, big, capitalized XYZ.

0:38:53.890,0:39:02.150
And then I'll say OK, XYZ[br]will be my setting here in 3D.

0:39:02.150,0:39:07.020


0:39:07.020,0:39:07.560
All right.

0:39:07.560,0:39:10.290
So how am I going[br]to re-parametrize

0:39:10.290,0:39:12.576
the whole surface?

0:39:12.576,0:39:22.220
Whole surface will be r of[br]xy equals in this case, well,

0:39:22.220,0:39:23.280
let's think about it.

0:39:23.280,0:39:29.020
In this case, I'm[br]going to have xy.

0:39:29.020,0:39:31.350
And where's the little f?

0:39:31.350,0:39:32.550
I just erased it.

0:39:32.550,0:39:35.085
I was smart, right,[br]that I erased f of xy.

0:39:35.085,0:39:37.830


0:39:37.830,0:39:46.010
So I have x, y, and[br]z, which is f of xy.

0:39:46.010,0:39:53.240


0:39:53.240,0:40:01.430
And this is the generic point[br]p of coordinates xy f of xy.

0:40:01.430,0:40:04.980


0:40:04.980,0:40:07.580
So I say, OK, what does it mean?

0:40:07.580,0:40:10.100
I will project this point.

0:40:10.100,0:40:13.175
And this is the point[br]when big x becomes little

0:40:13.175,0:40:17.860
x, when big y becomes--[br]where is my y-axis?

0:40:17.860,0:40:20.090
Somebody ate my y axis.

0:40:20.090,0:40:22.190
[INAUDIBLE]

0:40:22.190,0:40:28.400
So when big Y becomes[br]little y, little y

0:40:28.400,0:40:33.830
is just an instance of big Y.[br]And big Z will take what value?

0:40:33.830,0:40:35.630
Well, I need to project that.

0:40:35.630,0:40:39.120
How do you project from[br]a point to the z-axis?

0:40:39.120,0:40:42.680
You have to take the[br]parallel from the point

0:40:42.680,0:40:47.630
to the horizontal[br]plane until you

0:40:47.630,0:40:52.940
hit the-- [INAUDIBLE] the whole[br]plane parallel to the floor

0:40:52.940,0:40:54.210
through the point p.

0:40:54.210,0:40:55.450
And what do I get here?

0:40:55.450,0:40:56.410
STUDENT: [INAUDIBLE].

0:40:56.410,0:40:58.670
PROFESSOR TODA: Not[br]z0, but it's little z

0:40:58.670,0:41:03.120
equals f of xy, which is an[br]instance of the variable xz.

0:41:03.120,0:41:06.460
For you programmers, you know[br]that big z will be a variable

0:41:06.460,0:41:11.640
and little z will be[br][INAUDIBLE] a variable.

0:41:11.640,0:41:12.140
OK.

0:41:12.140,0:41:16.610
So I parametrized my graph[br]in a more general way,

0:41:16.610,0:41:18.578
general parametrization[br]for a graph.

0:41:18.578,0:41:25.960


0:41:25.960,0:41:33.420
And now, what are-- what's the[br]meaning of r sub x and r sub y?

0:41:33.420,0:41:34.489
What are they?

0:41:34.489,0:41:35.364
STUDENT: [INAUDIBLE].

0:41:35.364,0:41:38.180


0:41:38.180,0:41:41.660
PROFESSOR TODA: Now, we[br]don't say that in the book.

0:41:41.660,0:41:42.990
Shame on us.

0:41:42.990,0:41:43.630
Shame on us.

0:41:43.630,0:41:47.480
We should have because I was[br]browsing through the projects

0:41:47.480,0:41:49.900
about a year and a half ago.

0:41:49.900,0:41:52.970
The senior projects of[br]a few of my students

0:41:52.970,0:41:56.340
who are-- two of them were[br]in mechanical engineering.

0:41:56.340,0:42:00.660
One of them was in[br]petroleum engineering.

0:42:00.660,0:42:03.965
And he actually showed me[br]that they were doing this.

0:42:03.965,0:42:07.830
They were taking vectors[br]that depend on parameters--

0:42:07.830,0:42:11.250
this is a vector [INAUDIBLE]--[br]and differentiated them with

0:42:11.250,0:42:13.720
respect to those parameters.

0:42:13.720,0:42:17.215
And I was thinking OK, did we[br]do the partial derivatives r sub

0:42:17.215,0:42:17.960
x, r sub y?

0:42:17.960,0:42:19.340
Not so much.

0:42:19.340,0:42:22.380
But now I want to do it[br]because I think that prepares

0:42:22.380,0:42:24.640
you better as engineers.

0:42:24.640,0:42:29.070
So what is r sub x[br]and what is r sub y?

0:42:29.070,0:42:31.250
And you say, well,[br]OK. [INAUDIBLE],

0:42:31.250,0:42:34.860
I think I know how to do[br]that in my sleep, right.

0:42:34.860,0:42:36.780
If you want me to do[br]that theoretically

0:42:36.780,0:42:39.720
from this formula,[br]but on the picture,

0:42:39.720,0:42:42.450
I really don't know what it is.

0:42:42.450,0:42:45.590
So I'm asking you what[br]I'm going to have in terms

0:42:45.590,0:42:47.240
of r sub x and r sub y.

0:42:47.240,0:42:48.950
They will be vectors.

0:42:48.950,0:42:51.880
This should be a[br]vector as well, right.

0:42:51.880,0:42:56.620
And for me, vector triple[br]means the identification

0:42:56.620,0:42:59.930
between the three coordinates[br]and the physical vector.

0:42:59.930,0:43:01.960
So this is the physical vector.

0:43:01.960,0:43:06.032
Go ahead and write x prime[br]with respect to x is 1.

0:43:06.032,0:43:08.684


0:43:08.684,0:43:13.776
y prime with respect to x is 0.

0:43:13.776,0:43:15.970
The third [INAUDIBLE][br]prime with respect

0:43:15.970,0:43:20.190
to x is just whatever[br]this little f is,

0:43:20.190,0:43:21.984
it's not any of my business.

0:43:21.984,0:43:24.786
It's a [INAUDIBLE][br]function f sub x.

0:43:24.786,0:43:28.290


0:43:28.290,0:43:30.590
Well, what is the second vector?

0:43:30.590,0:43:32.285
STUDENT: 0, 1, f sub y.

0:43:32.285,0:43:34.810
PROFESSOR TODA: 0, 1, f sub y.

0:43:34.810,0:43:36.596
Now, are they slopes?

0:43:36.596,0:43:37.096
No.

0:43:37.096,0:43:38.010
These are slopes.

0:43:38.010,0:43:40.770
That's a slope and[br]that's a slope.

0:43:40.770,0:43:44.950
And we learned[br]about those in 11.3,

0:43:44.950,0:43:49.530
and we understood that those[br]are ski slopes, they were.

0:43:49.530,0:43:52.312
In the direction of x[br]and the direction of y,

0:43:52.312,0:44:00.030
the slopes of the tangents[br]to the coordinate lines.

0:44:00.030,0:44:04.980
But this looks like I have[br]a direction of a line,

0:44:04.980,0:44:08.610
and this would be the lope, and[br]that's the direction of a line,

0:44:08.610,0:44:10.310
and that would be the slope.

0:44:10.310,0:44:12.700
What are those lines?

0:44:12.700,0:44:16.314
STUDENT: [INAUDIBLE] to[br]the function [INAUDIBLE].

0:44:16.314,0:44:17.480
PROFESSOR TODA: Let me draw.

0:44:17.480,0:44:19.440
Then shall I erase[br]the whole thing?

0:44:19.440,0:44:20.160
No.

0:44:20.160,0:44:23.948
I'm just going to keep--[br]I'll erase the tangent.

0:44:23.948,0:44:27.470
Don't erase anything[br]on your notebooks.

0:44:27.470,0:44:28.922
So this is the point p.

0:44:28.922,0:44:29.630
It's still there.

0:44:29.630,0:44:30.570
This is the surface.

0:44:30.570,0:44:33.060
It's still there.

0:44:33.060,0:44:38.200
So my surface will be x,[br]slices of x, [? S ?] constant

0:44:38.200,0:44:39.590
are coming towards you.

0:44:39.590,0:44:45.800
They are these [? walls ?][br]like that, like this, yes.

0:44:45.800,0:44:47.606
It's like the CT scan.

0:44:47.606,0:44:52.190
I think that when they[br]slice up your body,

0:44:52.190,0:44:54.260
tch tch tch tch tch[br]tch, take pictures

0:44:54.260,0:44:57.590
of the slices of your body,[br]that's the same kind of thing.

0:44:57.590,0:44:59.508
So x0, x0, x0, x0.

0:44:59.508,0:45:05.414
I'm going to [INAUDIBLE][br]planes and I had x equals x0.

0:45:05.414,0:45:12.402
And in the other direction, I[br]cut and I get, what do I get?

0:45:12.402,0:45:18.400


0:45:18.400,0:45:20.226
Well, I started bad.

0:45:20.226,0:45:23.650


0:45:23.650,0:45:25.195
Great, Magdalena, this is--

0:45:25.195,0:45:27.226
What is this pink?

0:45:27.226,0:45:32.350
It's not Valentine's Day[br]anymore. y equals [INAUDIBLE].

0:45:32.350,0:45:34.810
And this is the point.

0:45:34.810,0:45:39.320
So, as Alex was[br]trying to tell you,

0:45:39.320,0:45:44.980
our sub x would represent the[br]vector, the physical vector

0:45:44.980,0:45:52.260
in 3D, that is originating[br]at p and tangent to which

0:45:52.260,0:45:55.760
of the two, to the purple[br]one or to the red one?

0:45:55.760,0:45:57.185
STUDENT: Red.

0:45:57.185,0:45:58.135
Uh, purple.

0:45:58.135,0:45:59.560
PROFESSOR TODA:[br]Make up your mind.

0:45:59.560,0:46:01.494
STUDENT: The purple one.

0:46:01.494,0:46:03.660
PROFESSOR TODA: [INAUDIBLE][br]constant and [INAUDIBLE]

0:46:03.660,0:46:06.770
constant in the red[br]one, y equals y0, right?

0:46:06.770,0:46:08.915
So, this depends on x.

0:46:08.915,0:46:11.010
So this has r sub x.

0:46:11.010,0:46:14.800


0:46:14.800,0:46:18.830
This is the velocity with[br]respect to the variable x.

0:46:18.830,0:46:23.200
And the other one, the[br]blue one, x equals x0,

0:46:23.200,0:46:27.640
means x0 is held fixed[br]and y is the variable.

0:46:27.640,0:46:30.505
So I have to do r sub y,[br]and what am I gonna get?

0:46:30.505,0:46:32.696
I'm gonna get the blue vector.

0:46:32.696,0:46:34.880
What's the property[br]of the blue vector?

0:46:34.880,0:46:37.830
It's tangent to the purple line.

0:46:37.830,0:46:44.160
So r sub y has to be[br]tangent to the curve.

0:46:44.160,0:46:47.440


0:46:47.440,0:46:55.310
x0, y, f of x0 and[br]y is the curve.

0:46:55.310,0:46:59.770
And r sub x is tangent[br]to which curve?

0:46:59.770,0:47:02.400
Who is telling me which curve?

0:47:02.400,0:47:12.020
x, y0 sub constant,[br]f of x and y0.

0:47:12.020,0:47:14.489
So that's a curve that[br]depends only on y,

0:47:14.489,0:47:16.854
y is the time in this case.

0:47:16.854,0:47:19.000
And that's the curve[br]that depends only on x.

0:47:19.000,0:47:21.210
x is the time in this case.

0:47:21.210,0:47:24.580
r sub x and r sub y are[br]the tangent vectors.

0:47:24.580,0:47:26.830
What's magical about them?

0:47:26.830,0:47:30.540
If I shape this[br]triangle between them,

0:47:30.540,0:47:32.172
that will be the tangent plane.

0:47:32.172,0:47:35.950


0:47:35.950,0:47:39.170
And I make a smile because I[br]discovered the tangent plane

0:47:39.170,0:47:43.230
in a different way than[br]we did it last time.

0:47:43.230,0:47:51.005
So the tangent plane represents[br]the plane of the vector r sub

0:47:51.005,0:47:54.532
x and r sub y.

0:47:54.532,0:48:02.290
The tangent plane[br]represents the plane

0:48:02.290,0:48:13.080
given by vectors r sub x and[br]r sub y with what conditions?

0:48:13.080,0:48:14.025
It's a conditional.

0:48:14.025,0:48:17.010


0:48:17.010,0:48:20.630
r sub x and r sub[br]y shouldn't be 0.

0:48:20.630,0:48:24.850
r sub x different from 0,[br]r sub y different from 0,

0:48:24.850,0:48:27.455
and r sub x and r sub[br]y are not collinear.

0:48:27.455,0:48:32.160


0:48:32.160,0:48:35.050
What's gonna happen[br]if they are collinear?

0:48:35.050,0:48:36.880
Well, they're gonna[br]collapse; they are not

0:48:36.880,0:48:38.190
gonna determine a plane.

0:48:38.190,0:48:40.770
So there will be[br]no tangent planes.

0:48:40.770,0:48:43.720
So they have to be[br]linearly independent.

0:48:43.720,0:48:47.940
For the people who are taking[br]now linear algebra, I'm saying.

0:48:47.940,0:48:50.940
So we have no other[br]choice, we have

0:48:50.940,0:48:54.820
to assume that these vectors,[br]called partial velocities,

0:48:54.820,0:49:04.120
by the way, for the[br]motion across the surface.

0:49:04.120,0:49:04.620
OK?

0:49:04.620,0:49:06.970
These are the partial[br]velocities, or partial velocity

0:49:06.970,0:49:08.630
vectors.

0:49:08.630,0:49:12.860
Partial velocity vectors[br]have to determine a plane,

0:49:12.860,0:49:16.560
so I have to assume[br]that they are non-zero,

0:49:16.560,0:49:20.120
they never become 0, and[br]they are not collinear.

0:49:20.120,0:49:23.270
If they are collinear,[br]life is over for you.

0:49:23.270,0:49:24.140
OK?

0:49:24.140,0:49:29.390
So I have to assume that I[br]throw away all the points where

0:49:29.390,0:49:35.100
the velocities become 0, and[br]all the points where--those are

0:49:35.100,0:49:39.710
singularity points--where[br]my velocity vectors are 0.

0:49:39.710,0:49:43.710


0:49:43.710,0:49:45.820
Have you ever studied design?

0:49:45.820,0:49:47.350
Any kind of experimental design.

0:49:47.350,0:49:52.310
Like, how do you design a car,[br]the coordinate lines on a car?

0:49:52.310,0:49:53.280
I'm just dreaming.

0:49:53.280,0:50:00.200
You have a car, a beautiful[br]car, and then you have-- Well,

0:50:00.200,0:50:04.790
I cannot draw really[br]well, but anyway.

0:50:04.790,0:50:08.730
I have these coordinate[br]lines on this car.

0:50:08.730,0:50:12.060
It's a mesh what I have there.

0:50:12.060,0:50:15.510
Actually, we do that in[br]animation all the time.

