1
00:00:00,000 --> 00:00:02,150
PROFESSOR TODA: Any
questions so far?

2
00:00:02,150 --> 00:00:06,450
I mean, conceptual,
theoretical questions first,

3
00:00:06,450 --> 00:00:08,754
and then we will
do the second part

4
00:00:08,754 --> 00:00:10,185
of [INAUDIBLE] applications.

5
00:00:10,185 --> 00:00:14,478
Then you can ask
for more questions.

6
00:00:14,478 --> 00:00:15,909
No questions so far?

7
00:00:15,909 --> 00:00:18,676
I have not finished 11-4.

8
00:00:18,676 --> 00:00:25,745
I still owe you a long
explanation about 11-4.

9
00:00:25,745 --> 00:00:28,270
Hopefully it's going to
make more sense today

10
00:00:28,270 --> 00:00:30,850
than it made last time.

11
00:00:30,850 --> 00:00:34,220
I was just saying
that I'm doing 11-4.

12
00:00:34,220 --> 00:00:36,350
This is a lot of chapter.

13
00:00:36,350 --> 00:00:47,958
So second part of 11-4 today--
tangent plane and applications.

14
00:00:47,958 --> 00:00:50,870


15
00:00:50,870 --> 00:00:53,739
Now, we don't say what
those applications are

16
00:00:53,739 --> 00:00:58,970
from the start, but these are
some very important concepts

17
00:00:58,970 --> 00:01:00,838
called the total differential.

18
00:01:00,838 --> 00:01:07,156


19
00:01:07,156 --> 00:01:13,960
And the linear
approximation number

20
00:01:13,960 --> 00:01:15,495
is going under the [INAUDIBLE].

21
00:01:15,495 --> 00:01:17,180
Thank you, sir.

22
00:01:17,180 --> 00:01:23,880
Linear approximation for
functions of the type z

23
00:01:23,880 --> 00:01:29,480
equals f of xy, which means
graphs of two variables.

24
00:01:29,480 --> 00:01:33,700
At the end of the chapter, I'll
take the notes copy from you.

25
00:01:33,700 --> 00:01:37,130
So don't give me
anything until it's over.

26
00:01:37,130 --> 00:01:38,910
When is that going to be over?

27
00:01:38,910 --> 00:01:41,862
We have four more
sections to go.

28
00:01:41,862 --> 00:01:47,140
So I guess right before
spring break you give me

29
00:01:47,140 --> 00:01:50,330
the notes for chapter 11.

30
00:01:50,330 --> 00:01:52,470
All right, and then
I'm thinking of making

31
00:01:52,470 --> 00:01:54,894
copies of both chapters.

32
00:01:54,894 --> 00:02:00,260
You get the-- I'm
distributing them to you.

33
00:02:00,260 --> 00:02:02,660
I haven't started
and yet go ahead.

34
00:02:02,660 --> 00:02:09,600
Could anybody tell
me what the equation

35
00:02:09,600 --> 00:02:13,520
that we used last time--
we proved it, actually.

36
00:02:13,520 --> 00:02:16,260


37
00:02:16,260 --> 00:02:20,982
What is the equation
of the tangent plane

38
00:02:20,982 --> 00:02:27,266
to a smooth surface or a patch
of a surface at the point

39
00:02:27,266 --> 00:02:33,784
m of coordinates x0, y0,
z0, where the graph is

40
00:02:33,784 --> 00:02:37,480
given by z equals f of x and y.

41
00:02:37,480 --> 00:02:40,860
I'm going to label it on
the patch of a surface.

42
00:02:40,860 --> 00:02:44,030
OK, imagine it
labeled brown there.

43
00:02:44,030 --> 00:02:52,540
And can somebody tell me the
equation of the other plane?

44
00:02:52,540 --> 00:02:54,450
But because you
have better memory,

45
00:02:54,450 --> 00:03:00,300
being much younger, about 25
years younger than me or so.

46
00:03:00,300 --> 00:03:05,650
So could you-- could anybody
tell me what the tangent

47
00:03:05,650 --> 00:03:08,620
planes equation-- I'll start.

48
00:03:08,620 --> 00:03:10,292
And it's going to come to you.

49
00:03:10,292 --> 00:03:14,880
z minus z0 equals.

50
00:03:14,880 --> 00:03:15,943
And now let's see.

51
00:03:15,943 --> 00:03:18,382
I'll pick a nice color.

52
00:03:18,382 --> 00:03:19,105
I'll wait.

53
00:03:19,105 --> 00:03:21,845


54
00:03:21,845 --> 00:03:23,650
STUDENT: fx of x.

55
00:03:23,650 --> 00:03:27,220
PROFESSOR TODA: f sub x, the
partial derivative measured

56
00:03:27,220 --> 00:03:35,780
at f0 i0 times the
quantity x minus x0 plus--

57
00:03:35,780 --> 00:03:36,980
STUDENT: f sub y.

58
00:03:36,980 --> 00:03:38,730
PROFESSOR TODA: f
sub y, excellent.

59
00:03:38,730 --> 00:03:40,676
f sub y.

60
00:03:40,676 --> 00:03:41,924
STUDENT: x0, y0.

61
00:03:41,924 --> 00:03:44,930
PROFESSOR TODA: x0,
y0 times y minus y0.

62
00:03:44,930 --> 00:03:49,259


63
00:03:49,259 --> 00:03:50,221
OK.

64
00:03:50,221 --> 00:03:51,630
All right.

65
00:03:51,630 --> 00:03:59,110
Now thinking of what those
quantities mean, x minus x0, y

66
00:03:59,110 --> 00:04:03,730
minus y0, z minus
z0, what are they?

67
00:04:03,730 --> 00:04:06,830
They are small
displacements, aren't they?

68
00:04:06,830 --> 00:04:10,380
I mean, what does it
mean small displacement?

69
00:04:10,380 --> 00:04:20,860
Imagine that you are near
the point on both surfaces.

70
00:04:20,860 --> 00:04:23,500
So what is a small
neighborhood--

71
00:04:23,500 --> 00:04:27,940
what's a typical small
neighborhood [INAUDIBLE]?

72
00:04:27,940 --> 00:04:30,280
It's a disk, right?

73
00:04:30,280 --> 00:04:32,672
There are many kinds of
neighborhoods, but one of them,

74
00:04:32,672 --> 00:04:36,956
I'd say, would be
this open disk, OK?

75
00:04:36,956 --> 00:04:38,860
I'll draw that.

76
00:04:38,860 --> 00:04:44,712
Now, if I have a
red point-- I don't

77
00:04:44,712 --> 00:04:53,190
know how to do that pink point--
somewhere nearby in planes--

78
00:04:53,190 --> 00:04:54,550
this is the plane.

79
00:04:54,550 --> 00:04:59,356
In plane, I have this
point that is close.