0:50:15.510,0:50:21.030
We have meshes over the[br]models we have in animation.

0:50:21.030,0:50:22.660
Think Avatar.

0:50:22.660,0:50:27.210
Now, those are all[br]coordinate lines.

0:50:27.210,0:50:33.650
Those coordinate lines would be,[br]even your singularities, where?

0:50:33.650,0:50:38.510
For example, if you take a body[br]in a mesh like that, in a net,

0:50:38.510,0:50:43.190
in, like, a fishnet, then[br]you pull from the fishnet,

0:50:43.190,0:50:52.980
all the coordinate lines[br]will come together,

0:50:52.980,0:50:55.310
and this would be a singularity.

0:50:55.310,0:50:57.890
We avoid this kind[br]of singularity.

0:50:57.890,0:51:00.430
So these are points where[br]something bad happened.

0:51:00.430,0:51:05.380
Either the velocity[br]vectors become collinear.

0:51:05.380,0:51:07.430
You see what I'm talking about?

0:51:07.430,0:51:11.260
Or the velocity[br]vectors shrank to 0.

0:51:11.260,0:51:14.190
So that's a bad point;[br]that's a singularity point.

0:51:14.190,0:51:16.870
They have this[br]problem when meshing.

0:51:16.870,0:51:20.670
So when they make[br]these models that

0:51:20.670,0:51:26.850
involve two-dimensional meshing[br]and three-dimensional ambient

0:51:26.850,0:51:31.490
space, like it is in[br]animation, the mesh

0:51:31.490,0:51:34.630
is called regular[br]if we don't have

0:51:34.630,0:51:39.770
this kind of singularity, where[br]the velocity vectors become 0,

0:51:39.770,0:51:42.000
or collinear.

0:51:42.000,0:51:45.795
It's very important for a[br]person who programs in animation

0:51:45.795,0:51:47.220
to know mathematics.

0:51:47.220,0:51:50.100
If they don't understand[br]these things, it's over.

0:51:50.100,0:51:55.910
Because you write the matrix,[br]and you will know the vectors

0:51:55.910,0:51:59.954
will become collinear when the[br]two vectors--let's say two rows

0:51:59.954,0:52:00.495
of a matrix--

0:52:00.495,0:52:00.810
STUDENT: Parallel.

0:52:00.810,0:52:01.880
PROFESSOR TODA:[br]Are proportional.

0:52:01.880,0:52:02.510
Or parallel.

0:52:02.510,0:52:03.870
Or proportional.

0:52:03.870,0:52:07.550
So, everything is numerical[br]in terms of those matrices,

0:52:07.550,0:52:12.890
but it's just a discretization[br]of a continuous phenomenon,

0:52:12.890,0:52:14.010
which is this one.

0:52:14.010,0:52:17.690


0:52:17.690,0:52:19.970
Do you remember Toy Story?

0:52:19.970,0:52:20.815
OK.

0:52:20.815,0:52:24.300
The Toy Story people,[br]the renderers,

0:52:24.300,0:52:27.010
the ones who did the rendering[br]techniques for Toy Story,

0:52:27.010,0:52:30.410
both have their[br]master's in mathematics.

0:52:30.410,0:52:33.970
And you realize why[br]now to do that you

0:52:33.970,0:52:38.860
have to know calc I, calc[br]II, calc III, linear algebra,

0:52:38.860,0:52:41.120
be able to deal with matrices.

0:52:41.120,0:52:45.610
Have a programming course[br]or two; that's essential.

0:52:45.610,0:52:50.042
They took advanced calculus[br]because some people

0:52:50.042,0:52:55.420
don't cover thi-- I was about to[br]skip it right now in calc III.

0:52:55.420,0:53:00.110
But they teach that in[br]advanced calculus 4350, 4351.

0:53:00.110,0:53:02.672
So that's about as[br]far as you can get,

0:53:02.672,0:53:05.870
and differential equation's[br]also very important.

0:53:05.870,0:53:09.510
So, if you master those and[br]you go into something else,

0:53:09.510,0:53:12.320
like programming,[br]electrical engineering,

0:53:12.320,0:53:14.250
you're ready for animation.

0:53:14.250,0:53:16.970
[INAUDIBLE] If you went[br]I want to be a rendering

0:53:16.970,0:53:20.140
guy for the next movie,[br]then they'll say no,

0:53:20.140,0:53:21.600
we won't take you.

0:53:21.600,0:53:23.920
I have a friend who[br]works for Disney.

0:53:23.920,0:53:26.780
She wanted to get a PhD.

0:53:26.780,0:53:29.384
At some point, she[br]changed her mind

0:53:29.384,0:53:31.967
and ended up just with a[br]master's in mathematics

0:53:31.967,0:53:33.800
while I was in Kansas,[br]University of Kansas,

0:53:33.800,0:53:36.565
and she said, "You know what?

0:53:36.565,0:53:41.620
Disney's just giving me[br]$65,000 as an intern."

0:53:41.620,0:53:45.582
And I was like OK and probably[br]asked [INAUDIBLE] $40,000 as

0:53:45.582,0:53:46.676
a postdoc.

0:53:46.676,0:53:48.050
And she said,[br]"Good luck to you."

0:53:48.050,0:53:49.420
Good luck to you, too.

0:53:49.420,0:53:52.520
But we stayed in touch,[br]and right now she's

0:53:52.520,0:53:57.460
making twice as much as[br]I'm making, for Disney.

0:53:57.460,0:53:58.676
Is she happy?

0:53:58.676,0:53:59.568
Yeah.

0:53:59.568,0:54:00.460
Would I be happy?

0:54:00.460,0:54:01.406
No.

0:54:01.406,0:54:05.764
Because she works[br]for 11 hours a day.

0:54:05.764,0:54:08.120
11 hours a day, on a chair.

0:54:08.120,0:54:09.090
That would kill me.

0:54:09.090,0:54:15.070
I mean, I spend about six hours[br]sitting on a chair every day

0:54:15.070,0:54:19.160
of the week, but[br]it's still too much.

0:54:19.160,0:54:20.800
She's a hard worker, though.

0:54:20.800,0:54:22.820
She loves what she's doing.

0:54:22.820,0:54:24.060
The problem is your eyes.

0:54:24.060,0:54:27.420
After a while, your[br]eyes are going bad.

0:54:27.420,0:54:33.600
So, what is the normal for[br]the plane in this case?

0:54:33.600,0:54:37.298
I'll try my best[br]ability to draw normal.

0:54:37.298,0:54:38.714
The normal has to[br]be perpendicular

0:54:38.714,0:54:41.950
to the tangent space, right?

0:54:41.950,0:54:43.700
Tangent plane.

0:54:43.700,0:54:46.230
So, n has to be[br]perpendicular to our sub

0:54:46.230,0:54:49.790
x and has to be[br]perpendicular to our sub y.

0:54:49.790,0:54:53.045


0:54:53.045,0:54:56.240
So, can you have any[br]guess how in the world

0:54:56.240,0:54:59.470
I'm gonna get n vector?

0:54:59.470,0:55:01.454
STUDENT: [INAUDIBLE]

0:55:01.454,0:55:02.870
PROFESSOR TODA:[br][INAUDIBLE] That's

0:55:02.870,0:55:05.070
why you need to[br]know linear algebra

0:55:05.070,0:55:09.040
sort of at the same time, but[br]you guys are making it fine.

0:55:09.040,0:55:10.460
It's not a big deal.

0:55:10.460,0:55:16.450
You have a matrix, i, j, k[br]in the front row vectors,

0:55:16.450,0:55:21.570
and then you have r sub x that[br]you gave me, and I erased it.

0:55:21.570,0:55:23.605
1, 0, f sub x.

0:55:23.605,0:55:26.590


0:55:26.590,0:55:29.150
0, 1, f sub y.

0:55:29.150,0:55:40.604
And you have exactly 18[br]seconds to compute this vector.

0:55:40.604,0:55:47.627


0:55:47.627,0:55:48.460
STUDENT: [INAUDIBLE]

0:55:48.460,0:55:52.890


0:55:52.890,0:55:55.690
PROFESSOR TODA: You want k, but[br]I want to leave k at the end

0:55:55.690,0:55:58.540
because I always[br]order my vectors.

0:55:58.540,0:56:02.137
Something i plus something[br]j plus something k.

0:56:02.137,0:56:02.970
[INTERPOSING VOICES]

0:56:02.970,0:56:05.276


0:56:05.276,0:56:06.400
PROFESSOR TODA: Am I right?

0:56:06.400,0:56:07.025
Minus f sub x--

0:56:07.025,0:56:09.910
STUDENT: Minus f of x plus k.

0:56:09.910,0:56:11.892
PROFESSOR TODA: --times i.

0:56:11.892,0:56:14.262
For j, do I have to change sign?

0:56:14.262,0:56:18.370
Yeah, because 1 plus 2 is odd.

0:56:18.370,0:56:21.273
So I go minus 1.

0:56:21.273,0:56:22.600
And do it slowly.

0:56:22.600,0:56:25.740
You're not gonna make fun of[br]me; I gotta make fun of you, OK?

0:56:25.740,0:56:28.100
And minus 1 times--

0:56:28.100,0:56:29.440
STUDENT: Did you forget f y?

0:56:29.440,0:56:37.150
PROFESSOR TODA: --f sub y--I go[br]like that--sub y times j plus

0:56:37.150,0:56:39.228
k.

0:56:39.228,0:56:42.120
As you said very well[br]in the most elegant way

0:56:42.120,0:56:45.750
without being like yours,[br]but I say it like this.

0:56:45.750,0:56:49.870
So you have minus f[br]sub x, minus f sub y,

0:56:49.870,0:56:54.580
and 1 as a triple with angular[br]brackets--You love that.

0:56:54.580,0:57:00.250
I don't; I like it parentheses[br][INAUDIBLE]--equals n.

0:57:00.250,0:57:03.485
But n is non-unitary,[br]but I don't care.

0:57:03.485,0:57:04.730
Why don't I care?

0:57:04.730,0:57:08.270
I can write the[br]tangent plane very well

0:57:08.270,0:57:13.216
without that n being[br]unitary, right?

0:57:13.216,0:57:14.540
It doesn't matter in the end.

0:57:14.540,0:57:17.680
These would be my a, b, c.

0:57:17.680,0:57:18.860
Now I know my ABC.

0:57:18.860,0:57:20.400
I know my ABC.

0:57:20.400,0:57:26.315
So, the tangent plane[br]is your next guess.

0:57:26.315,0:57:30.140
The tangent plane would[br]be perpendicular to n.

0:57:30.140,0:57:32.150
So this is n.

0:57:32.150,0:57:35.515
The tangent plane passes[br]through the point p

0:57:35.515,0:57:37.350
and is perpendicular to n.

0:57:37.350,0:57:43.147
So, what is the equation[br]of the tangent plane?

0:57:43.147,0:57:44.730
STUDENT: Do you want[br]scalar equations?

0:57:44.730,0:57:49.160
PROFESSOR TODA: A by x minus 0.

0:57:49.160,0:57:50.220
Very good.

0:57:50.220,0:57:56.330
That's exactly what I[br]wanted you to write.

0:57:56.330,0:58:01.390
All right, so, does[br]it look familiar?

0:58:01.390,0:58:01.920
Not yet.

0:58:01.920,0:58:02.387
[STUDENT SNEEZES]

0:58:02.387,0:58:02.854
STUDENT: Bless you.

0:58:02.854,0:58:03.757
STUDENT: Bless you.

0:58:03.757,0:58:04.840
PROFESSOR TODA: Bless you.

0:58:04.840,0:58:05.994
Who sneezed?

0:58:05.994,0:58:08.800
OK.

0:58:08.800,0:58:10.370
Am I almost done?

0:58:10.370,0:58:11.700
Well, I am almost done.

0:58:11.700,0:58:14.930
I have to go backwards,[br]and whatever I get

0:58:14.930,0:58:17.760
I'll put it big here in[br]a big formula on top.

0:58:17.760,0:58:22.370
I'm gonna say oh, my God.

0:58:22.370,0:58:24.020
No, that's not[br]what I'm gonna say.

0:58:24.020,0:58:33.330
I'm gonna say minus f sub x at[br]my point p--that is a, right?

0:58:33.330,0:58:37.089
Times x minus x0.

0:58:37.089,0:58:45.956
Plus minus f sub y at[br]the point p; that's b.

0:58:45.956,0:58:54.660
y minus y0 plus--c is 1, right?

0:58:54.660,0:58:55.360
c is 1.

0:58:55.360,0:58:58.020
I'm not gonna write[br]it because if I write

0:58:58.020,0:59:03.990
it you'll want to make fun[br]of me. z minus z0 equals 0.

0:59:03.990,0:59:08.560
Now it starts looking like[br]something familiar, finally.

0:59:08.560,0:59:14.961
Now we discovered[br]that the tangent plane

0:59:14.961,0:59:20.630
can be written as z minus z0.

0:59:20.630,0:59:24.630
I'm keeping the guys z minus[br]z0 on the left-hand side.

0:59:24.630,0:59:28.630
And these guys are gonna[br]move to the right-hand side.

0:59:28.630,0:59:33.570
So, I'm gonna have[br]again, my friend,

0:59:33.570,0:59:45.460
the equation of the tangent[br]plane for the graph z equals f

0:59:45.460,0:59:46.180
of x,y.

0:59:46.180,0:59:51.940


0:59:51.940,0:59:54.870
But you will say[br]OK, I think by now

0:59:54.870,0:59:57.340
we've learned these[br]by heart, we know

0:59:57.340,1:00:00.480
the equation of the tangent[br]plane, and now we're asleep.

1:00:00.480,1:00:06.160
But what if your surface[br]would be implicit the way

1:00:06.160,1:00:08.760
you gave it to us at first.

1:00:08.760,1:00:11.840
Maybe you remember the sphere[br]that was an implicit equation,

1:00:11.840,1:00:14.720
x squared plus x squared[br]plus x squared equals--

1:00:14.720,1:00:16.030
What do you want it to be?

1:00:16.030,1:00:16.777
STUDENT: 16.

1:00:16.777,1:00:17.610
PROFESSOR TODA: Huh?

1:00:17.610,1:00:18.800
STUDENT: 16.

1:00:18.800,1:00:20.920
PROFESSOR TODA: 16.

1:00:20.920,1:00:22.385
So, radius should be 4.

1:00:22.385,1:00:26.798


1:00:26.798,1:00:31.060
And in such a case, the equation[br]is of the type f of x, y, z

1:00:31.060,1:00:33.190
equals constant.

1:00:33.190,1:00:35.740
Can we write again the[br]equation [INAUDIBLE]?

1:00:35.740,1:00:39.770


1:00:39.770,1:00:42.240
Well, you say well,[br]you just taught

1:00:42.240,1:00:51.240
us some theory that says I have[br]to think of u and v, but not x

1:00:51.240,1:00:51.850
and y.

1:00:51.850,1:00:55.190
Because if I think of x[br]and y, what would they be?

1:00:55.190,1:00:57.960
I think the sphere[br]as being an apple.

1:00:57.960,1:01:01.880
Not an apple, something[br]you can cut easily.

1:01:01.880,1:01:05.480
Well, an apple, an[br]orange, something.

1:01:05.480,1:01:07.080
A round piece of soft cheese.

1:01:07.080,1:01:09.510
I started being hungry,[br]and I'm dreaming.

1:01:09.510,1:01:14.190
So, this is a huge something[br]you're gonna slice up.

1:01:14.190,1:01:19.100
If you are gonna[br]do it with x and y,

1:01:19.100,1:01:21.580
the slices would be like this.

1:01:21.580,1:01:24.630
Like that and like this, right?

1:01:24.630,1:01:27.120
And in that case,[br]your coordinate curves

1:01:27.120,1:01:30.540
are sort of weird.

1:01:30.540,1:01:33.610
If you want to do it in[br]different coordinates,

1:01:33.610,1:01:35.080
so we want to[br]change coordinates,

1:01:35.080,1:01:39.810
and those coordinates should[br]be plotted to the longitude,

1:01:39.810,1:01:43.628
then we cannot use x and y.

1:01:43.628,1:01:44.990
Am I right?

1:01:44.990,1:01:46.590
We cannot use x and y.

1:01:46.590,1:01:50.630
So those u and v will be[br]different coordinates,

1:01:50.630,1:01:55.160
and then we can do it[br]like that, latitude.

1:01:55.160,1:01:57.790


1:01:57.790,1:02:00.010
[INAUDIBLE] minus [INAUDIBLE].

1:02:00.010,1:02:00.805
And longitude.

1:02:00.805,1:02:03.080
We are gonna talk about[br]spherical coordinates

1:02:03.080,1:02:05.202
later, not today.

1:02:05.202,1:02:06.160
Latitude and longitude.

1:02:06.160,1:02:10.340


1:02:10.340,1:02:12.890
1 point extra credit,[br]because eventually we

1:02:12.890,1:02:16.884
are gonna get[br]there, chapter 12.7.

1:02:16.884,1:02:20.650
12.7 comes way[br]after spring break.

1:02:20.650,1:02:27.390
But before we get there, who[br]is in mechanical engineering

1:02:27.390,1:02:28.830
again?

1:02:28.830,1:02:32.710
You know about Euler's[br]angles, and stuff like that.

1:02:32.710,1:02:33.550
OK.

1:02:33.550,1:02:40.330
Can you write me[br]the equations of x

1:02:40.330,1:02:47.850
and y and z of the sphere[br]with respect to u and v,

1:02:47.850,1:02:51.200
u being latitude and[br]v being longitude?

1:02:51.200,1:02:53.980


1:02:53.980,1:02:58.641
These have to be[br]trigonometric functions.

1:02:58.641,1:03:03.860


1:03:03.860,1:03:10.770
In terms of u and v, when u is[br]latitude and v is longitude.

1:03:10.770,1:03:15.310
1 point extra credit[br]until a week from today.

1:03:15.310,1:03:16.280
How about that?

1:03:16.280,1:03:20.650


1:03:20.650,1:03:23.850
U and v are latitude[br]and longitude.

1:03:23.850,1:03:33.800
And express the xyz point in[br]the ambient space on the sphere.

1:03:33.800,1:03:36.460
x squared plus x squared[br]plus x squared would be 16.

1:03:36.460,1:03:40.020
So you'll have lots of[br]cosines and sines [INAUDIBLE]

1:03:40.020,1:03:46.024
of those angles, the latitude[br]angle and the longitude angle.

1:03:46.024,1:03:49.800
And I would suggest to you that[br]you take--for the extra credit

1:03:49.800,1:03:54.910
thing--you take the longitude[br]angle to be from 0 to 2pi,

1:03:54.910,1:04:00.150
from the Greenwich 0 meridian[br]going back to himself,

1:04:00.150,1:04:07.725
and--well, there are two ways[br]we do this in mathematics

1:04:07.725,1:04:09.810
because mathematicians[br]are so diverse.

1:04:09.810,1:04:14.850
Some of us, say, for me,[br]I measure the latitude

1:04:14.850,1:04:17.100
starting from the North Pole.

1:04:17.100,1:04:20.270
I think that's because we all[br]believe in Santa or something.

1:04:20.270,1:04:23.440
So, we start measuring[br]always from the North Pole

1:04:23.440,1:04:27.030
because that's the most[br]important place on Earth.

1:04:27.030,1:04:35.633
They go 0, pi over 2, and then--[br]what is our lat--shame on me.

1:04:35.633,1:04:36.480
STUDENT: It's 33.

1:04:36.480,1:04:37.271
PROFESSOR TODA: 33?

1:04:37.271,1:04:39.220
OK.

1:04:39.220,1:04:44.060
Then pi would be the[br]equator, and then pi

1:04:44.060,1:04:45.834
would be the South Pole.

1:04:45.834,1:04:50.625
But some other mathematicians,[br]especially biologists

1:04:50.625,1:04:54.530
and differential geometry[br]people, I'm one of them,

1:04:54.530,1:04:56.090
we go like that.

1:04:56.090,1:05:01.620
Minus pi over 2, South Pole[br]0, pi over 2 North Pole.

1:05:01.620,1:05:06.820
So we shift that[br]kind of interval.

1:05:06.820,1:05:10.280
Then for us, the trigonometric[br]functions of these angles

1:05:10.280,1:05:12.020
would be a little[br]bit different when we

1:05:12.020,1:05:14.395
do the spherical coordinates.

1:05:14.395,1:05:16.335
OK, that's just extra credit.

1:05:16.335,1:05:19.070
It has nothing to do with[br]what I'm gonna do right now.

1:05:19.070,1:05:22.960
What I'm gonna do right now[br]is to pick a point on Earth.

1:05:22.960,1:05:26.000
We have to find Lubbock.

1:05:26.000,1:05:27.210
STUDENT: It's on the left.

1:05:27.210,1:05:28.740
PROFESSOR TODA: Here?

1:05:28.740,1:05:29.870
Is that a good point?

1:05:29.870,1:05:32.400


1:05:32.400,1:05:34.486
This is LBB.

1:05:34.486,1:05:38.430
That's Lubbock[br]International Airport.

1:05:38.430,1:05:47.530
So, for Lubbock--let's call it[br]p as well--draw the r sub u,

1:05:47.530,1:05:52.550
r sub v. So, u was latitude.

1:05:52.550,1:05:55.750
So if I fix the latitude,[br]that means I fix

1:05:55.750,1:05:58.650
the 33 point whatever you said.

1:05:58.650,1:06:00.060
u equals u0.

1:06:00.060,1:06:09.630
It is fixed, so I have u[br]fixed, and v equals v0 is that.

1:06:09.630,1:06:14.345
I fixed the meridian[br]where we are.

1:06:14.345,1:06:15.990
What is this tangent vector?

1:06:15.990,1:06:20.518


1:06:20.518,1:06:22.950
To the pink parallel,[br]the tangent vector

1:06:22.950,1:06:25.660
would be r sub what?

1:06:25.660,1:06:26.160
STUDENT: v.

1:06:26.160,1:06:27.785
PROFESSOR TODA: r[br]sub v. You are right.

1:06:27.785,1:06:28.920
You've got the idea.

1:06:28.920,1:06:33.370
And the blue vector would[br]be the partial velocity.

1:06:33.370,1:06:39.466
That's the tangent vector[br]to the blue meridian,

1:06:39.466,1:06:43.920
which is r sub u.

1:06:43.920,1:06:48.675
And what is n gonna be? n's[br]gonna be r sub u [INAUDIBLE].

1:06:48.675,1:06:53.370
But is there any other way[br]to do it in a simpler way

1:06:53.370,1:06:55.515
without you guys going oh, man.

1:06:55.515,1:06:58.055
Suppose some of you don't[br]wanna do the extra credit

1:06:58.055,1:07:00.332
and then say the[br]heck with it; I don't

1:07:00.332,1:07:03.610
care about her stinking extra[br]credit until chapter 12,

1:07:03.610,1:07:07.700
when I have to study the[br]spherical coordinates,

1:07:07.700,1:07:11.170
and is there another[br]way to get n.

1:07:11.170,1:07:13.408
I told you another way to get n.

1:07:13.408,1:07:15.384
Well, we are getting there.

1:07:15.384,1:07:21.750
n was the gradient of f[br]over the length of that.

1:07:21.750,1:07:26.490
And if we want it unitary,[br]the length of f was what?

1:07:26.490,1:07:31.720
f sub x, f sub y, f[br]sub z vector, where

1:07:31.720,1:07:36.530
the implicit equation of[br]the surface was f of x, y, z

1:07:36.530,1:07:38.400
equals c.

1:07:38.400,1:07:40.240
So now we've done this before.

1:07:40.240,1:07:42.470
You say Magdalena, you're[br]repeating yourself.

1:07:42.470,1:07:47.210
I know I'm repeating myself, but[br]I want you to learn this twice

1:07:47.210,1:07:49.260
so you can remember it.

1:07:49.260,1:07:52.410
What is f of x, y, z?

1:07:52.410,1:07:56.700
In my case, it's x squared[br]plus y squared plus z squared

1:07:56.700,1:07:59.930
minus 16, or even nothing.

1:07:59.930,1:08:01.850
Because the constant[br]doesn't matter anyway

1:08:01.850,1:08:04.434
when I do the gradient.

1:08:04.434,1:08:05.600
You guys are doing homework.

1:08:05.600,1:08:08.210
You saw how the gradient goes.

1:08:08.210,1:08:13.730
So gradient of f would[br]be 2x times-- and that's

1:08:13.730,1:08:19.384
the partial derivative times i[br]plus 2y times j plus 2z times

1:08:19.384,1:08:22.964
k-- that's very important.

1:08:22.964,1:08:28.270
[? Lovett ?] has some[br]coordinates we plug in.

1:08:28.270,1:08:33.500
Now, can we write-- two things.

1:08:33.500,1:08:35.620
I want two things from you.

1:08:35.620,1:08:41.340
Write me a total[br]differential b tangent plane

1:08:41.340,1:08:46.140
at the point-- so, a, write[br]the total differential.

1:08:46.140,1:08:50.970


1:08:50.970,1:08:53.670
I'm not going to ask you you[br]to do a linear approximation.

1:08:53.670,1:08:55.810
I could.

1:08:55.810,1:09:23.660
B, write the tangent plane[br]to the sphere at the point

1:09:23.660,1:09:25.189
that-- I don't know.

1:09:25.189,1:09:26.870
I don't want one that's trivial.

1:09:26.870,1:09:30.040


1:09:30.040,1:09:37.770
Let's take this 0, square root[br]of 8, and square root of 8.

1:09:37.770,1:09:39.640
I just have to make[br]sure that I don't

1:09:39.640,1:09:41.700
come with some[br]nonsensical point that's

1:09:41.700,1:09:43.290
not going to be on the sphere.

1:09:43.290,1:09:45.863
This will be because I[br]plugged it in in my mind.

1:09:45.863,1:09:50.229
I get 8 plus 8 is 16 last[br]time I checked, right?

1:09:50.229,1:09:54.980
So after we do this[br]we take a break.

1:09:54.980,1:09:58.282
Suppose that this is a[br]problem on your midterm,

1:09:58.282,1:10:00.742
or on your final or[br]on your homework,

1:10:00.742,1:10:04.326
or on somebody [? YouTubed it ?][br]for a lot of money,

1:10:04.326,1:10:10.010
you asked them, $25 an hour[br]for me to work that problem.

1:10:10.010,1:10:10.570
That's good.

1:10:10.570,1:10:16.730
I mean-- it's-- it's a[br]class that you're taking

1:10:16.730,1:10:20.030
for your general requirement[br]because your school wants you

1:10:20.030,1:10:22.470
to take calc 3.

1:10:22.470,1:10:25.570
But it gives you-- and[br]I know from experience,

1:10:25.570,1:10:27.670
some of my students came[br]back to me and said,

1:10:27.670,1:10:30.160
after I took calc[br]3, I understood it

1:10:30.160,1:10:33.380
so well that I was able to[br]tutor calc 1, calc 2, calc 3,

1:10:33.380,1:10:35.840
so I got a double job.

1:10:35.840,1:10:38.060
Several hours a week,[br]the tutoring center,

1:10:38.060,1:10:39.557
math department,[br]and several hours

1:10:39.557,1:10:40.640
at the [INAUDIBLE] center.

1:10:40.640,1:10:42.670
You know what I'm talking about?

1:10:42.670,1:10:46.220
So I've had students who did[br]well and ended up liking this,

1:10:46.220,1:10:49.276
and said I can tutor[br]this in my sleep.

1:10:49.276,1:10:53.760
So-- and also private tutoring[br]is always a possibility.

1:10:53.760,1:10:55.200
OK.

1:10:55.200,1:10:58.670
Write total differential.

1:10:58.670,1:11:04.260
df equals, and now[br]I'll say at any point.

1:11:04.260,1:11:06.965
So I don't care what[br]the value will be.

1:11:06.965,1:11:08.820
I didn't say at what point.

1:11:08.820,1:11:09.923
It means in general.

1:11:09.923,1:11:12.010
Why is that?

1:11:12.010,1:11:14.900
You tell me, you[br]know that by now.

1:11:14.900,1:11:18.410
2x times what?

1:11:18.410,1:11:20.305
Now, you learned[br]your lesson, you're

1:11:20.305,1:11:21.980
never gonna make mistakes.

1:11:21.980,1:11:25.490
2y plus 2z dz.

1:11:25.490,1:11:26.450
That is very good.

1:11:26.450,1:11:28.010
That's the total differential.

1:11:28.010,1:11:33.960
Now, what is the equation[br]of the tangent plane?

1:11:33.960,1:11:37.040
It's not gonna be that.

1:11:37.040,1:11:40.670
Because I'm not[br]considering a graph.

1:11:40.670,1:11:44.590
I'm considering an[br]implicitly given surface

1:11:44.590,1:11:52.720
by this implicit equation f of[br]x, y, z, equals c, your friend.

1:11:52.720,1:11:57.734
So what was, in that case,[br]the equation of the plane

1:11:57.734,1:11:59.630
written as?

1:11:59.630,1:12:02.480
STUDENT: [INAUDIBLE]

1:12:02.480,1:12:05.130
PROFESSOR TODA: I'm--[br]yeah, you guys are smart.

1:12:05.130,1:12:06.440
I mean, you are fast.

1:12:06.440,1:12:07.790
Let's do it in general.

1:12:07.790,1:12:11.635
F sub x-- we did that last[br]time, [INAUDIBLE] times--

1:12:11.635,1:12:14.260
do you guys remember?

1:12:14.260,1:12:16.470
x minus x0.

1:12:16.470,1:12:20.660
And this is at the point plus[br]big F sub y at the point times

1:12:20.660,1:12:25.590
y minus y0 plus big F sub[br]z at the point z minus z0.

1:12:25.590,1:12:26.810
This is just review.

1:12:26.810,1:12:27.990
Equals 0.

1:12:27.990,1:12:28.490
Stop.

1:12:28.490,1:12:31.466
Where do these guys come from?

1:12:31.466,1:12:32.954
From the gradient.

1:12:32.954,1:12:34.830
From the gradient.

1:12:34.830,1:12:40.150
Which are the a,b,c, now I[br]know my ABCs, from the normal.