80
00:04:59,356 --> 00:05:01,030
And that point is xyz.

81
00:05:01,030 --> 00:05:04,350


82
00:05:04,350 --> 00:05:08,750
And you think, OK, can
I visualize that better?

83
00:05:08,750 --> 00:05:11,790
Well, guys, it's hard to
visualize that better.

84
00:05:11,790 --> 00:05:14,692
But I'll draw a triangle
[? doing ?] a better job.

85
00:05:14,692 --> 00:05:17,437


86
00:05:17,437 --> 00:05:18,145
That's the frame.

87
00:05:18,145 --> 00:05:22,404


88
00:05:22,404 --> 00:05:24,869
This is a surface.

89
00:05:24,869 --> 00:05:27,450
Imagine it's a surface, OK?

90
00:05:27,450 --> 00:05:32,060
That's the point of x0, y0.

91
00:05:32,060 --> 00:05:34,720
[? It's ?] the 0 and that.

92
00:05:34,720 --> 00:05:36,690
Where is the point xyz again?

93
00:05:36,690 --> 00:05:40,640
The point xyz is not
on the pink stuff.

94
00:05:40,640 --> 00:05:41,790
This is a pink surface.

95
00:05:41,790 --> 00:05:45,330
It looks like Pepto
Bismol or something.

96
00:05:45,330 --> 00:05:46,310
You shaded it.

97
00:05:46,310 --> 00:05:47,065
No.

98
00:05:47,065 --> 00:05:48,230
That's not what I want.

99
00:05:48,230 --> 00:05:55,560
I want the close enough
point on the blue plane.

100
00:05:55,560 --> 00:06:01,180
It's actually in the blue plane
pie and this guy would be xyz.

101
00:06:01,180 --> 00:06:05,190
So now say, OK, how
far I x be from x0?

102
00:06:05,190 --> 00:06:06,090
Well, I don't know.

103
00:06:06,090 --> 00:06:13,510
We would have to check
the points, the set 0,

104
00:06:13,510 --> 00:06:15,948
check the blue point.

105
00:06:15,948 --> 00:06:18,470
This is x.

106
00:06:18,470 --> 00:06:23,940
So between x and x0, I
have this difference,

107
00:06:23,940 --> 00:06:34,070
which is delta x displacement,
displacement along the x-axis,

108
00:06:34,070 --> 00:06:38,915
away from the
point, fixed point.

109
00:06:38,915 --> 00:06:41,825


110
00:06:41,825 --> 00:06:44,735
This is the fixed
point, this point.

111
00:06:44,735 --> 00:06:47,160
This point is p.

112
00:06:47,160 --> 00:06:48,150
OK.

113
00:06:48,150 --> 00:06:51,233
y minus y0, let's call
that delta y, which

114
00:06:51,233 --> 00:06:53,426
is the displacement
along the y-axis.

115
00:06:53,426 --> 00:06:56,425


116
00:06:56,425 --> 00:07:02,020
And then the z minus z0 can be.

117
00:07:02,020 --> 00:07:05,700
Just because I'm a mathematician
and I don't like writing down

118
00:07:05,700 --> 00:07:11,280
a lot, I would use
s batch as I can,

119
00:07:11,280 --> 00:07:16,780
compact symbols, to
speed up my computation.

120
00:07:16,780 --> 00:07:19,200
So I can rewrite
this whole thing

121
00:07:19,200 --> 00:07:27,560
as a delta z equals f sub
x, x0 y0, which is a number.

122
00:07:27,560 --> 00:07:28,520
It's a slope.

123
00:07:28,520 --> 00:07:31,570
We discussed about
that last time.

124
00:07:31,570 --> 00:07:33,875
We even went skiing
last time, when

125
00:07:33,875 --> 00:07:38,290
we said that's like the slope
in-- what's the x direction?

126
00:07:38,290 --> 00:07:41,970
Slope in the x direction
and slope in the y direction

127
00:07:41,970 --> 00:07:49,000
on the graph that was the
white covered with snow hill.

128
00:07:49,000 --> 00:07:50,886
That was what we had last time.

129
00:07:50,886 --> 00:07:54,546
Delta x plus f sub
0, another slope

130
00:07:54,546 --> 00:07:56,742
in the y direction, delta y.

131
00:07:56,742 --> 00:08:01,622


132
00:08:01,622 --> 00:08:07,890
And fortunately-- OK, the book
is a very good book, obviously,

133
00:08:07,890 --> 00:08:09,250
right?

134
00:08:09,250 --> 00:08:15,630
But I wish we could've done
certain things better in terms

135
00:08:15,630 --> 00:08:21,800
of comparisons between
this notion in Calc III

136
00:08:21,800 --> 00:08:27,260
and some corresponding
notion in Calc I.

137
00:08:27,260 --> 00:08:29,610
So you're probably
thinking, what the heck

138
00:08:29,610 --> 00:08:31,240
is this witch thinking about?

139
00:08:31,240 --> 00:08:34,590
Well, I'm thinking
of something that you

140
00:08:34,590 --> 00:08:39,730
may want to remember
from Calc I.

141
00:08:39,730 --> 00:08:42,880
And that's going to come
into place beautifully

142
00:08:42,880 --> 00:08:47,810
right now because you have the
Calc I, Calc III comparison.

143
00:08:47,810 --> 00:08:52,799
And that's why it would be
great-- the books don't even

144
00:08:52,799 --> 00:08:55,270
talk about this comparison.

145
00:08:55,270 --> 00:08:59,810
In Calc I, I reminded
you about Mr. Leibniz.

146
00:08:59,810 --> 00:09:01,110
He was a very nice guy.

147
00:09:01,110 --> 00:09:02,570
I have no idea, right?

148
00:09:02,570 --> 00:09:04,130
Never met him.

149
00:09:04,130 --> 00:09:06,660
One of the fathers of calculus.

150
00:09:06,660 --> 00:09:10,630
And he introduced the
so-called Leibniz notation.

151
00:09:10,630 --> 00:09:15,720
And one of you in office
hours last Wednesday

152
00:09:15,720 --> 00:09:19,280
told me, so the
Leibnitz notation

153
00:09:19,280 --> 00:09:23,455
for a function g of
x-- I'm intentionally

154
00:09:23,455 --> 00:09:26,273
changing notation-- is what?

155
00:09:26,273 --> 00:09:31,627
Well, this is just
the derivative

156
00:09:31,627 --> 00:09:34,000
which is the limit of
the different quotients

157
00:09:34,000 --> 00:09:38,480
of your delta g over
delta x-- as done by some

158
00:09:38,480 --> 00:09:43,180
blutches-- 0, right, which
would be the same as lim

159
00:09:43,180 --> 00:09:50,840
of g of x minus g of x0 over
x minus x0 as x approaches x0,

160
00:09:50,840 --> 00:09:52,070
right?