1:12:40.150,1:12:41.925
My ABCs from the normal.

1:12:41.925,1:12:46.636
So in this case-- I[br]don't want to erase

1:12:46.636,1:12:49.016
this beautiful picture.

1:12:49.016,1:12:54.910
The last thing I have to do[br]before the break is-- you

1:12:54.910,1:12:56.900
said 0.

1:12:56.900,1:12:58.960
I'm a lazy person by definition.

1:12:58.960,1:13:02.990
Can you tell me why[br]you said 0 times?

1:13:02.990,1:13:05.000
STUDENT: Because the[br]x value is [INAUDIBLE]

1:13:05.000,1:13:07.430
PROFESSOR TODA: You said[br]2x, plug in and x equals 0

1:13:07.430,1:13:10.050
from your point,[br]Magdalena, so you don't

1:13:10.050,1:13:12.410
have to write down everything.

1:13:12.410,1:13:19.670
But I'm gonna write down 0[br]times x minus 0 plus-- what's

1:13:19.670,1:13:20.613
next for me?

1:13:20.613,1:13:21.710
STUDENT: 2 square root 8.

1:13:21.710,1:13:23.630
PROFESSOR TODA: 2y, 2 root 8.

1:13:23.630,1:13:26.187
Is root 8 beautiful?

1:13:26.187,1:13:28.095
It looks like heck.

1:13:28.095,1:13:32.870
At the end I'm gonna[br]brush it up a little bit.

1:13:32.870,1:13:39.120
This is the partial-- f sub y of[br]t times y minus-- who is y, z?

1:13:39.120,1:13:40.782
Root 8.

1:13:40.782,1:13:41.734
Do I like it?

1:13:41.734,1:13:43.638
I hate it, but it[br]doesn't matter.

1:13:43.638,1:13:45.542
Because I'm gonna simplify.

1:13:45.542,1:13:52.470
Plus again, 2 root 8, thank you.

1:13:52.470,1:13:56.710
This is my c guy.

1:13:56.710,1:14:02.440
Times z minus root 8 equals 0.

1:14:02.440,1:14:05.430
I picked another example[br]from the one from the book,

1:14:05.430,1:14:08.830
because you are gonna[br]read the book anyway.

1:14:08.830,1:14:11.960
I'm gonna erase that.

1:14:11.960,1:14:14.830
And I'm gonna brush[br]this up because it

1:14:14.830,1:14:17.490
looks horrible to me.

1:14:17.490,1:14:19.890
Thank God this goes away.

1:14:19.890,1:14:21.980
So the plane will[br]simply be a combination

1:14:21.980,1:14:24.250
of my y and z in a constant.

1:14:24.250,1:14:28.000
And if I want to[br]make my life easier,

1:14:28.000,1:14:30.466
I'm gonna divide by what?

1:14:30.466,1:14:32.280
By this.

1:14:32.280,1:14:34.390
So in the end, it[br]doesn't matter.

1:14:34.390,1:14:35.916
Come on.

1:14:35.916,1:14:42.300
I'll get y minus root 8 plus[br]c minus root 8 equals 0.

1:14:42.300,1:14:44.020
Do I like it?

1:14:44.020,1:14:44.770
I hate it.

1:14:44.770,1:14:46.660
No, you know, I don't like it.

1:14:46.660,1:14:49.040
Why don't I like it?

1:14:49.040,1:14:50.430
It's not simplified.

1:14:50.430,1:14:56.000
So in any case, if this[br]were multiple choice,

1:14:56.000,1:14:59.450
it would not be written[br]like that, right?

1:14:59.450,1:15:03.990
So what would be the[br]simplified claim in this case?

1:15:03.990,1:15:09.270
The way I would write[br]it-- a y plus a z minus--

1:15:09.270,1:15:11.486
think, what is root 8?

1:15:11.486,1:15:12.530
STUDENT: 2 root 2.

1:15:12.530,1:15:13.738
PROFESSOR TODA: And 2 root 2.

1:15:13.738,1:15:20.990
And 2 root 2, how[br]much-- minus 4 root 2.

1:15:20.990,1:15:28.840
And this is how you are expected[br]to leave this answer boxed.

1:15:28.840,1:15:37.812
This is that tangent[br]plane at the point.

1:15:37.812,1:15:41.200


1:15:41.200,1:15:42.652
To the sphere.

1:15:42.652,1:15:45.570


1:15:45.570,1:15:48.570
There are programs--[br]one time I was teaching

1:15:48.570,1:15:53.970
advance geometry, 4331, and one[br]thing I gave my students to do,

1:15:53.970,1:15:58.920
which was a lot of fun--[br]using a parametrization,

1:15:58.920,1:16:02.710
plot the entire[br]sphere with MathLab.

1:16:02.710,1:16:04.060
We did it with MathLab.

1:16:04.060,1:16:06.930
Some people said they know[br][INAUDIBLE] I didn't care.

1:16:06.930,1:16:09.326
So MathLab for me[br]was easier, so we

1:16:09.326,1:16:11.800
plotted the sphere in MathLab.

1:16:11.800,1:16:14.940
We picked a point,[br]and we drew-- well,

1:16:14.940,1:16:21.960
we drew-- with MathLab we[br]drew the tangent plane that

1:16:21.960,1:16:25.950
was tangent to the[br]sphere at that point.

1:16:25.950,1:16:27.220
And they liked it.

1:16:27.220,1:16:29.550
It was-- you know[br]what this class is,

1:16:29.550,1:16:31.750
is-- if you're math[br]majors you take it.

1:16:31.750,1:16:34.250
It's called advanced geometries.

1:16:34.250,1:16:35.630
Mainly it's theoretical.

1:16:35.630,1:16:38.540
It teaches you Euclidian[br]axioms and stuff,

1:16:38.540,1:16:41.540
and then some[br]non-Euclidian geometries.

1:16:41.540,1:16:45.830
But I thought that I would[br]do it into an honors class.

1:16:45.830,1:16:49.270
And I put one third of that[br]last class visualization

1:16:49.270,1:16:50.850
with MathLab of geometry.

1:16:50.850,1:16:54.020
And I think that was what[br]they liked the most, not so

1:16:54.020,1:16:56.070
much the axiomatic[br]part and the proofs,

1:16:56.070,1:17:03.270
but the hands-on computation[br]and visualization in the lab.

1:17:03.270,1:17:04.980
We have this lab, 113.

1:17:04.980,1:17:07.340
We used to have two labs,[br]but now we are poor,

1:17:07.340,1:17:09.090
we only have one.

1:17:09.090,1:17:10.510
No, we lost the lab.

1:17:10.510,1:17:13.660
The undergraduate[br]lab-- 009, next to you,

1:17:13.660,1:17:18.560
is lost because-- I used[br]to each calc 3 there.

1:17:18.560,1:17:21.536
Not because-- that's[br]not why we lost it.

1:17:21.536,1:17:24.830
We lost it because we-- we[br]put some 20 graduate students

1:17:24.830,1:17:25.330
there.

1:17:25.330,1:17:26.686
We have no space.

1:17:26.686,1:17:30.810
And we have 130 graduate[br]students in mathematics.

1:17:30.810,1:17:32.430
Where do you put them?

1:17:32.430,1:17:34.165
We just cram them into cubicles.

1:17:34.165,1:17:37.590
So they made 20 cubicles[br]here, and they put some,

1:17:37.590,1:17:40.010
so we lost the lab.

1:17:40.010,1:17:41.860
It's sad.

1:17:41.860,1:17:42.730
All right.

1:17:42.730,1:17:45.090
So that's it for now.

1:17:45.090,1:17:47.540
We are gonna take a[br]short break, and we

1:17:47.540,1:17:52.120
will continue for one more hour,[br]which is mostly application.

1:17:52.120,1:17:54.660
I'm sort of done with 11.4.

1:17:54.660,1:17:57.838
I'll jump into 11.5 next.

1:17:57.838,1:18:00.832
Take a short break.

1:18:00.832,1:18:02.828
Thanks for the attendance.

1:18:02.828,1:18:04.824
Oh, and you did the calculus.

1:18:04.824,1:18:05.822
Very good.

1:18:05.822,1:19:51.584


1:19:51.584,1:19:55.077
Did this homework give you[br]a lot of headaches, troubles

1:19:55.077,1:19:56.075
or anything, or not?

1:19:56.075,1:19:57.572
Not too much?

1:19:57.572,1:19:59.069
It's a long homework.

1:19:59.069,1:20:00.566
49 problems-- 42 problems.

1:20:00.566,1:20:05.556


1:20:05.556,1:20:07.053
It wasn't bad?

1:20:07.053,1:22:39.080


1:22:39.080,1:22:45.831
OK, questions from the-- what[br]was it, the first part-- mainly

1:22:45.831,1:22:47.620
the first part of chapter 11.

1:22:47.620,1:22:49.510
This is where we are.

1:22:49.510,1:22:56.690
Right now we hit the[br]half point because 11.8

1:22:56.690,1:22:59.250
is the last section.

1:22:59.250,1:23:03.310
And we will do that, that's[br]Lagrange multipliers.

1:23:03.310,1:23:06.900
So, let's do a little[br]bit of a review.

1:23:06.900,1:23:08.836
Questions about homework.

1:23:08.836,1:23:11.160
Do you have them?

1:23:11.160,1:23:13.990
Imagine this would[br]be office hour.

1:23:13.990,1:23:15.116
What would you ask?

1:23:15.116,1:23:17.804


1:23:17.804,1:23:19.510
STUDENT: I know it's[br]a stupid question,

1:23:19.510,1:23:22.210
but my visualization [INAUDIBLE][br]coming along, and question

1:23:22.210,1:23:26.910
three about the sphere passing[br]the plane and passing the line.

1:23:26.910,1:23:31.590
So you have a 3, 5,[br]and 4 x, y, and z,

1:23:31.590,1:23:34.335
and you have a radius of 5.

1:23:34.335,1:23:36.275
Is it passing the x, y plane?

1:23:36.275,1:23:40.710
Is it passing [INAUDIBLE][br]x plane and [INAUDIBLE]

1:23:40.710,1:23:42.220
passing the other plane.

1:23:42.220,1:23:44.010
PROFESSOR TODA: So-- say again.

1:23:44.010,1:23:46.050
So you have 3 and 4 and 5--

1:23:46.050,1:23:47.520
STUDENT: x minus-- yes.

1:23:47.520,1:23:49.380
PROFESSOR TODA: What[br]are the coordinates?

1:23:49.380,1:23:50.691
STUDENT: 3, 4, and 5.

1:23:50.691,1:23:53.410
PROFESSOR TODA: 3, 4, and[br]5, just as you said them.

1:23:53.410,1:23:54.190
You can--

1:23:54.190,1:23:55.730
STUDENT: And the radius is 5.

1:23:55.730,1:23:56.840
PROFESSOR TODA: Radius of?

1:23:56.840,1:23:57.340
STUDENT: 5.

1:23:57.340,1:23:59.640
Radius is equal to 5.

1:23:59.640,1:24:00.600
[INAUDIBLE]

1:24:00.600,1:24:02.110
PROFESSOR TODA: Yeah, well, OK.

1:24:02.110,1:24:07.779
So assume you have a[br]sphere of radius 5, which

1:24:07.779,1:24:09.270
means you have 25.

1:24:09.270,1:24:14.710
If you do the 3 squared plus[br]4 squared plus 5 squared,

1:24:14.710,1:24:16.465
what is that?

1:24:16.465,1:24:17.090
For this point.

1:24:17.090,1:24:18.750
You have two separate points.

1:24:18.750,1:24:22.919
For this point you[br]have 25 plus 25.

1:24:22.919,1:24:24.875
Are you guys with me?

1:24:24.875,1:24:30.254
So you have the[br]specific x0, y0, z0.

1:24:30.254,1:24:39.060
You do the sum of the[br]squares, and you get 50.

1:24:39.060,1:24:43.900
My question is, is this point[br]outside, inside the sphere

1:24:43.900,1:24:45.370
or on the sphere?

1:24:45.370,1:24:47.010
On the sphere,[br]obviously, it's not,

1:24:47.010,1:24:54.140
because it does not verify the[br]equation of the sphere, right?

1:24:54.140,1:24:59.140
STUDENT: [INAUDIBLE] those the[br]location of the center point.

1:24:59.140,1:25:01.307
STUDENT: Where's the[br]center of the sphere?

1:25:01.307,1:25:02.140
STUDENT: [INAUDIBLE]

1:25:02.140,1:25:05.640


1:25:05.640,1:25:09.140
PROFESSOR TODA: The center[br]of the sphere would be at 0.

1:25:09.140,1:25:11.640
STUDENT: [INAUDIBLE]

1:25:11.640,1:25:13.520
PROFESSOR TODA: We are[br]making up a question.

1:25:13.520,1:25:14.730
So, right?

1:25:14.730,1:25:16.784
So practically, I am[br]making up a question.

1:25:16.784,1:25:17.450
STUDENT: Oh, OK.

1:25:17.450,1:25:22.930
PROFESSOR TODA: So I'm saying if[br]you have a sphere of radius 5,

1:25:22.930,1:25:27.175
and somebody gives you this[br]point of coordinates 3, 4,

1:25:27.175,1:25:29.240
and 5, where is the point?

1:25:29.240,1:25:34.934
Is it inside the sphere, outside[br]the sphere or on the sphere?

1:25:34.934,1:25:37.100
On the sphere it cannot be[br]because it doesn't verify

1:25:37.100,1:25:39.776
the sphere.

1:25:39.776,1:25:44.580
Ah, it looks like a Mr. Egg.

1:25:44.580,1:25:47.280
I don't like it.

1:25:47.280,1:25:50.610
I'm sorry, it's a sphere.

1:25:50.610,1:25:54.880
So a point on a sphere that[br]will have-- that's a hint.

1:25:54.880,1:25:58.470
A point on a sphere that[br]will have coordinates 3 and 4

1:25:58.470,1:26:02.490
would be exactly 3, 4, and 0.

1:26:02.490,1:26:05.960
So it would be where?

1:26:05.960,1:26:07.760
STUDENT: 16, 4.

1:26:07.760,1:26:11.480
PROFESSOR TODA: 3 squared plus[br]4 squared is 5 squared, right?

1:26:11.480,1:26:13.300
So those are[br]Pythagorean numbers.

1:26:13.300,1:26:15.162
That's the beauty of them.

1:26:15.162,1:26:22.602


1:26:22.602,1:26:27.570
I'm trying to draw well.

1:26:27.570,1:26:28.410
Right.

1:26:28.410,1:26:29.980
This is the point a.

1:26:29.980,1:26:33.235


1:26:33.235,1:26:36.620
You go up how many?

1:26:36.620,1:26:38.940
You shift by 5.

1:26:38.940,1:26:41.329
So are you inside or outside?

1:26:41.329,1:26:42.307
STUDENT: Outside.

1:26:42.307,1:26:43.182
PROFESSOR TODA: Yeah.

1:26:43.182,1:26:50.131


1:26:50.131,1:26:55.020
STUDENT: Are you outside[br]or are you exactly on-- oh.

1:26:55.020,1:26:55.770
Sorry, I thought--

1:26:55.770,1:26:56.400
PROFESSOR TODA: You go--

1:26:56.400,1:26:58.286
STUDENT: I thought you[br]were saying point a.

1:26:58.286,1:26:59.960
Point a is like[br]exactly-- [INAUDIBLE]

1:26:59.960,1:27:00.810
PROFESSOR TODA: You[br]are on the equator,

1:27:00.810,1:27:02.260
and from the Equator[br]of the Earth,

1:27:02.260,1:27:05.750
you're going parallel to the[br]z-axis, then you stay outside.

1:27:05.750,1:27:08.570
But the question is[br]more subtle than that.

1:27:08.570,1:27:12.000
This is pretty--[br]you figured it out.

1:27:12.000,1:27:15.310
1 point-- 0.5 extra credit.

1:27:15.310,1:27:18.580
That we don't have--[br]I wish we had-- maybe

1:27:18.580,1:27:19.900
we'll find some time.

1:27:19.900,1:27:23.030
When I-- when we rewrite the[br]book, maybe we should do that.

1:27:23.030,1:27:38.636
So express the points outside[br]the sphere, inside the sphere,

1:27:38.636,1:27:50.210
and on the sphere[br]using exclusively

1:27:50.210,1:27:51.636
equalities and inequalities.

1:27:51.636,1:27:57.900


1:27:57.900,1:27:58.900
And that's extra credit.

1:27:58.900,1:28:01.000
So, of course, the[br][INAUDIBLE] is obvious.

1:28:01.000,1:28:06.800
The sphere is the set of[br]the triples x, y, z in R3.

1:28:06.800,1:28:09.680


1:28:09.680,1:28:13.480
OK, I'm teaching you a little[br]bit of mathematical language.

1:28:13.480,1:28:19.560
x, y, z belongs to R3,[br]R3 being the free space,

1:28:19.560,1:28:23.810
with the property that x squared[br]plus y squared plus z squared

1:28:23.810,1:28:26.720
equals given a squared.

1:28:26.720,1:28:29.840
What if you have less than,[br]what if you have greater than?

1:28:29.840,1:28:31.836
Ah, shut up, Magdalena.

1:28:31.836,1:28:33.400
This is all up to you.

1:28:33.400,1:28:35.910
You will figure[br]out how the points

1:28:35.910,1:28:40.920
on the outside and the points[br]on the inside are characterized.

1:28:40.920,1:28:47.060
And unfortunately we don't[br]emphasize that in the textbook.

1:28:47.060,1:28:49.510
I'll erase.

1:28:49.510,1:28:51.682
You figured it out.

1:28:51.682,1:28:53.265
And now I want to[br]move on to something

1:28:53.265,1:28:57.056
a little bit challenging,[br]but not very challenging.

1:28:57.056,1:29:11.203


1:29:11.203,1:29:12.494
STUDENT: Professor, [INAUDIBLE]

1:29:12.494,1:29:19.830


1:29:19.830,1:29:21.330
PROFESSOR TODA: The[br]last requirement

1:29:21.330,1:29:22.630
on the extra credit?

1:29:22.630,1:29:26.660
So I said the sphere[br]represents the set of all

1:29:26.660,1:29:29.532
triples x, y, z in[br]R3 with the property

1:29:29.532,1:29:31.990
that x squared plus y squared[br]plus y squared plus z squared

1:29:31.990,1:29:33.880
equals a squared.

1:29:33.880,1:29:36.840
With the equality sign.

1:29:36.840,1:29:40.020
Represent the points on[br]the inside of the sphere

1:29:40.020,1:29:44.560
and the outside of the sphere[br]using just inequalities.

1:29:44.560,1:29:45.280
Mathematics.

1:29:45.280,1:29:48.710
No writing, no words,[br]just mathematics.

1:29:48.710,1:29:50.150
In set theory symbols.

1:29:50.150,1:29:54.824
Like, the set of points[br]with braces like that.

1:29:54.824,1:29:57.680
OK.

1:29:57.680,1:30:02.620
I'll help you review a little[br]bit of stuff from the chain

1:30:02.620,1:30:12.150
rule in-- in chapter--[br]I don't know, guys,

1:30:12.150,1:30:14.770
it was a long time ago.

1:30:14.770,1:30:15.730
Shame on me.

1:30:15.730,1:30:19.320
Chapter 3, calc 1.

1:30:19.320,1:30:38.180
Versus chain rule rules in[br]calc in-- chapter 5 calc 3.

1:30:38.180,1:30:40.550
This is a little[br]bit of a warmup.

1:30:40.550,1:30:42.325
I don't want to[br][INAUDIBLE] again

1:30:42.325,1:30:44.330
next time when we[br]meet on Thursday.

1:30:44.330,1:30:45.990
Bless you.

1:30:45.990,1:30:48.924
The bless you was[br]out of the context.

1:30:48.924,1:30:51.580
What was the chain rule?

1:30:51.580,1:30:53.670
We did compositions[br]of functions,

1:30:53.670,1:31:01.090
and we had a diagram that we[br]don't show you, but we should.

1:31:01.090,1:31:05.050
There is practically a function[br]that comes from a set A

1:31:05.050,1:31:08.490
to a set B to a set[br]C. These are the sets.

1:31:08.490,1:31:12.760
And we have g and an f.

1:31:12.760,1:31:17.480
And we have g of f of t.

1:31:17.480,1:31:22.450
t is your favorite letter here.

1:31:22.450,1:31:26.790
How do you do the[br]derivative with respect

1:31:26.790,1:31:28.940
to g composed with f?

1:31:28.940,1:31:32.920


1:31:32.920,1:31:36.850
I asked the same question to[br]my Calc 1 and Calc 2 students,

1:31:36.850,1:31:42.470
and they really had a hard[br]time expressing themselves,

1:31:42.470,1:31:44.710
expressing the chain rule.

1:31:44.710,1:31:46.530
And when I gave them[br]an example, they

1:31:46.530,1:31:49.600
said, oh, I know how to[br]do it on the example.

1:31:49.600,1:31:55.090
I just don't know how to do it[br]on the-- I like the numbers,

1:31:55.090,1:31:57.510
but I don't like them letters.

1:31:57.510,1:32:02.345
So how do we do[br]it in an example?

1:32:02.345,1:32:05.340


1:32:05.340,1:32:09.140
I chose natural log,[br]which you find everywhere.

1:32:09.140,1:32:14.442
So how do you do d[br]dt of this animal?

1:32:14.442,1:32:15.888
It's an animal.

1:32:15.888,1:32:18.310
STUDENT: [INAUDIBLE]

1:32:18.310,1:32:21.490
PROFESSOR TODA: So the idea[br]is you go from the outside

1:32:21.490,1:32:23.140
to the inside, one at a time.

1:32:23.140,1:32:24.680
My students know that.

1:32:24.680,1:32:27.480
You prime the function,[br]the outer function,

1:32:27.480,1:32:30.572
the last one you applied,[br]to the function inside.

1:32:30.572,1:32:33.770
And you prime that with[br]respect to the argument.

1:32:33.770,1:32:37.040
This is called the[br]argument in that case.

1:32:37.040,1:32:40.694
Derivative of natural[br]log is 1 over what?

1:32:40.694,1:32:43.610
The argument.

1:32:43.610,1:32:46.315
And you cover up natural[br]log with your hand,

1:32:46.315,1:32:47.170
and you keep going.

1:32:47.170,1:32:51.868
And you say, next I go,[br]times the derivative

1:32:51.868,1:32:55.700
of this square, plus 1,[br]prime with respect to t.

1:32:55.700,1:32:58.460
So I go times 2t.

1:32:58.460,1:33:01.290
And that's what we have.

1:33:01.290,1:33:04.620
And they say, when you explain[br]it like that, they said to me,

1:33:04.620,1:33:06.350
I can understand it.

1:33:06.350,1:33:09.050
But I'm having a[br]problem understanding it

1:33:09.050,1:33:12.960
when you express this diagram--[br]that it throws me off.

1:33:12.960,1:33:19.242
So in order to avoid that kind[br]of theoretical misconception,

1:33:19.242,1:33:24.990
I'm saying, let us see[br]what the heck this is.

1:33:24.990,1:33:32.805
d dt of g of f of t, because[br]this is what you're doing,

1:33:32.805,1:33:34.680
has to have some understanding.

1:33:34.680,1:33:38.615
The problem is that Mister[br]f of t, that lives here,

1:33:38.615,1:33:40.370
has a different argument.

1:33:40.370,1:33:45.410
The letter in B should[br]be, let's say, u.

1:33:45.410,1:33:48.510


1:33:48.510,1:33:51.684
That doesn't say[br]anything practically.

1:33:51.684,1:33:54.000
How do you differentiate[br]with respect to what?

1:33:54.000,1:33:56.240
You cannot say d dt here.

1:33:56.240,1:34:00.940
So you have to call f[br]of t something generic.

1:34:00.940,1:34:05.210
You have to have a[br]generic variable for that.

1:34:05.210,1:34:13.690
So you have then dg du, at[br]what specific value of u?

1:34:13.690,1:34:18.050
At the specific value of[br]u that we have as f of t.

1:34:18.050,1:34:21.570
Do you understand the[br]specificity of this?

1:34:21.570,1:34:26.700
Times-- that's the chain[br]rule, the product coming

1:34:26.700,1:34:31.764
from the chain rule-- df pt.

1:34:31.764,1:34:33.890
You take du dt or d of dt.

1:34:33.890,1:34:34.940
It is the same thing.

1:34:34.940,1:34:36.884
Say it again, df dt.

1:34:36.884,1:34:41.260


1:34:41.260,1:34:43.540
I had a student ask me,[br]what if I put du dt?

1:34:43.540,1:34:44.790
Would it be wrong?

1:34:44.790,1:34:50.079
No, as long as you understand[br]that u is a-something,

1:34:50.079,1:34:54.867
as the image of this t.

1:34:54.867,1:34:55.950
Do you know what he liked?

1:34:55.950,1:34:58.935


1:34:58.935,1:35:01.790
He said, do you know[br]what I like about that?

1:35:01.790,1:35:07.165
I like that I can imagine[br]that these are two cowboys-- I

1:35:07.165,1:35:09.450
told the same thing to my son.

1:35:09.450,1:35:12.510
He was so excited,[br]not about that,

1:35:12.510,1:35:14.596
but about these two cowboys.

1:35:14.596,1:35:17.329
Of course, he is 10.

1:35:17.329,1:35:18.245
These are the cowboys.

1:35:18.245,1:35:20.420
They are across.

1:35:20.420,1:35:22.640
One is on top of[br]the building there,

1:35:22.640,1:35:24.850
shooting at this[br]guy, who is here

1:35:24.850,1:35:28.480
across the street on the bottom.

1:35:28.480,1:35:31.390
So they are[br]annihilating each other.

1:35:31.390,1:35:33.290
They shoot and they die.

1:35:33.290,1:35:37.080
And they die, and[br]you're left with 1/3.

1:35:37.080,1:35:41.780
The same idea is that, actually,[br]these guys do not simplify.

1:35:41.780,1:35:46.100
du and-- [? du, ?] they're not[br]cowboys who shoot at each other

1:35:46.100,1:35:48.580
at the same time and both[br]die at the same time.

1:35:48.580,1:35:53.220
It is not so romantic.

1:35:53.220,1:35:59.640
But the idea of remembering[br]this formula is the same.

1:35:59.640,1:36:03.700
Because practically, if you want[br]to annihilate the two cowboys

1:36:03.700,1:36:06.330
and put your hands over them[br]so you don't see them anymore,

1:36:06.330,1:36:10.580
du dt, you would[br]have to remember, oh,

1:36:10.580,1:36:12.430
so that was the[br]derivative with respect

1:36:12.430,1:36:15.930
to t that I initially[br]have of the guy on top,

1:36:15.930,1:36:19.110
which was g of f of[br]the composed function.

1:36:19.110,1:36:22.850
So if you view g of f of t[br]as the composed function,

1:36:22.850,1:36:23.838
who is that?

1:36:23.838,1:36:28.980
The composition g[br]composed with f of t

1:36:28.980,1:36:31.900
is the function g of f of t.

1:36:31.900,1:36:34.940
This is the function that[br]you want to differentiate

1:36:34.940,1:36:37.370
with respect to time, t.

1:36:37.370,1:36:40.980
This is this, prime[br]with respect to t.

1:36:40.980,1:36:46.024
It's like they would be killing[br]each other, and you would die.

1:36:46.024,1:36:48.200
And I liked this[br]idea, and I said,

1:36:48.200,1:36:50.390
I should tell that to my[br]students and to my son.

1:36:50.390,1:36:52.925
And, of course, my son[br]started jumping around

1:36:52.925,1:36:56.485
and said that he understands[br]multiplication of fractions

1:36:56.485,1:36:57.890
better now.

1:36:57.890,1:37:01.390
They don't learn about[br]simplifications-- I don't

1:37:01.390,1:37:03.042
know how they teach these kids.

1:37:03.042,1:37:06.320


1:37:06.320,1:37:07.830
It became so complicated.

1:37:07.830,1:37:10.900
It's as if mathematics--[br]mathematics is the same.

1:37:10.900,1:37:12.110
It hasn't changed.

1:37:12.110,1:37:14.310
It's the people[br]who make the rules

1:37:14.310,1:37:17.415
on how to teach it that change.

1:37:17.415,1:37:21.530
So he simply doesn't see[br]that this simplifies.

1:37:21.530,1:37:24.690
And when I tell him simplify,[br]he's like, what is simplify?

1:37:24.690,1:37:25.850
What is this word simplify?

1:37:25.850,1:37:27.235
My teacher doesn't use it.

1:37:27.235,1:37:31.705
So I feel like sometimes[br]I want to shoot myself.

1:37:31.705,1:37:35.380
But he went over that and[br]he understood about the idea

1:37:35.380,1:37:37.420
of simplification.

1:37:37.420,1:37:39.370
[? He ?] composing[br]something on top

1:37:39.370,1:37:43.190
and the bottom finding the[br]common factors up and down,

1:37:43.190,1:37:44.820
crossing them out, and so on.

1:37:44.820,1:37:47.260
And so now he knows[br]what it means.

1:37:47.260,1:37:50.576
But imagine going to[br]college without having

1:37:50.576,1:37:51.450
this early knowledge.

1:37:51.450,1:37:55.132
You come to college,[br]you were good in school,

1:37:55.132,1:37:57.090
and you've never learned[br]enough simplification.

1:37:57.090,1:38:00.220
And then somebody like me,[br]and tells you simplification.

1:38:00.220,1:38:03.010
You say, she is a foreigner.

1:38:03.010,1:38:07.770
She has a language barrier[br]that is [INAUDIBLE] she has

1:38:07.770,1:38:10.050
that I've never heard before.

1:38:10.050,1:38:15.250
So I wish the people who[br]really re-conceive, re-write

1:38:15.250,1:38:18.820
the curriculum for K12[br]would be a little bit

1:38:18.820,1:38:21.730
more respectful of the history.

1:38:21.730,1:38:25.590
Imagine that I[br]would teach calculus

1:38:25.590,1:38:28.813
without ever telling you[br]anything about Leibniz, who

1:38:28.813,1:38:31.200
was Leibniz, he doesn't exist.