161
00:09:52,070 --> 00:09:52,570
Right.

162
00:09:52,570 --> 00:09:57,720
So we've done that in Calc I.
But it was a long time ago.

163
00:09:57,720 --> 00:10:00,630
My mission is to teach
you all Calc III,

164
00:10:00,630 --> 00:10:03,835
but I feel that
my mission is also

165
00:10:03,835 --> 00:10:08,635
to teach you what you may not
remember very well from Calc I,

166
00:10:08,635 --> 00:10:11,640
because everything is related.

167
00:10:11,640 --> 00:10:17,690
So what was the way we
could have written this,

168
00:10:17,690 --> 00:10:21,260
not delta g over delta
x equals g prime.

169
00:10:21,260 --> 00:10:22,486
No.

170
00:10:22,486 --> 00:10:29,040
But it's an approximation of
g prime around a very small

171
00:10:29,040 --> 00:10:33,745
[INAUDIBLE], very close to x0.

172
00:10:33,745 --> 00:10:36,550


173
00:10:36,550 --> 00:10:39,676
So if you wanted to
rewrite this approximation,

174
00:10:39,676 --> 00:10:42,091
how would you have rewritten it?

175
00:10:42,091 --> 00:10:47,410


176
00:10:47,410 --> 00:10:48,140
Delta g--

177
00:10:48,140 --> 00:10:54,866


178
00:10:54,866 --> 00:10:57,310
STUDENT: g prime sub x.

179
00:10:57,310 --> 00:11:02,490
PROFESSOR TODA: g prime
of x0 times delta x.

180
00:11:02,490 --> 00:11:03,930
OK?

181
00:11:03,930 --> 00:11:08,280
Now, why this approximation?

182
00:11:08,280 --> 00:11:11,710
What if I had put equal?

183
00:11:11,710 --> 00:11:14,120
If I had put equal, it
would be all nonsense.

184
00:11:14,120 --> 00:11:15,405
Why?

185
00:11:15,405 --> 00:11:19,210
Well, say, Magdalena, if you
put equal, it's another object.

186
00:11:19,210 --> 00:11:19,770
What object?

187
00:11:19,770 --> 00:11:20,330
OK.

188
00:11:20,330 --> 00:11:22,120
Let's look at the objects.

189
00:11:22,120 --> 00:11:22,995
Let's draw a picture.

190
00:11:22,995 --> 00:11:25,730


191
00:11:25,730 --> 00:11:27,176
This is g.

192
00:11:27,176 --> 00:11:28,622
This is x0.

193
00:11:28,622 --> 00:11:30,560
This is g of x.

194
00:11:30,560 --> 00:11:32,440
What's g prime?

195
00:11:32,440 --> 00:11:39,420
g prime-- thank god-- is the
slope of g prime x0 over here.

196
00:11:39,420 --> 00:11:46,610
So if I want to write the
line, the line is exactly this.

197
00:11:46,610 --> 00:11:50,170
The red object is the line.

198
00:11:50,170 --> 00:11:52,770
So what is the red object again?

199
00:11:52,770 --> 00:11:58,350
It's y minus y over x
minus x0 equals m, which

200
00:11:58,350 --> 00:12:00,290
is g prime number 0.

201
00:12:00,290 --> 00:12:01,750
m is the slope.

202
00:12:01,750 --> 00:12:05,210
That's the point slope
formula, thank you very much.

203
00:12:05,210 --> 00:12:06,770
So the red object is this.

204
00:12:06,770 --> 00:12:08,990
This is the line.

205
00:12:08,990 --> 00:12:10,770
Attention is not the same.

206
00:12:10,770 --> 00:12:15,625
The blue thing is my
curve, more precisely

207
00:12:15,625 --> 00:12:17,600
a tiny portion of my curve.

208
00:12:17,600 --> 00:12:21,610
This neighborhood around the
point is what I have here.

209
00:12:21,610 --> 00:12:22,805
What I'm actually-- what?

210
00:12:22,805 --> 00:12:26,010


211
00:12:26,010 --> 00:12:29,510
I'm trying to
approximate my curve

212
00:12:29,510 --> 00:12:32,310
function with a little line.

213
00:12:32,310 --> 00:12:36,420
And I say, I would rather
approximate with a red line

214
00:12:36,420 --> 00:12:38,582
because this is the
best approximation

215
00:12:38,582 --> 00:12:44,200
to the blue arc of a curve
which is on the curve, right?

216
00:12:44,200 --> 00:12:46,985
So this is what it is
is just an approximation

217
00:12:46,985 --> 00:12:54,620
of a curve, approximation of
a curve of an arc of a curve.

218
00:12:54,620 --> 00:12:57,590
But Magdalena's lazy
today-- approximation

219
00:12:57,590 --> 00:13:03,550
of an arc of a curve
with a segment of a line,

220
00:13:03,550 --> 00:13:07,102
with a segment of
the tangent line

221
00:13:07,102 --> 00:13:10,735
of the tangent [INAUDIBLE].

222
00:13:10,735 --> 00:13:13,360
How do we call
such a phenomenon?

223
00:13:13,360 --> 00:13:17,650
An approximation of
an arc of a circle

224
00:13:17,650 --> 00:13:23,115
with a little segment
of a tangent line

225
00:13:23,115 --> 00:13:26,040
is like a discretization, right?

226
00:13:26,040 --> 00:13:29,416
But we call it
linear approximation.

227
00:13:29,416 --> 00:13:32,460
It's called a linear
approximation.

228
00:13:32,460 --> 00:13:36,590


229
00:13:36,590 --> 00:13:40,220
A-P-P, approx.

230
00:13:40,220 --> 00:13:42,460
Have you ever seen a
linear approximation

231
00:13:42,460 --> 00:13:46,880
before coming from Calc II?

232
00:13:46,880 --> 00:13:49,700
Well, in Calc II you've
seen the Taylor's formula.

233
00:13:49,700 --> 00:13:51,510
What is the Taylor's formula?

234
00:13:51,510 --> 00:13:55,246
It's a beautiful
thing that said what?

235
00:13:55,246 --> 00:13:55,990
I don't know.

236
00:13:55,990 --> 00:13:56,990
Let's remember together.

237
00:13:56,990 --> 00:14:00,206
So relationship
with Calc II, I'm

238
00:14:00,206 --> 00:14:04,670
going to go and make an arrow--
relationship with Calc II,

239
00:14:04,670 --> 00:14:08,160
because everything
is actually related.

240
00:14:08,160 --> 00:14:13,750
In Calc II-- how did we
introduce Taylor's formula?

241
00:14:13,750 --> 00:14:16,930
Well, instead of little a that
you're so used to in Calc II,

242
00:14:16,930 --> 00:14:21,170
we are going to put x0
is the same thing, right?

243
00:14:21,170 --> 00:14:23,550
So what was Taylor's
formula saying?