1:38:31.200,1:38:34.100
Or Euler, or one[br]of these fathers.

1:38:34.100,1:38:37.660
They are the ones who[br]created these notations.

1:38:37.660,1:38:42.630
And if we never tell you[br]about them, that I guess,

1:38:42.630,1:38:47.400
wherever they are, it is an[br]injustice that we are doing.

1:38:47.400,1:38:48.180
All right.

1:38:48.180,1:38:53.520
Chain rule in[br]Chapter 5 of Calc 3.

1:38:53.520,1:38:56.310
This is a little bit[br]more complicated,

1:38:56.310,1:38:59.690
but I'm going to teach it[br]to you because I like it.

1:38:59.690,1:39:05.582
Imagine that you have z equals[br]x squared plus y squared.

1:39:05.582,1:39:06.570
What is that?

1:39:06.570,1:39:08.052
It's an example of a graph.

1:39:08.052,1:39:10.614
And I just taught[br]you what a graph is.

1:39:10.614,1:39:13.490


1:39:13.490,1:39:22.939
But imagine that[br]xy follow a curve.

1:39:22.939,1:39:25.900


1:39:25.900,1:39:27.884
[INAUDIBLE] with[br]respect to time.

1:39:27.884,1:39:37.780


1:39:37.780,1:39:41.330
And you will say, Magdalena,[br]can you draw that?

1:39:41.330,1:39:45.670
What in the world do you mean[br]that x and y follow a curve?

1:39:45.670,1:39:46.770
I'll try to draw it.

1:39:46.770,1:39:48.640
First of all, you are on a walk.

1:39:48.640,1:39:50.480
You are in a beautiful valley.

1:39:50.480,1:39:51.480
It's not a vase.

1:39:51.480,1:39:57.088
It's a circular[br]paraboloid, as an example.

1:39:57.088,1:40:00.944


1:40:00.944,1:40:01.908
It's like an egg shell.

1:40:01.908,1:40:05.290


1:40:05.290,1:40:07.050
You have a curve on that.

1:40:07.050,1:40:08.050
You draw that.

1:40:08.050,1:40:10.340
You have nothing better[br]to do than decorating eggs

1:40:10.340,1:40:10.960
for Easter.

1:40:10.960,1:40:12.190
Hey, wait.

1:40:12.190,1:40:14.695
Easter is far, far away.

1:40:14.695,1:40:17.355
But let's say you want to[br]decorate eggs for Easter.

1:40:17.355,1:40:22.800
You take some color of paint[br]and put paint on the egg.

1:40:22.800,1:40:28.400
You are actually describing[br]an arc of a curve.

1:40:28.400,1:40:38.240
And x and y, their[br]projection on the floor

1:40:38.240,1:40:39.605
will be x of t, y of t.

1:40:39.605,1:40:42.790


1:40:42.790,1:40:45.470
Because you paint in time.

1:40:45.470,1:40:46.220
You paint in time.

1:40:46.220,1:40:48.090
You describe this in time.

1:40:48.090,1:40:54.090
Now, if x of ty of t is[br]being projected on the floor.

1:40:54.090,1:40:58.820
Of course, you have a curve[br]here as well, which is what?

1:40:58.820,1:41:05.700
Which it will be x[br]of t, y of t, z of t.

1:41:05.700,1:41:06.620
Oh, my god.

1:41:06.620,1:41:11.910
Yes, because the altitude also[br]depends on the motion in time.

1:41:11.910,1:41:13.810
All right.

1:41:13.810,1:41:16.460
So what's missing here?

1:41:16.460,1:41:18.880
It's missing the third[br]coordinate, duh, that's

1:41:18.880,1:41:21.382
0 because I'm on the floor.

1:41:21.382,1:41:26.500
I'm on the xy plane, which[br]is the floor z equals z.

1:41:26.500,1:41:28.560
But now let's[br]suppose that I want

1:41:28.560,1:41:36.570
to say this is f of x and y,[br]and I want to differentiate

1:41:36.570,1:41:39.400
f with respect to t.

1:41:39.400,1:41:40.730
And you go, say what?

1:41:40.730,1:41:41.440
Oh, my god.

1:41:41.440,1:41:42.480
What is that?

1:41:42.480,1:41:45.840
I differentiate f[br]with respect to t.

1:41:45.840,1:41:48.730
By differentiating[br]f with respect to t,

1:41:48.730,1:41:54.780
I mean that I have f of[br]x and y differentiated

1:41:54.780,1:41:56.145
with respect to t.

1:41:56.145,1:41:58.070
And you say, wait, Magdalena.

1:41:58.070,1:41:59.820
This doesn't make any sense.

1:41:59.820,1:42:03.640
And you would be right to say[br]it doesn't make any sense.

1:42:03.640,1:42:07.280
Can somebody tell me why[br]it doesn't make any sense?

1:42:07.280,1:42:13.580
It's not clear where in the[br]world the variable t is inside.

1:42:13.580,1:42:17.220
So I'm going to say, OK,[br]x are themselves functions

1:42:17.220,1:42:19.500
of t, functions of that.

1:42:19.500,1:42:21.342
x of t, y of t.

1:42:21.342,1:42:23.950
If I don't do that,[br]it's not clear.

1:42:23.950,1:42:27.722
So this is a composed[br]function just like this one.

1:42:27.722,1:42:28.680
Look at the similarity.

1:42:28.680,1:42:31.100
It's really beautiful.

1:42:31.100,1:42:35.670
This is a function of[br]a function, g of f.

1:42:35.670,1:42:38.520
This is a function[br]of two functions.

1:42:38.520,1:42:43.304
Say it again, f is a function[br]of two functions, x and y.

1:42:43.304,1:42:45.232
This was a function[br]of a function of t.

1:42:45.232,1:42:47.642
This was a function[br]of two functions of t.

1:42:47.642,1:42:48.606
Oh, my God.

1:42:48.606,1:42:52.470


1:42:52.470,1:42:55.080
How do we compute this?

1:42:55.080,1:42:56.851
There is a rule.

1:42:56.851,1:42:58.204
It can be proved.

1:42:58.204,1:43:01.950
We will look a little bit into[br]the theoretical justification

1:43:01.950,1:43:03.403
of this proof later.

1:43:03.403,1:43:05.720
But practically what[br]you do, you say,

1:43:05.720,1:43:07.955
I have to have some[br]order in my life.

1:43:07.955,1:43:09.090
OK.?

1:43:09.090,1:43:12.880
So the way we do that,[br]we differentiate first

1:43:12.880,1:43:17.150
with respect to the first[br]location, which is x.

1:43:17.150,1:43:21.515
I go there, but I cannot write[br]df dx because f is a mother

1:43:21.515,1:43:23.110
of two babies.

1:43:23.110,1:43:26.520
f is a function of two[br]variables, x and y.

1:43:26.520,1:43:28.800
She has to be a mother[br]to both of them;

1:43:28.800,1:43:31.620
otherwise, they get[br]jealous of one another.

1:43:31.620,1:43:37.630
So I have to say, partial[br]of f with respect to x,

1:43:37.630,1:43:38.860
I cannot use d.

1:43:38.860,1:43:43.510
Like Leibniz, I have[br]to use del, d of dx.

1:43:43.510,1:43:49.030
At the point x of dy of t,[br]this is the location I have.

1:43:49.030,1:43:50.630
Times what?

1:43:50.630,1:43:51.970
I keep derivation.

1:43:51.970,1:43:55.640
I keep derivating, like[br]don't drink and derive.

1:43:55.640,1:43:56.630
What is that?

1:43:56.630,1:43:58.981
The chain rule.

1:43:58.981,1:44:05.430
Prime again, this guy x[br]with respect to t, dx dt.

1:44:05.430,1:44:09.320
And then you go,[br]plus because she has

1:44:09.320,1:44:11.570
to be a mother to both kids.

1:44:11.570,1:44:14.670
The same thing for[br]the second child.

1:44:14.670,1:44:17.690
So you go, the derivative[br]of f with respect

1:44:17.690,1:44:26.990
to y, add x of ty[br]of t times dy dt.

1:44:26.990,1:44:30.230


1:44:30.230,1:44:35.440
So you see on the surface, x and[br]y are moving according to time.

1:44:35.440,1:44:39.000
And somehow we want to[br]measure the derivative

1:44:39.000,1:44:42.792
of the resulting function,[br]or composition function,

1:44:42.792,1:44:44.610
with respect to time.

1:44:44.610,1:44:46.330
This is a very[br]important chain rule

1:44:46.330,1:44:50.020
that I would like[br]you to memorize.

1:44:50.020,1:44:53.430
A chain rule.

1:44:53.430,1:44:54.060
Chain Rule No.

1:44:54.060,1:44:54.560
1.

1:44:54.560,1:44:58.720


1:44:58.720,1:44:59.920
Is it hard?

1:44:59.920,1:45:01.490
No, but for me it was.

1:45:01.490,1:45:04.690
When I was 21 and I saw[br]that-- and, of course,

1:45:04.690,1:45:06.020
my teacher was good.

1:45:06.020,1:45:10.350
And he told me, Magdalena,[br]imagine that instead of del you

1:45:10.350,1:45:13.530
would have d's.

1:45:13.530,1:45:16.680
So you have d and d and d and d.

1:45:16.680,1:45:21.300
The dx dx here, dy dy here,[br]they should be in your mind.

1:45:21.300,1:45:22.720
They are facing each other.

1:45:22.720,1:45:25.850
They are across on a diagonal.

1:45:25.850,1:45:29.140
And then, of course, I didn't[br]tell my teacher my idea

1:45:29.140,1:45:31.770
with the cowboys,[br]but it was funny.

1:45:31.770,1:45:38.810
So this is the chain rule[br]that re-makes, or generalizes

1:45:38.810,1:45:42.870
this idea to two variables.

1:45:42.870,1:45:47.970
Let's finish the example[br]because we didn't do it.

1:45:47.970,1:45:53.310
What is the derivative[br]of f in our case?

1:45:53.310,1:46:01.660
df dt will be-- oh, my god--[br]at any point p, how arbitary,

1:46:01.660,1:46:03.848
would be what?

1:46:03.848,1:46:07.640
First, you write[br]with respect to x.

1:46:07.640,1:46:10.501
2x, right?

1:46:10.501,1:46:11.000
2x.

1:46:11.000,1:46:16.900
But then you have to compute[br]this dx, add the pair you give.

1:46:16.900,1:46:19.650
And the pair they[br]gave you has a t.

1:46:19.650,1:46:23.450
So 2x is add x of[br]ty-- if you're going

1:46:23.450,1:46:25.335
to write it first[br]like that, you're

1:46:25.335,1:46:29.730
going to find it weird-- times,[br]I'm done with the first guy.

1:46:29.730,1:46:32.795
Then I'm going to take[br]the second guy in red,

1:46:32.795,1:46:35.310
and I'll put it here.

1:46:35.310,1:46:39.278
dx dt, but dx dt[br]everybody knows.

1:46:39.278,1:46:45.080
[INAUDIBLE] Let me[br]write it like this.

1:46:45.080,1:46:52.186
Plus [INAUDIBLE] that[br]guy again with green-- dy

1:46:52.186,1:46:59.146
computed at the pair x[br]of dy of [? t ?] times,

1:46:59.146,1:47:01.511
again, in red, dy dt.

1:47:01.511,1:47:06.730


1:47:06.730,1:47:08.772
So how do we write[br]the whole thing?

1:47:08.772,1:47:10.990
Could I have written it[br]from the beginning better?

1:47:10.990,1:47:11.490
Yeah.

1:47:11.490,1:47:20.630
2x of t, dx dt plus 2y of t dy.

1:47:20.630,1:47:21.610
Is it hard?

1:47:21.610,1:47:25.150
No, this is the idea.

1:47:25.150,1:47:28.070
Let's have something[br]more specific.

1:47:28.070,1:47:30.230
I'm going to erase[br]the whole thing.

1:47:30.230,1:47:36.230


1:47:36.230,1:47:40.205
I'll give you a problem[br]that we gave on the final

1:47:40.205,1:47:41.630
a few years ago.

1:47:41.630,1:47:44.890
And I'll show you how my[br]students cheated on that.

1:47:44.890,1:47:53.386
And I let them cheat, in[br]a way, because in the end

1:47:53.386,1:47:54.060
they were smart.

1:47:54.060,1:47:59.350
It didn't matter how they did[br]the problem, as long as they

1:47:59.350,1:48:01.790
got the correct answer.

1:48:01.790,1:48:03.330
So the problem was like that.

1:48:03.330,1:48:10.155
And my colleague did that many[br]years ago, several years ago,

1:48:10.155,1:48:11.980
did that several times.

1:48:11.980,1:48:19.700
So he said, let's do f of[br]t, dt squared and g of t.

1:48:19.700,1:48:27.000
I'll I'll do this[br]one, dq plus 1.

1:48:27.000,1:48:42.980
And then let's[br][INAUDIBLE] the w of u

1:48:42.980,1:48:54.100
and B, exactly the same thing I[br]gave you before, [INAUDIBLE] I

1:48:54.100,1:48:56.040
remember that.

1:48:56.040,1:49:05.708
And he said, compute the[br]derivative of w of f of t,

1:49:05.708,1:49:10.280
and g of t with respect to t.

1:49:10.280,1:49:12.250
And you will ask,[br]wait a minute here.

1:49:12.250,1:49:14.560
Why do you put d and not del?

1:49:14.560,1:49:17.850
Because this is a composed[br]function that in the end

1:49:17.850,1:49:20.580
is a function of t only.

1:49:20.580,1:49:22.680
So if you do it as[br]a composed function,

1:49:22.680,1:49:26.040
because this goes like this.

1:49:26.040,1:49:31.560
t goes to two[br]functions, f of t and u.

1:49:31.560,1:49:34.454


1:49:34.454,1:49:40.850
And there is a function w[br]that takes both of them, that

1:49:40.850,1:49:42.870
is a function of both of them.

1:49:42.870,1:49:46.835
In the end, this composition[br]that's straight from here

1:49:46.835,1:49:50.826
to here, is a function[br]of one variable only.

1:49:50.826,1:49:54.850


1:49:54.850,1:49:58.280
So my students then-- it was in[br]the beginning of the examine,

1:49:58.280,1:49:59.020
I remember.

1:49:59.020,1:50:02.300
And they said, well,[br]I forgot, they said.

1:50:02.300,1:50:03.860
I stayed up almost all night.

1:50:03.860,1:50:05.432
Don't do that.

1:50:05.432,1:50:06.390
Don't do what they did.

1:50:06.390,1:50:08.330
Many of my students[br]stay up all night

1:50:08.330,1:50:11.090
before the final because[br]I think I scare people,

1:50:11.090,1:50:12.700
and that's not what I mean.

1:50:12.700,1:50:15.480
I just want you to study.

1:50:15.480,1:50:18.670
But they stay up before[br]the final and the next day,

1:50:18.670,1:50:19.380
I'm a vegetable.

1:50:19.380,1:50:21.410
I don't even remember[br]the chain rule.