244
00:14:23,550 --> 00:14:28,150
You have this kind of
smooth, beautiful curve.

245
00:14:28,150 --> 00:14:30,860
But being smooth is not enough.

246
00:14:30,860 --> 00:14:33,860
You have that real analytic.

247
00:14:33,860 --> 00:14:36,070
Real analytic means
that the function can be

248
00:14:36,070 --> 00:14:41,100
expanded in Taylor's formula.

249
00:14:41,100 --> 00:14:42,250
So what does it mean?

250
00:14:42,250 --> 00:14:53,000
It means that we have f of x
prime is f of x0 equals-- or g.

251
00:14:53,000 --> 00:14:54,920
You want-- it doesn't matter.

252
00:14:54,920 --> 00:15:01,160
f prime of x0 times
x minus x0 plus

253
00:15:01,160 --> 00:15:06,010
dot, dot, dot, dot something
that I'm going to put.

254
00:15:06,010 --> 00:15:09,300
This is [? O. ?] It's a small
quantity that's maybe not

255
00:15:09,300 --> 00:15:12,898
so small, but I declare
it to be negligible.

256
00:15:12,898 --> 00:15:14,690
And so they're going
to be negligible.

257
00:15:14,690 --> 00:15:18,920
I have to make a face,
a smiley face and eyes,

258
00:15:18,920 --> 00:15:23,530
meaning that it's OK to
neglect the second order

259
00:15:23,530 --> 00:15:25,420
term, the third order term.

260
00:15:25,420 --> 00:15:28,370
So what happens, that
little h, when I square it,

261
00:15:28,370 --> 00:15:29,336
say the heck with it.

262
00:15:29,336 --> 00:15:30,800
It's going to be very small.

263
00:15:30,800 --> 00:15:36,700
Like if h is 0.1 and then
h squared will be 0.0001.

264
00:15:36,700 --> 00:15:40,445
And I have a certain range
of error that I allow,

265
00:15:40,445 --> 00:15:41,540
a threshold.

266
00:15:41,540 --> 00:15:43,470
I say that's negligible.

267
00:15:43,470 --> 00:15:47,430
If h squared and h cubed and h
to the fourth are negligible,

268
00:15:47,430 --> 00:15:49,930
then I'm fine.

269
00:15:49,930 --> 00:15:53,440
If I take all the
other spot, that's

270
00:15:53,440 --> 00:15:55,960
the linear approximation.

271
00:15:55,960 --> 00:15:59,730
And that's exactly
what I wrote here

272
00:15:59,730 --> 00:16:02,140
with little g instead of f.

273
00:16:02,140 --> 00:16:05,120
The only difference is this is
little f and this is little g.

274
00:16:05,120 --> 00:16:09,340
But it's the same exact
formula, linear approximation.

275
00:16:09,340 --> 00:16:14,596
Do you guys remember then next
terms of the Taylor's formula?

276
00:16:14,596 --> 00:16:15,310
STUDENT: fw--

277
00:16:15,310 --> 00:16:16,437
PROFESSOR TODA: fw--

278
00:16:16,437 --> 00:16:19,920
STUDENT: w over--

279
00:16:19,920 --> 00:16:23,430
PROFESSOR TODA: So
fw prime at x0 over--

280
00:16:23,430 --> 00:16:24,384
STUDENT: 1 factorial.

281
00:16:24,384 --> 00:16:25,550
PROFESSOR TODA: 2 factorial.

282
00:16:25,550 --> 00:16:26,625
This was 1 factorial.

283
00:16:26,625 --> 00:16:28,950
This was over 1 factorial.

284
00:16:28,950 --> 00:16:30,573
But I don't write
it because it's one.

285
00:16:30,573 --> 00:16:31,197
STUDENT: Right.

286
00:16:31,197 --> 00:16:35,823
PROFESSOR TODA: Here I would
have f double prime of blah,

287
00:16:35,823 --> 00:16:41,100
blah, blah over-- what did
you say-- 2 factorial times x

288
00:16:41,100 --> 00:16:44,376
minus x0 squared plus, plus,
plus, the cubic [INAUDIBLE]

289
00:16:44,376 --> 00:16:49,730
of the-- this is the quadratic
term that I neglect, right?

290
00:16:49,730 --> 00:16:51,180
So that was Taylor's formula.

291
00:16:51,180 --> 00:16:54,790
Do I mention anything
about it now?

292
00:16:54,790 --> 00:16:55,905
We should.

293
00:16:55,905 --> 00:16:58,250
But practically, the
authors of the book

294
00:16:58,250 --> 00:17:00,400
thought, well, everything
is in the book.

295
00:17:00,400 --> 00:17:02,120
You can go back and forth.

296
00:17:02,120 --> 00:17:05,300
It's not like that unless
somebody opens your eyes.

297
00:17:05,300 --> 00:17:09,930
For example, I didn't
see that when I was 21.

298
00:17:09,930 --> 00:17:13,040
I couldn't make any connection
between these Calc I,

299
00:17:13,040 --> 00:17:14,920
Calc II, Calc III notions.

300
00:17:14,920 --> 00:17:17,886
Because nobody told me, hey,
Magdalena, open your eyes

301
00:17:17,886 --> 00:17:20,118
and look at that in
perspective and make

302
00:17:20,118 --> 00:17:24,720
a comparison between what you
learned in different chapters.

303
00:17:24,720 --> 00:17:26,220
I had to grow.

304
00:17:26,220 --> 00:17:29,030
After 20 years, I
said, oh, I finally

305
00:17:29,030 --> 00:17:33,680
see the picture of linearization
of a function of, let's say,

306
00:17:33,680 --> 00:17:35,390
n variables.

307
00:17:35,390 --> 00:17:38,480
So all these total
differentials will come in place

308
00:17:38,480 --> 00:17:41,050
when time comes.

309
00:17:41,050 --> 00:17:46,410
You have a so-called
differential in Calc I.

310
00:17:46,410 --> 00:17:47,920
And that's not delta g.

311
00:17:47,920 --> 00:17:49,890
Some people say, OK,
no, that's delta g.

312
00:17:49,890 --> 00:17:52,000
No, no, no, no.

313
00:17:52,000 --> 00:17:53,610
The delta x is a displacement.

314
00:17:53,610 --> 00:17:57,305
The delta g is the
induced displacement.

315
00:17:57,305 --> 00:17:59,985
If you want this to be
come a differential,

316
00:17:59,985 --> 00:18:02,840
then you shrink
that displacement

317
00:18:02,840 --> 00:18:05,640
to infinitesimally small.

318
00:18:05,640 --> 00:18:06,230
OK?

319
00:18:06,230 --> 00:18:09,684
So it's like going from
a molecule to an atom

320
00:18:09,684 --> 00:18:13,990
to an electron to subatomic
particles but even more,

321
00:18:13,990 --> 00:18:16,060
something infinitesimally small.

322
00:18:16,060 --> 00:18:17,070
So what do we do?

323
00:18:17,070 --> 00:18:22,810
We shrink delta x into dx
which is infinitesimally small.

324
00:18:22,810 --> 00:18:26,390


325
00:18:26,390 --> 00:18:28,932
It's like the notion of
God but microscopically

326
00:18:28,932 --> 00:18:33,920
or like microbiology
compared to the universe, OK?

327
00:18:33,920 --> 00:18:42,210
So dx is multiplied
by g prime of x0.

328
00:18:42,210 --> 00:18:46,430
And instead of delta g, I'm
going to have a so-called dg,

329
00:18:46,430 --> 00:18:49,060
and that's a form.

330
00:18:49,060 --> 00:18:53,260
In mathematics, this is
called a form or a one form.

331
00:18:53,260 --> 00:18:58,520
And it's a special
kind of object, OK?

332
00:18:58,520 --> 00:19:01,550
So Mr. Leibniz was very smart.

333
00:19:01,550 --> 00:19:09,720
He said, but I can rewrite this
form like dg dx equals g prime.

334
00:19:09,720 --> 00:19:13,450
So if you ever forget
about this form which

335
00:19:13,450 --> 00:19:18,169
is called differential,
differential form,

336
00:19:18,169 --> 00:19:20,780
you remember Mr.
Leibniz, he taught you

337
00:19:20,780 --> 00:19:25,322
how to write the derivative in
two different ways, dg dx or g

338
00:19:25,322 --> 00:19:26,630
prime.

339
00:19:26,630 --> 00:19:30,220
What you do is just formally
multiply g prime by dx

340
00:19:30,220 --> 00:19:31,670
and you get dg.

341
00:19:31,670 --> 00:19:34,700
Say it again, Magdalena--
multiply g prime by dx

342
00:19:34,700 --> 00:19:35,880
and you get dg.

343
00:19:35,880 --> 00:19:38,890
And that's your
so-called differential.

344
00:19:38,890 --> 00:19:42,500
Now, why do you say total
differential-- total

345
00:19:42,500 --> 00:19:46,870
differential, my god, like
complete differentiation?

346
00:19:46,870 --> 00:19:52,280
In 11.4, we deal with
functions of two variables.

347
00:19:52,280 --> 00:19:54,750
So can we say differentials?

348
00:19:54,750 --> 00:19:57,290
Mmm, it's a little bit
like a differential

349
00:19:57,290 --> 00:20:00,030
with respect to what variable?

350
00:20:00,030 --> 00:20:02,590
If you say with respect
to all the variables,

351
00:20:02,590 --> 00:20:08,960
then you have to be thinking
to be smart and event,

352
00:20:08,960 --> 00:20:11,690
create this new object.

353
00:20:11,690 --> 00:20:17,312
If one would write
Taylor's formula,

354
00:20:17,312 --> 00:20:22,720
there is a Taylor's
formula that we don't give.

355
00:20:22,720 --> 00:20:23,260
OK.

356
00:20:23,260 --> 00:20:26,210
Now, you guys are looking
at me with excitement.

357
00:20:26,210 --> 00:20:30,740
For one point extra
credit, on the internet,

358
00:20:30,740 --> 00:20:35,310
find Taylor's formula for
n variables, functions

359
00:20:35,310 --> 00:20:38,590
of n variables or at
least two variables,

360
00:20:38,590 --> 00:20:43,720
which was going to look
like z minus z0 equals

361
00:20:43,720 --> 00:20:49,140
f sub x at the point x0
at 0 times x minus x0 plus

362
00:20:49,140 --> 00:21:00,200
f sub y at x0 y0 times x minus
x0 plus second order terms

363
00:21:00,200 --> 00:21:04,010
plus third order terms
plus fourth order terms.

364
00:21:04,010 --> 00:21:06,720
And the video cannot see me.

365
00:21:06,720 --> 00:21:08,850
So what do we do?

366
00:21:08,850 --> 00:21:13,830
We just truncate this
part of Taylor's I say,

367
00:21:13,830 --> 00:21:18,170
I already take the Taylor
polynomial of degree one.

368
00:21:18,170 --> 00:21:21,470
And the quadratic terms and
everything else, the heck

369
00:21:21,470 --> 00:21:22,850
with that.

370
00:21:22,850 --> 00:21:25,020
And I call that a
linear approximation,

371
00:21:25,020 --> 00:21:28,330
but it's actually Taylor's
formula being discussed.

372
00:21:28,330 --> 00:21:30,680
We don't tell you in
the book because we

373
00:21:30,680 --> 00:21:31,740
don't want to scare you.

374
00:21:31,740 --> 00:21:34,865
I think we would better
tell you at some point,

375
00:21:34,865 --> 00:21:38,010
so I decided to tell you now.

376
00:21:38,010 --> 00:21:38,850
All right.

377
00:21:38,850 --> 00:21:42,440
So this is Taylor's formula
for functions of two variables.

378
00:21:42,440 --> 00:21:45,630
We have to create
not out of nothing

379
00:21:45,630 --> 00:21:49,810
but out of this the
total differential.

380
00:21:49,810 --> 00:21:51,190
Who tells me?

381
00:21:51,190 --> 00:21:54,033
Shrink the
displacement, Magdalena.

382
00:21:54,033 --> 00:21:58,141
The delta x shrunk to
an infinitesimally small

383
00:21:58,141 --> 00:21:58,640
will be dx.

384
00:21:58,640 --> 00:22:01,110
Delta y will become dy.

385
00:22:01,110 --> 00:22:06,390
The line is a smiley from the
skies, just looking at us.

386
00:22:06,390 --> 00:22:08,040
He loves our notations.

387
00:22:08,040 --> 00:22:10,896
And this is dz.

388
00:22:10,896 --> 00:22:18,970
So I'm going to write dz or df's
the same thing equals f sub x.

389
00:22:18,970 --> 00:22:22,420
At the point, you
could be at any point

390
00:22:22,420 --> 00:22:29,780
you are taking in particular,
dx plus f sub y xy dy.

391
00:22:29,780 --> 00:22:34,010
So this is at any point
at the arbitrary point xy

392
00:22:34,010 --> 00:22:39,310
in the domain where your
function e is at least c1.

393
00:22:39,310 --> 00:22:40,730
What does it mean, c1?

394
00:22:40,730 --> 00:22:43,280
It means the function
is differentiable

395
00:22:43,280 --> 00:22:47,400
and the partial
derivatives are continuous.

396
00:22:47,400 --> 00:22:50,850
I said several times, I
want even more than that.

397
00:22:50,850 --> 00:22:56,790
I want it maybe second
order derivatives

398
00:22:56,790 --> 00:23:02,868
to exist and be continuous
and so on and so forth.