1:50:21.410,1:50:23.450
So they did not[br]remember the chain rule

1:50:23.450,1:50:25.000
that I've just wrote.

1:50:25.000,1:50:28.490
And they said, oh, but I[br]think I know how to do it.

1:50:28.490,1:50:29.990
And I said, shh.

1:50:29.990,1:50:31.980
Just don't say anything.

1:50:31.980,1:50:34.840
Let me show you how the[br]course coordinator wanted

1:50:34.840,1:50:37.170
that done several years ago.

1:50:37.170,1:50:40.165
So he wanted it done[br]by the chain rule.

1:50:40.165,1:50:41.550
He didn't say how you do it.

1:50:41.550,1:50:42.050
OK?

1:50:42.050,1:50:44.160
He said just get to[br]the right answer.

1:50:44.160,1:50:45.614
It doesn't matter.

1:50:45.614,1:50:46.780
He wanted it done like that.

1:50:46.780,1:50:55.700
He said, dw of f of tg[br]of p with respect to t,

1:50:55.700,1:51:06.738
would be dw du, instead[br]of u you have f of t.

1:51:06.738,1:51:16.730
f of tg of t times df[br]dt plus dw with respect

1:51:16.730,1:51:18.940
to the second variable.

1:51:18.940,1:51:25.137
So this would be u, and[br]this would be v with respect

1:51:25.137,1:51:27.300
to the variable v,[br]the second variable

1:51:27.300,1:51:30.726
where [? measure ?][br]that f of dg of t.

1:51:30.726,1:51:39.410
Evaluate it there times dg dt.

1:51:39.410,1:51:46.136
So it's like dv dt, which is dg[br]dt. [INAUDIBLE] So he did that,

1:51:46.136,1:51:48.045
and he expected[br]people to do what?

1:51:48.045,1:51:51.192
He expected people to take[br]a u squared the same 2 times

1:51:51.192,1:51:54.251
u, just like you[br]did before, 2 times.

1:51:54.251,1:51:57.715
And instead of u, since u is[br]f of t to [INAUDIBLE] puts

1:51:57.715,1:52:13.171
2f of t, this is the first[br]squiggly thing, times v of dt.

1:52:13.171,1:52:19.932
2t is this smiley face.

1:52:19.932,1:52:31.200
This is 2t plus--[br]what is the f dv?

1:52:31.200,1:52:37.600
Dw with respect to dv is[br]going to be 2v 2 time gf t.

1:52:37.600,1:52:46.794
When I evaluate add gf[br]t, this funny fellow

1:52:46.794,1:52:57.580
with this funny fellow, times qg[br]d, which, with your permission

1:52:57.580,1:53:00.890
I'm going to erase[br]and write 3p squared.

1:53:00.890,1:53:04.030


1:53:04.030,1:53:07.340
And the last row he expected[br]my students to write

1:53:07.340,1:53:22.135
was 2t squared times 2t plus[br]2pq plus 1, times 3t squared.

1:53:22.135,1:53:27.580


1:53:27.580,1:53:31.540
Are you guys with me?

1:53:31.540,1:53:43.210
So [INAUDIBLE] 2t 2x[br]2t squared, correct.

1:53:43.210,1:53:49.730
I forgot to identify[br]this as that.

1:53:49.730,1:53:50.230
All right.

1:53:50.230,1:53:52.650
So in the end, the answer[br]is a simplified answer.

1:53:52.650,1:53:53.930
Can you tell me what it is?

1:53:53.930,1:53:55.250
I'm too lazy to write it down.

1:53:55.250,1:53:56.935
You compute it.

1:53:56.935,1:53:58.930
How much is it simplified?

1:53:58.930,1:54:00.421
Find it as a polynomial.

1:54:00.421,1:54:01.415
STUDENT: [INAUDIBLE].

1:54:01.415,1:54:04.397


1:54:04.397,1:54:08.870
PROFESSOR TODA:[br]So you have 6, 6--

1:54:08.870,1:54:10.150
STUDENT: 16 cubed plus 3--

1:54:10.150,1:54:15.300
PROFESSOR TODA: T[br]to the 5th plus--

1:54:15.300,1:54:17.240
STUDENT: [INAUDIBLE].

1:54:17.240,1:54:19.350
PROFESSOR TODA: In[br]order, in order.

1:54:19.350,1:54:20.420
What's the next guy?

1:54:20.420,1:54:21.636
STUDENT: [INAUDIBLE].

1:54:21.636,1:54:22.830
PROFESSOR TODA: 4t cubed.

1:54:22.830,1:54:23.975
And the last guy--

1:54:23.975,1:54:24.925
STUDENT: 6t squared.

1:54:24.925,1:54:26.050
PROFESSOR TODA: 6t squared.

1:54:26.050,1:54:31.220


1:54:31.220,1:54:31.720
Yes?

1:54:31.720,1:54:33.320
Did you get the same thing?

1:54:33.320,1:54:34.220
OK.

1:54:34.220,1:54:37.145
Now, how did my students do it?

1:54:37.145,1:54:37.644
[INAUDIBLE]

1:54:37.644,1:54:40.171


1:54:40.171,1:54:41.420
Did they apply the chain rule?

1:54:41.420,1:54:41.920
No.

1:54:41.920,1:54:44.100
They said OK, this[br]is how it goes.

1:54:44.100,1:54:46.960


1:54:46.960,1:54:58.210
W of U of T and V of T is U is[br]F. So this guy is T squared,

1:54:58.210,1:55:01.886
T squared squared,[br]plus this guy is T

1:55:01.886,1:55:08.960
cubed plus 1 taken and[br]shaken and squared.

1:55:08.960,1:55:13.710
And then when I do the[br]whole thing, derivative

1:55:13.710,1:55:22.640
of this with respect[br]to T, I get--

1:55:22.640,1:55:27.570
I'm too lazy-- T to the[br]4 prime is 40 cubed.

1:55:27.570,1:55:28.870
I'm not going to do on the map.

1:55:28.870,1:55:37.380
2 out T cubed plus 1 times[br]chain rule, 3t squared.

1:55:37.380,1:55:49.610
40 cubed plus 16 to the 5 plus--[br][INAUDIBLE] 2 and 6t squared.

1:55:49.610,1:55:56.450
So you realize that I[br]have to give them 100%.

1:55:56.450,1:55:59.465
Although they were very[br]honest and said, we blanked.

1:55:59.465,1:56:01.130
We don't remember[br]the chain rule.

1:56:01.130,1:56:02.926
We don't remember the formula.

1:56:02.926,1:56:03.550
So that's fine.

1:56:03.550,1:56:05.080
Do whatever you can.

1:56:05.080,1:56:06.920
So I gave them 100% for that.

1:56:06.920,1:56:11.280
But realize that the[br]author of the problem

1:56:11.280,1:56:14.080
was a little bit naive.

1:56:14.080,1:56:16.520
Because you could have[br]done this differently.

1:56:16.520,1:56:22.190
I mean if you wanted to[br]actually test the whole thing,

1:56:22.190,1:56:26.170
you wouldn't have given-- let's[br]say you wouldn't have given

1:56:26.170,1:56:32.400
the actual-- yeah, you wouldn't[br]have given the actual functions

1:56:32.400,1:56:37.530
and say write the chain[br]formula symbolically

1:56:37.530,1:56:44.875
for this function applied[br]for F of T and G of T.

1:56:44.875,1:56:49.340
So it was-- they[br]were just lucky.

1:56:49.340,1:56:52.470
Remember that you need[br]to know this chain rule.

1:56:52.470,1:56:53.970
It's going to be[br]one of the problems

1:56:53.970,1:56:56.900
to be emphasized in the exams.

1:56:56.900,1:57:02.408
Maybe one of the top 15 or[br]16 most important topics.

1:57:02.408,1:57:07.170


1:57:07.170,1:57:07.900
Is that OK?

1:57:07.900,1:57:09.396
Can I erase the whole thing?

1:57:09.396,1:57:09.896
OK.

1:57:09.896,1:57:11.393
Let me erase the whole thing.

1:57:11.393,1:57:44.326


1:57:44.326,1:57:44.826
OK.

1:57:44.826,1:57:45.824
Any other questions?

1:57:45.824,1:58:02.291


1:58:02.291,1:58:03.661
No?

1:58:03.661,1:58:05.285
I'm not going to let[br]you go right away,

1:58:05.285,1:58:07.780
we're going to work one[br]more problem or two more

1:58:07.780,1:58:08.778
simple problems.

1:58:08.778,1:58:10.773
And then we are going to go.

1:58:10.773,1:58:11.273
OK?

1:58:11.273,1:58:22.750


1:58:22.750,1:58:26.480
So question.

1:58:26.480,1:58:27.986
A question.

1:58:27.986,1:58:32.946


1:58:32.946,1:58:39.890
What do you think the[br]gradient is good at?

1:58:39.890,1:58:49.314


1:58:49.314,1:58:50.980
Two reasons, right.

1:58:50.980,1:58:54.340
Review number one.

1:58:54.340,1:58:59.120
If you have an increasingly[br]defined function,

1:58:59.120,1:59:02.860
then the gradient of F was what?

1:59:02.860,1:59:21.904
Equals direction of the[br]normal to the surface S--

1:59:21.904,1:59:26.395
let's say S is given[br]increasingly at the point

1:59:26.395,1:59:27.393
with [INAUDIBLE].

1:59:27.393,1:59:31.884


1:59:31.884,1:59:33.381
But any other reason?

1:59:33.381,2:00:00.327


2:00:00.327,2:00:01.824
Let's take that again.

2:00:01.824,2:00:05.816
Z equals x squared[br]plus y squared.

2:00:05.816,2:00:07.812
Let's compute a few[br]partial derivatives.

2:00:07.812,2:00:09.309
Let's compute the gradient.

2:00:09.309,2:00:21.120
The gradient is Fs of x, Fs[br]of y, where this is F of xy

2:00:21.120,2:00:24.888
or Fs of xi plus Fs of yj.

2:00:24.888,2:00:28.880


2:00:28.880,2:00:31.375
[INAUDIBLE]

2:00:31.375,2:00:34.369
And we drew it.

2:00:34.369,2:00:42.353
I drew this case, and we also[br]drew another related example,

2:00:42.353,2:00:45.846
where we took Z equals 1 minus[br]x squared minus y squared.

2:00:45.846,2:00:46.844
And we went skiing.

2:00:46.844,2:00:52.333
And we were so happy last week[br]to go skiing, because we still

2:00:52.333,2:00:57.650
had snow in New[br]Mexico, and we-- and we

2:00:57.650,2:01:02.546
said now we computed the[br]Z to be minus 2x minus 2y.

2:01:02.546,2:01:06.018


2:01:06.018,2:01:09.597
And we said, I'm[br]looking at the slopes.

2:01:09.597,2:01:12.936
This is the x duration[br]and the y duration.

2:01:12.936,2:01:18.670
And I'm looking at the slopes of[br]the lines of these two curves.

2:01:18.670,2:01:23.630
So one that goes[br]down, like that.

2:01:23.630,2:01:25.010
So this was for what?

2:01:25.010,2:01:27.540
For y equals 0.

2:01:27.540,2:01:32.190
And this was for x equals 0.

2:01:32.190,2:01:36.645


2:01:36.645,2:01:39.580
Curve, x equals[br]0 curve in plane.

2:01:39.580,2:01:40.360
Right?

2:01:40.360,2:01:42.742
We just cross-section[br]our surface,

2:01:42.742,2:01:43.950
and we have this [INAUDIBLE].

2:01:43.950,2:01:51.594
And then we have the two[br]tangents, two slopes.

2:01:51.594,2:01:54.064
And we computed them everywhere.

2:01:54.064,2:02:00.486


2:02:00.486,2:02:01.968
At every point.

2:02:01.968,2:02:06.910


2:02:06.910,2:02:10.845
But realize that to go[br]up or down these hills,

2:02:10.845,2:02:15.095
I can go on a curve[br]like that, or I

2:02:15.095,2:02:17.950
can go-- remember the[br]train of Mickey Mouse going

2:02:17.950,2:02:20.182
on the hilly point on the hill?

2:02:20.182,2:02:22.174
We try to take different paths.

2:02:22.174,2:02:24.166
We are going hiking.

2:02:24.166,2:02:28.648
We are going hiking, and we'll[br]take hiking through the pass.

2:02:28.648,2:02:38.608


2:02:38.608,2:02:41.098
OK.

2:02:41.098,2:03:01.420
How do we get the maximum[br]rate of change of the function

2:03:01.420,2:03:03.600
Z equals F of x1?

2:03:03.600,2:03:05.870
So now I'm[br]anticipating something.

2:03:05.870,2:03:10.680
I'd like to see your intuition,[br]your inborn sense of I

2:03:10.680,2:03:12.300
know what's going to happen.

2:03:12.300,2:03:14.092
And you know what[br]that from Mister--

2:03:14.092,2:03:14.842
STUDENT: Heinrich.

2:03:14.842,2:03:17.590
PROFESSOR TODA: [? Heinrich ?][br]from high school.

2:03:17.590,2:03:21.280
So I'm asking-- let me[br]rephrase the question

2:03:21.280,2:03:23.130
like a non-mathematician.

2:03:23.130,2:03:24.230
Let's go hiking.

2:03:24.230,2:03:30.274
This is [INAUDIBLE] we[br]go to the lighthouse.

2:03:30.274,2:03:33.790
Which path shall I take[br]on my mountain, my hill,

2:03:33.790,2:03:37.570
my god knows what[br]geography, in order

2:03:37.570,2:03:40.440
to obtain the maximum[br]rate of change?

2:03:40.440,2:03:43.515
That means the[br]highest derivative.

2:03:43.515,2:03:46.470
In what direction do I get[br]the highest derivative?

2:03:46.470,2:03:49.110
STUDENT: In what direction you[br]get the highest derivative--

2:03:49.110,2:03:50.735
PROFESSOR TODA: So[br]in which direction--

2:03:50.735,2:03:53.330
in which direction[br]on this hill do

2:03:53.330,2:03:55.098
I get the highest derivative?

2:03:55.098,2:03:57.014
The highest rate of change.

2:03:57.014,2:04:03.740
Rate of change means I want to[br]get the fastest possible way

2:04:03.740,2:04:04.850
somewhere.

2:04:04.850,2:04:08.230
STUDENT: The shortest slope?

2:04:08.230,2:04:09.857
Along just the straight line up.

2:04:09.857,2:04:10.831
PROFESSOR TODA: Along--

2:04:10.831,2:04:12.292
STUDENT: You don't want[br]to take any [INAUDIBLE].

2:04:12.292,2:04:13.270
PROFESSOR TODA: Right.

2:04:13.270,2:04:13.910
STUDENT: [INAUDIBLE].

2:04:13.910,2:04:15.140
It could be along any axis.

2:04:15.140,2:04:17.710
PROFESSOR TODA: So could[br]you see which direction

2:04:17.710,2:04:19.070
those are-- very good.

2:04:19.070,2:04:21.230
Actually you were getting[br]to the same direction.

2:04:21.230,2:04:24.370
So [INAUDIBLE] says[br]Magdalena, don't be silly.