399
00:23:02,868 --> 00:23:08,465
And I will assume
that the function can

400
00:23:08,465 --> 00:23:11,585
be expanded [INAUDIBLE] series.

401
00:23:11,585 --> 00:23:14,445


402
00:23:14,445 --> 00:23:17,440
All right, now example
of a final problem

403
00:23:17,440 --> 00:23:22,260
that was my first problem
on the final many times

404
00:23:22,260 --> 00:23:26,310
and also on the common
final departmental final.

405
00:23:26,310 --> 00:23:28,320
And many students
screwed up, and I

406
00:23:28,320 --> 00:23:32,380
don't want you to ever
make such a mistake.

407
00:23:32,380 --> 00:23:37,322
So this is a mistake not
to make, OK, mistake not

408
00:23:37,322 --> 00:23:43,730
to make because after 20
something years of teaching,

409
00:23:43,730 --> 00:23:46,010
I'm quite familiar with
the mistakes students

410
00:23:46,010 --> 00:23:49,230
make in general and I don't
want you to make them.

411
00:23:49,230 --> 00:23:50,664
You are too good to do this.

412
00:23:50,664 --> 00:23:52,098
So problem 1.

413
00:23:52,098 --> 00:23:56,600
On the final, I said-- we
said-- the only difference was

414
00:23:56,600 --> 00:24:00,900
on some departmental finals,
we gave a more sophisticated

415
00:24:00,900 --> 00:24:02,470
function.

416
00:24:02,470 --> 00:24:06,580
I'm going to give only
some simple function

417
00:24:06,580 --> 00:24:07,820
for this polynomial.

418
00:24:07,820 --> 00:24:09,770
That's beautiful.

419
00:24:09,770 --> 00:24:18,930
And then I said we said
write the differential

420
00:24:18,930 --> 00:24:28,090
of this function at an
arbitrary point x, y.

421
00:24:28,090 --> 00:24:28,610
And done.

422
00:24:28,610 --> 00:24:31,080
And [INAUDIBLE].

423
00:24:31,080 --> 00:24:34,642
Well, let me tell you what
some of my students-- some

424
00:24:34,642 --> 00:24:36,350
of my studentss-- don't do that.

425
00:24:36,350 --> 00:24:38,302
I'm going to cross it with red.

426
00:24:38,302 --> 00:24:41,770
And some of my students
wrote me very beautifully df

427
00:24:41,770 --> 00:24:44,390
equals 2x plus 2y.

428
00:24:44,390 --> 00:24:47,550
And that can send
me to the hospital.

429
00:24:47,550 --> 00:24:53,320
If you want to go to the ER
soon, do this on the exam

430
00:24:53,320 --> 00:24:55,960
because this is nonsense.

431
00:24:55,960 --> 00:24:57,480
Why is this nonsense?

432
00:24:57,480 --> 00:24:58,360
This is not--

433
00:24:58,360 --> 00:24:59,840
STUDENT: [INAUDIBLE] dx or dy.

434
00:24:59,840 --> 00:25:00,840
PROFESSOR TODA: Exactly.

435
00:25:00,840 --> 00:25:06,980
So the most important thing
is that the df is like-- OK,

436
00:25:06,980 --> 00:25:09,060
let me come back to driving.

437
00:25:09,060 --> 00:25:14,480
I'm driving to Amarillo-- and I
give this example to my calc 1

438
00:25:14,480 --> 00:25:18,201
students all the time because
it's a linear motion in terms

439
00:25:18,201 --> 00:25:18,700
of time.

440
00:25:18,700 --> 00:25:21,090
And let's say I'm on
cruise control or not.

441
00:25:21,090 --> 00:25:22,780
It doesn't matter.

442
00:25:22,780 --> 00:25:30,190
When we drive and I'm looking at
the speedometer and I see 60--

443
00:25:30,190 --> 00:25:37,000
I didn't want to say more, but
let's say 80, 80 miles an hour.

444
00:25:37,000 --> 00:25:38,620
That is a miles an hour.

445
00:25:38,620 --> 00:25:43,344
That means the hour is a huge
chunk delta h or delta t.

446
00:25:43,344 --> 00:25:45,010
Let's call it delta
t because it's time.

447
00:25:45,010 --> 00:25:45,640
I'm silly.

448
00:25:45,640 --> 00:25:47,660
Delta t is 1.

449
00:25:47,660 --> 00:25:51,310
Delta s, the space,
the space, is going

450
00:25:51,310 --> 00:25:54,970
to be the chunk of 60 miles.

451
00:25:54,970 --> 00:26:00,360
But then that is the
average speed that I had.

452
00:26:00,360 --> 00:26:02,130
So that's why I said 60.

453
00:26:02,130 --> 00:26:04,806
That's the average
speed I had in my trip,

454
00:26:04,806 --> 00:26:05,930
during my trip [INAUDIBLE].

455
00:26:05,930 --> 00:26:10,600
There were moments when my
speed was 0 or close to 0.

456
00:26:10,600 --> 00:26:12,390
Let's assume it was never 0.

457
00:26:12,390 --> 00:26:14,931
But that means there were many
moments when my speed could've

458
00:26:14,931 --> 00:26:18,990
been 100, and nobody knows
because they didn't catch me.

459
00:26:18,990 --> 00:26:21,450
So I was just lucky.

460
00:26:21,450 --> 00:26:26,300
So in average, if somebody is
asking you what is the average,

461
00:26:26,300 --> 00:26:30,440
that doesn't tell them anything.

462
00:26:30,440 --> 00:26:34,090
That reminds me of that
joke-- overall I'm good,

463
00:26:34,090 --> 00:26:38,190
the statistician joke
who was, are you cold?

464
00:26:38,190 --> 00:26:39,000
Are you warm?

465
00:26:39,000 --> 00:26:44,142
And he was actually sitting
on with one half of him

466
00:26:44,142 --> 00:26:47,090
on a block of ice and the
other half on the stove,

467
00:26:47,090 --> 00:26:49,172
and he says, in
average, I'm fine.

468
00:26:49,172 --> 00:26:52,400
But he was dying.

469
00:26:52,400 --> 00:26:53,910
This is the same kind of thing.

470
00:26:53,910 --> 00:26:58,360
My average was 60 miles
an hour, but I almost

471
00:26:58,360 --> 00:27:02,110
got caught when I was
driving almost 100.

472
00:27:02,110 --> 00:27:06,250
But nobody knows because I'm
not giving you that information.

473
00:27:06,250 --> 00:27:12,440
That's the infinitesimally small
information that I have not

474
00:27:12,440 --> 00:27:16,610
put correctly here
means that what is

475
00:27:16,610 --> 00:27:18,990
what I see on the speedometer?

476
00:27:18,990 --> 00:27:21,060
It's the instantaneous
rate of change

477
00:27:21,060 --> 00:27:23,880
that I see that
fraction of second.