2:04:24.370,2:04:28.295
The actual maximum rate of[br]change for the function Z

2:04:28.295,2:04:31.070
is obviously, because[br]it is common sense,

2:04:31.070,2:04:36.630
it's obviously happening if[br]you take the so-called-- what

2:04:36.630,2:04:37.880
are these guys?

2:04:37.880,2:04:40.810
[INAUDIBLE], not meridians.

2:04:40.810,2:04:42.280
STUDENT: Longtitudes?

2:04:42.280,2:04:43.260
PROFESSOR TODA: OK.

2:04:43.260,2:04:44.730
That is-- OK.

2:04:44.730,2:04:47.810
Suppose that we don't hike,[br]because it's too tiring.

2:04:47.810,2:04:51.170
We go down from the[br]top of the hill.

2:04:51.170,2:04:53.310
Ah, there's also very good idea.

2:04:53.310,2:04:58.870
So when you let yourself[br]go down on a sleigh,

2:04:58.870,2:05:02.560
don't think bobsled or[br]anything-- just a sleigh,

2:05:02.560,2:05:04.110
think of a child's sleigh.

2:05:04.110,2:05:07.680
No, take a plastic bag[br]and put your butt in it

2:05:07.680,2:05:10.590
and let yourself go.

2:05:10.590,2:05:14.140
What is their[br]direction actually?

2:05:14.140,2:05:19.925
Your body will find the[br]fastest way to get down.

2:05:19.925,2:05:23.065
The fastest way to get[br]down will happen exactly

2:05:23.065,2:05:27.710
in the same[br]directions going down

2:05:27.710,2:05:29.600
in the directions[br]of these meridians.

2:05:29.600,2:05:34.100


2:05:34.100,2:05:35.512
OK?

2:05:35.512,2:05:37.000
And now, [INAUDIBLE].

2:05:37.000,2:05:46.424


2:05:46.424,2:05:58.576
The maximum rate of[br]change will always

2:05:58.576,2:06:07.256
happen in the direction[br]of the gradient.

2:06:07.256,2:06:14.696


2:06:14.696,2:06:18.950
You can get a little[br]bit ahead of time

2:06:18.950,2:06:21.860
by just-- I would like this[br]to [INAUDIBLE] in your heads

2:06:21.860,2:06:23.860
until we get to that section.

2:06:23.860,2:06:26.880
In one section we will be there.

2:06:26.880,2:06:40.430
We also-- it's also reformulated[br]as the highest, the steepest,

2:06:40.430,2:06:42.079
ascent or descent.

2:06:42.079,2:06:44.574
The steepest.

2:06:44.574,2:06:58.546
The steepest ascent or[br]the steepest descent

2:06:58.546,2:07:09.524
always happens in the[br]direction of the gradient.

2:07:09.524,2:07:14.550


2:07:14.550,2:07:17.110
Ascent is when you hike[br]to the top of the hill.

2:07:17.110,2:07:21.450
Descent is when you let yourself[br]go in the plastic [INAUDIBLE]

2:07:21.450,2:07:25.270
bag in the snow.

2:07:25.270,2:07:26.080
Right?

2:07:26.080,2:07:30.030
Can you verify this happens[br]just on this example?

2:07:30.030,2:07:32.540
It's true in general,[br]for any smooth function.

2:07:32.540,2:07:36.010
Our smooth function is[br]a really nice function.

2:07:36.010,2:07:39.816
So what is the gradient?

2:07:39.816,2:07:42.720
Well again, it was 2x 2y, right?

2:07:42.720,2:07:45.850


2:07:45.850,2:07:50.508
And that means at a certain[br]point, x0 y0, whenever you are,

2:07:50.508,2:07:52.300
guys you don't[br]necessarily have to start

2:07:52.300,2:07:54.750
from the top of the hill.

2:07:54.750,2:07:58.870
You can be-- OK,[br]this is your cabin.

2:07:58.870,2:08:01.970
And here you are with[br]friends, or with mom and dad,

2:08:01.970,2:08:05.110
or whoever, on the hill.

2:08:05.110,2:08:09.030
You get out, you take the[br]sleigh, and you go down.

2:08:09.030,2:08:14.320
So no matter where[br]you are, there you go.

2:08:14.320,2:08:22.933
You have 2x0 times[br]i plus 2y0 times j.

2:08:22.933,2:08:31.640
And the direction of the[br]gradient will be 2x0 2y0.

2:08:31.640,2:08:34.520
Do you like this one?

2:08:34.520,2:08:39.240
Well in this case,[br]if you were-- suppose

2:08:39.240,2:08:42.300
you were at the[br]point [INAUDIBLE].

2:08:42.300,2:08:49.230


2:08:49.230,2:08:53.596
You are at the point[br]of coordinates--

2:08:53.596,2:08:55.130
do you want to be here?

2:08:55.130,2:08:57.000
You want to be here, right?

2:08:57.000,2:08:58.770
So we've done that before.

2:08:58.770,2:09:02.560
I'll take it as 1 over[br][? square root of ?]

2:09:02.560,2:09:09.620
2-- I'm trying to be creative[br]today-- [INAUDIBLE] y equals 0,

2:09:09.620,2:09:14.580
and Z equals-- what's left?

2:09:14.580,2:09:16.460
1/2, right?

2:09:16.460,2:09:17.750
Where am I?

2:09:17.750,2:09:20.390
Guys, do you realize where I am?

2:09:20.390,2:09:21.640
I'll [? take a ?] [INAUDIBLE].

2:09:21.640,2:09:24.340


2:09:24.340,2:09:25.140
y0.

2:09:25.140,2:09:28.656
So I need to be on this[br]meridian on the red thingy.

2:09:28.656,2:09:33.920


2:09:33.920,2:09:37.356
And somewhere here.

2:09:37.356,2:09:40.250


2:09:40.250,2:09:43.385
What's the duration[br]of the gradient here?

2:09:43.385,2:09:45.830
Delta z at this p.

2:09:45.830,2:09:56.610


2:09:56.610,2:09:58.760
Then you say ah,[br]well, I don't get it.

2:09:58.760,2:10:04.040
I have-- the second guy will[br]become 0, because y0 is 0.

2:10:04.040,2:10:06.980
The first guy will become[br]1 over square root of 2.

2:10:06.980,2:10:15.280
So I have 2 times 1 over square[br]root of 2 times i plus 0j.

2:10:15.280,2:10:29.416
It means in the direction of i--[br]in the direction of i-- from p,

2:10:29.416,2:10:39.201
I have the fastest-- fastest,[br]Magdalena, fastest-- descent

2:10:39.201,2:10:39.700
possible.

2:10:39.700,2:10:43.010


2:10:43.010,2:10:46.730
But we don't say in[br]the direction of i

2:10:46.730,2:10:49.640
in our everyday life, right?

2:10:49.640,2:10:53.270
Let's say geographic points.

2:10:53.270,2:10:58.610
We are-- I'm a bug,[br]and this is north.

2:10:58.610,2:11:00.086
This is south.

2:11:00.086,2:11:05.006


2:11:05.006,2:11:05.990
This is east.

2:11:05.990,2:11:08.960


2:11:08.960,2:11:11.470
And this is west.

2:11:11.470,2:11:18.140
So if I go east, going east[br]means going in the direction i.

2:11:18.140,2:11:23.040


2:11:23.040,2:11:25.510
Now suppose-- I'm going[br]to finish with this one.

2:11:25.510,2:11:28.870
Suppose that my house[br]is not on the prairie

2:11:28.870,2:11:31.710
but my house is here.

2:11:31.710,2:11:34.400
House, h.

2:11:34.400,2:11:37.834
Find me a wood[br]point to be there.

2:11:37.834,2:11:39.738
STUDENT: Northeast.

2:11:39.738,2:11:41.170
Or to get further down.

2:11:41.170,2:11:45.047
PROFESSOR TODA: Anything, what[br]would look like why I'm here?

2:11:45.047,2:11:48.041
x0, y0, z0.

2:11:48.041,2:11:50.040
Hm.

2:11:50.040,2:11:57.800
1/2, 1/2, and I[br]need the minimum.

2:11:57.800,2:12:02.580
So I want to be on the[br]bisecting plane between the two.

2:12:02.580,2:12:03.420
You understand?

2:12:03.420,2:12:04.400
This is my quarter.

2:12:04.400,2:12:06.870
And I want to be in[br]this bisecting plane.

2:12:06.870,2:12:10.420
So I'll take 1/2, 1/2, and[br]what results from here?

2:12:10.420,2:12:11.540
I have to do math.

2:12:11.540,2:12:16.110
1 minus 1/4 minus 1/4 is 1/2.

2:12:16.110,2:12:17.902
Right?

2:12:17.902,2:12:19.740
1/2, 1/2, 1/2.

2:12:19.740,2:12:22.440
This is where my house[br]is [? and so on. ?]

2:12:22.440,2:12:24.106
And this is full of smoke.

2:12:24.106,2:12:29.940
And what is the[br]maximum rate of change?

2:12:29.940,2:12:34.500
What is the steepest[br]descent is the trajectory

2:12:34.500,2:12:37.920
that my body will take[br]when I let myself go down

2:12:37.920,2:12:39.402
on the sleigh.

2:12:39.402,2:12:40.884
How do I compute that?

2:12:40.884,2:12:43.624
I will just do the same thing.

2:12:43.624,2:12:49.900
Delta z at the point x0[br]equals 1/2, y0 equals 1/2,

2:12:49.900,2:12:52.100
z0 equals 1/2.

2:12:52.100,2:12:54.060
Well what do I get as direction?

2:12:54.060,2:12:57.490
That will be the[br]direction of the gradient.

2:12:57.490,2:13:02.920
2 times 1/2-- you[br]guys with me still?

2:13:02.920,2:13:09.240
i plus 2 times 1/2 with j.

2:13:09.240,2:13:14.311
And there is no Mr.[br]z0 In the picture.

2:13:14.311,2:13:14.810
Why?

2:13:14.810,2:13:17.100
Because that will[br]give me the direction

2:13:17.100,2:13:22.000
like on-- in a geographic way.

2:13:22.000,2:13:24.420
North, west, east, south.

2:13:24.420,2:13:26.490
These are the[br]direction in plane.

2:13:26.490,2:13:28.120
I'm not talking[br]directions on the hill,

2:13:28.120,2:13:31.496
I'm talking[br]directions on the map.

2:13:31.496,2:13:33.530
These are directions on the map.

2:13:33.530,2:13:35.930
So what is the direction[br]i plus j on the map?

2:13:35.930,2:13:39.820
If you show this to a[br]geography major and say,

2:13:39.820,2:13:43.340
I'm going in the direction[br]i plus j on the map,

2:13:43.340,2:13:45.700
he will say you are crazy.

2:13:45.700,2:13:47.980
He doesn't understand the thing.

2:13:47.980,2:13:50.280
But you know what you mean.

2:13:50.280,2:13:54.100
East for you is the[br]direction of i in the x-axis.

2:13:54.100,2:13:56.210
[INAUDIBLE]

2:13:56.210,2:13:58.430
And this is north.

2:13:58.430,2:13:59.630
Are you guys with me?

2:13:59.630,2:14:01.790
The y direction is north.

2:14:01.790,2:14:06.410
So I'm going perfectly[br]northeast at a 45-degree angle.

2:14:06.410,2:14:08.093
If I tell the[br]geography major I'm

2:14:08.093,2:14:10.725
going northeast perfectly in[br]the middle, he will say I know.

2:14:10.725,2:14:13.560
But you will know that[br]for you, that is i plus j.

2:14:13.560,2:14:15.649
Because you are[br]the mathematician.

2:14:15.649,2:14:17.100
Right?

2:14:17.100,2:14:18.980
So you go down.

2:14:18.980,2:14:20.780
And this is where you are.

2:14:20.780,2:14:22.500
And you're on the meridian.

2:14:22.500,2:14:25.090
This is the direction i plus j.

2:14:25.090,2:14:29.610
So if I want to project my[br]trajectory-- I went down

2:14:29.610,2:14:33.260
with the sleigh, all the way[br]down-- project the trajectory,

2:14:33.260,2:14:36.810
my trajectory is a[br]body on the snow.

2:14:36.810,2:14:39.320
Projecting it on the[br]ground is this one.

2:14:39.320,2:14:43.800
So it is exactly the[br]direction i plus j.

2:14:43.800,2:14:44.320
Right, guys?

2:14:44.320,2:14:48.170
So exactly northeast[br]perfectly at 45-degree angles.

2:14:48.170,2:14:51.150
Now one caveat.

2:14:51.150,2:14:53.455
One caveat, because[br]when we get there,

2:14:53.455,2:14:59.060
you should be ready[br]already, in 11.6 and 11.7.

2:14:59.060,2:15:02.930
When we will say direction,[br]we are also crazy people.

2:15:02.930,2:15:04.920
I told you, mathematicians[br]are not normal.

2:15:04.920,2:15:07.108
You have to be a[br]little bit crazy

2:15:07.108,2:15:11.460
to want to do all the stuff[br]in your head like that.

2:15:11.460,2:15:16.420
i plus j for us is not a[br]direction most of the time.

2:15:16.420,2:15:20.015
When we say direction, we mean[br]we normalize that direction.

2:15:20.015,2:15:23.055
We take the unit[br]vector, which is unique,

2:15:23.055,2:15:25.940
for responding to i plus j.

2:15:25.940,2:15:28.890
So what is that[br]unique unit vector?

2:15:28.890,2:15:32.510
You learned in Chapter 9[br]everything is connected.

2:15:32.510,2:15:33.890
It's a big circle.

2:15:33.890,2:15:35.340
i plus j, very good.

2:15:35.340,2:15:40.190
So direction is a unit vector[br]for most mathematicians,

2:15:40.190,2:15:45.390
which means you will be i[br]plus j over square root of 2.

2:15:45.390,2:15:51.966
So in Chapter 5, please[br]remember, unlike Chapter 9,

2:15:51.966,2:15:55.613
direction is a unit vector.

2:15:55.613,2:15:59.609
In Chapter 9, Chapter 10,[br]it said direction lmn,

2:15:59.609,2:16:00.650
direction god knows what.

2:16:00.650,2:16:05.900
But in Chapter 11, direction[br]is a vector in plane,

2:16:05.900,2:16:07.860
like this one, i[br]plus [INAUDIBLE]

2:16:07.860,2:16:12.015
has to be a unique[br]normal-- a unique vector.

2:16:12.015,2:16:12.514
OK?

2:16:12.514,2:16:14.318
And we-- keep that in mind.

2:16:14.318,2:16:16.050
Next time, when we[br]meet on Thursday,

2:16:16.050,2:16:19.956
you will understand why[br]we need to normalize it.

2:16:19.956,2:16:23.220
Now can we say goodbye to[br]the snow and everything?

2:16:23.220,2:16:25.810
It's not going to[br]show up much anymore.

2:16:25.810,2:16:27.555
Remember this example.

2:16:27.555,2:16:30.660
But we will start with[br]flowers next time.

2:16:30.660,2:16:31.260
OK.

2:16:31.260,2:16:32.760
Have a nice day.

2:16:32.760,2:16:33.959
Yes, sir?

2:16:33.959,2:16:36.410
Let me stop the video.

2:16:36.410,2:16:37.245