478
00:27:23,880 --> 00:27:30,940
So that means maybe a few feet
per a fraction of a second.

479
00:27:30,940 --> 00:27:33,920
It means how many
feet did I travel

480
00:27:33,920 --> 00:27:36,470
in that fraction of a second?

481
00:27:36,470 --> 00:27:41,240
And if that fraction of a second
is very tiny that I cannot even

482
00:27:41,240 --> 00:27:44,000
express it properly, that's
what I'm going to have--

483
00:27:44,000 --> 00:27:46,610
df equals f prime dx.

484
00:27:46,610 --> 00:27:52,010
So df and dx have to be small
because their ratio will be

485
00:27:52,010 --> 00:27:56,180
a good number, like 60, like
80, but [? them in ?] themselves

486
00:27:56,180 --> 00:27:58,635
delta m delta [? srv, ?]
very tiny things.

487
00:27:58,635 --> 00:28:03,420
It's the ratio that matters
in the end to be 60, or 80,

488
00:28:03,420 --> 00:28:04,470
or whatever.

489
00:28:04,470 --> 00:28:08,520
So I have 2x dx plus 2y dy.

490
00:28:08,520 --> 00:28:10,920
Never say that the
differential, which

491
00:28:10,920 --> 00:28:13,160
is something
infinitesimally small,

492
00:28:13,160 --> 00:28:17,376
is equal to this scalar
function that it doesn't even

493
00:28:17,376 --> 00:28:18,160
make any sense.

494
00:28:18,160 --> 00:28:20,060
Don't do that because
you get 0 points

495
00:28:20,060 --> 00:28:21,900
and then we argue,
and I don't want

496
00:28:21,900 --> 00:28:25,450
you to get 0 points on
this problem, right.

497
00:28:25,450 --> 00:28:27,250
So it's a very simple problem.

498
00:28:27,250 --> 00:28:31,080
All I want to test you on
would be this definition.

499
00:28:31,080 --> 00:28:36,000
Remember, you're going to
see that again on the midterm

500
00:28:36,000 --> 00:28:39,020
and on the final, or
just on the final.

501
00:28:39,020 --> 00:28:41,650
Any questions about that?

502
00:28:41,650 --> 00:28:42,250
All right.

503
00:28:42,250 --> 00:28:53,978
So I want to give you the
following homework out

504
00:28:53,978 --> 00:29:00,680
of section 11.4 on
top of the web work.

505
00:29:00,680 --> 00:29:07,250


506
00:29:07,250 --> 00:29:16,640
Read all the solved
examples of the section.

507
00:29:16,640 --> 00:29:23,530


508
00:29:23,530 --> 00:29:24,030
OK.

509
00:29:24,030 --> 00:29:30,470
So for example,
somebody tells you

510
00:29:30,470 --> 00:29:40,110
I have to apply this
knowing that I have

511
00:29:40,110 --> 00:29:44,610
an error of measurement of
some sort in the s direction

512
00:29:44,610 --> 00:29:48,210
and an error of measurement of
some sort in the y direction.

513
00:29:48,210 --> 00:29:51,010
There are two or three
examples like that.

514
00:29:51,010 --> 00:29:54,910
They will give you all this
data, including the error

515
00:29:54,910 --> 00:29:55,640
measurement.

516
00:29:55,640 --> 00:29:58,490
For delta, it should be 0.1.

517
00:29:58,490 --> 00:30:04,240
Don't confuse the 0.1 with
dx. dx is not a quantity.

518
00:30:04,240 --> 00:30:08,608
dx is something like
micro cosmic thing.

519
00:30:08,608 --> 00:30:14,134
It's like infinitely
[? small ?].

520
00:30:14,134 --> 00:30:15,050
Infinitesimally small.

521
00:30:15,050 --> 00:30:19,560
So saying that dx should be
0.1 doesn't make any sense,

522
00:30:19,560 --> 00:30:22,880
but delta x being
0.1 make sense.

523
00:30:22,880 --> 00:30:26,350
Delta y being 0.3 makes sense.

524
00:30:26,350 --> 00:30:29,560
And they ask you to
plug it in and find

525
00:30:29,560 --> 00:30:32,130
the general difference.

526
00:30:32,130 --> 00:30:33,730
For example, where
could that happen?

527
00:30:33,730 --> 00:30:35,760
And you see examples
in the book.

528
00:30:35,760 --> 00:30:40,910
Somebody measures something--
an area of a rectangle

529
00:30:40,910 --> 00:30:42,970
or a volume of a cube.

530
00:30:42,970 --> 00:30:46,110
But when you measure,
you make mistakes.

531
00:30:46,110 --> 00:30:48,270
You have measurement errors.

532
00:30:48,270 --> 00:30:53,250
In the delta x, you have
an error of plus minus 0.1.

533
00:30:53,250 --> 00:31:00,870
In the y direction, you have
displacement error 0.2 or 0.3,

534
00:31:00,870 --> 00:31:02,220
something like that.

535
00:31:02,220 --> 00:31:05,090
What is the overall
error you are

536
00:31:05,090 --> 00:31:08,100
going to make when you measure
that function of two variables?

537
00:31:08,100 --> 00:31:09,730
That's what you have.

538
00:31:09,730 --> 00:31:12,140
So you plug in all
those displacements

539
00:31:12,140 --> 00:31:14,790
and you come up with the
computational problem.

540
00:31:14,790 --> 00:31:20,200
Several of you Wednesday we
discussed in my office already

541
00:31:20,200 --> 00:31:24,700
solved those problems through
web work and came to me,

542
00:31:24,700 --> 00:31:27,510
and I said, how did you know
to plug in those [? numbers ?]?

543
00:31:27,510 --> 00:31:28,900
Well, it's not so hard.

544
00:31:28,900 --> 00:31:30,120
It's sort of common sense.

545
00:31:30,120 --> 00:31:32,990
Plus, I looked in the book
and that gave me the idea

546
00:31:32,990 --> 00:31:34,517
to remind you to
look in the book

547
00:31:34,517 --> 00:31:37,250
for those numerical examples.

548
00:31:37,250 --> 00:31:40,370
You will have to
use your calculator.

549
00:31:40,370 --> 00:31:42,990
So you don't have it with
you, you generally, we

550
00:31:42,990 --> 00:31:45,000
don't use in the classroom,
but it's very easy.

551
00:31:45,000 --> 00:31:48,392
All you have to do is use the
calculator and [INAUDIBLE]

552
00:31:48,392 --> 00:31:51,310
examples and see how it goes.

553
00:31:51,310 --> 00:31:57,430
I wanted to show you
something more interesting

554
00:31:57,430 --> 00:32:09,410
even, more beautiful
regarding something

555
00:32:09,410 --> 00:32:12,930
we don't show in the
book until later on,

556
00:32:12,930 --> 00:32:18,240
and I'm uncomfortable with the
idea of not showing this to you

557
00:32:18,240 --> 00:32:19,610
now.

558
00:32:19,610 --> 00:32:26,560
An alternate way, or
more advanced way,

559
00:32:26,560 --> 00:32:38,390
more advanced way, to
define the tangent plane--

560
00:32:38,390 --> 00:32:49,190
the tangent plane-- to a
surface S at the point p.

561
00:32:49,190 --> 00:32:51,690
And I'll draw again.

562
00:32:51,690 --> 00:32:56,470
Half of my job is drawing
in this class, which I like.

563
00:32:56,470 --> 00:32:59,910
I mean, I was having an argument
with one of my colleagues who

564
00:32:59,910 --> 00:33:03,480
said, I hate when they are
giving me to teach calculus 3

565
00:33:03,480 --> 00:33:07,660
because I cannot draw.

566
00:33:07,660 --> 00:33:09,910
I think that the
most beautiful part

567
00:33:09,910 --> 00:33:15,450
is that we can represent
things visually,

568
00:33:15,450 --> 00:33:20,262
and this is just pi, the
tangent plane I'm after,

569
00:33:20,262 --> 00:33:24,880
and p will be a
coordinate 0 by 0, z0.

570
00:33:24,880 --> 00:33:26,900
And what was the label?

571
00:33:26,900 --> 00:33:27,790
Oh, the label.

572
00:33:27,790 --> 00:33:28,365
The label.

573
00:33:28,365 --> 00:33:34,330
The label was internal
where z equals f of xy.

574
00:33:34,330 --> 00:33:40,160
But more generally, I'll say
this time plus more generally,

575
00:33:40,160 --> 00:33:58,970
what if you have f of xyz
equals c for that surface.

576
00:33:58,970 --> 00:34:00,560
Let's call it [INAUDIBLE].

577
00:34:00,560 --> 00:34:04,800
F of xy is [INAUDIBLE].

578
00:34:04,800 --> 00:34:08,210
And somebody even said, can
you have a parametrization?

579
00:34:08,210 --> 00:34:10,440
And this is where
I wanted to go.

580
00:34:10,440 --> 00:34:14,469


581
00:34:14,469 --> 00:34:16,230
Ryan was the first
one who asked me,

582
00:34:16,230 --> 00:34:18,870
but then there were
three more of you

583
00:34:18,870 --> 00:34:21,159
who have restless
minds plus you--

584
00:34:21,159 --> 00:34:25,670
because that's the essence
of being active here.

585
00:34:25,670 --> 00:34:29,840
We don't lose our connections.

586
00:34:29,840 --> 00:34:34,300
We lose neurons anyway, but
we don't lose our connections

587
00:34:34,300 --> 00:34:37,949
if we think, and
anticipate things,

588
00:34:37,949 --> 00:34:40,080
and try to relate concepts.

589
00:34:40,080 --> 00:34:42,590
So if you don't want to
get Alzheimer's, just

590
00:34:42,590 --> 00:34:45,730
think about the parametrization.

591
00:34:45,730 --> 00:34:49,699
So can I have a
parametrization for a surface?

592
00:34:49,699 --> 00:34:52,179
All righty, what do you mean?

593
00:34:52,179 --> 00:34:58,240
What if somebody says for a
curve, we have r of t, right,

594
00:34:58,240 --> 00:34:59,075
which was what?

595
00:34:59,075 --> 00:35:06,500
It was x of ti plus y of tj plus
z of tk, and we were so happy

596
00:35:06,500 --> 00:35:09,825
and we were happy
because we were traveling

597
00:35:09,825 --> 00:35:12,320
in time with respect
to the origin,

598
00:35:12,320 --> 00:35:15,640
and this was r of t at time t.

599
00:35:15,640 --> 00:35:18,330
[INAUDIBLE]

600
00:35:18,330 --> 00:35:20,210
But somebody asked
me, [INAUDIBLE],

601
00:35:20,210 --> 00:35:27,010
can you have such a position
vector moving on a surface?

602
00:35:27,010 --> 00:35:30,240
Like look, it's a rigid motion.

603
00:35:30,240 --> 00:35:32,770
If you went to the
robotics science

604
00:35:32,770 --> 00:35:36,340
fair, Texas Tech, or something
like that, you know about that.

605
00:35:36,340 --> 00:35:37,180
Yeah, cities.

606
00:35:37,180 --> 00:35:39,977
So how do we introduce
such a parametrization?

607
00:35:39,977 --> 00:35:44,470
We have an origin of course.

608
00:35:44,470 --> 00:35:46,390
An origin is always important.

609
00:35:46,390 --> 00:35:48,326
Everybody has an origin.

610
00:35:48,326 --> 00:35:53,170


611
00:35:53,170 --> 00:35:57,610
And I take that position
vector, and where does it start?

612
00:35:57,610 --> 00:36:02,120
It starts at the origin, and
the tip of it is on the surface,

613
00:36:02,120 --> 00:36:05,382
And it's gliding on the
surface, the tip of it.

614
00:36:05,382 --> 00:36:10,500
And that's going to be r, but
it's not going to be r of t.

615
00:36:10,500 --> 00:36:12,930
It's going to be r of
longitude and latitude.

616
00:36:12,930 --> 00:36:16,110
Like imagine, that would
be the radius coming

617
00:36:16,110 --> 00:36:18,360
from the center of the earth.

618
00:36:18,360 --> 00:36:20,980
And it depends on
two parameters.

619
00:36:20,980 --> 00:36:24,780
One of them would be latitude.

620
00:36:24,780 --> 00:36:26,140
Am I drawing this right?

621
00:36:26,140 --> 00:36:26,640
Latitude--

622
00:36:26,640 --> 00:36:28,730
STUDENT: [INAUDIBLE] longitude.

623
00:36:28,730 --> 00:36:30,870
PROFESSOR TODA:
--from a latitude 0.

624
00:36:30,870 --> 00:36:32,010
I'm at the equator.

625
00:36:32,010 --> 00:36:33,760
Then latitude 90 degrees.

626
00:36:33,760 --> 00:36:35,970
I'm at the North Pole.

627
00:36:35,970 --> 00:36:37,760
In mathematics, we are funny.

628
00:36:37,760 --> 00:36:40,880
We say latitude 0,
latitude 90 North Pole,

629
00:36:40,880 --> 00:36:45,165
latitude negative 90,
which is South Pole.

630
00:36:45,165 --> 00:36:49,290
And longitude from 0 to 2 pi.

631
00:36:49,290 --> 00:36:53,740
Meridian 0 to all around.

632
00:36:53,740 --> 00:36:58,200
So r will be not a function of
t but a function of u and b,

633
00:36:58,200 --> 00:37:02,240
thank god, because u and b
are the latitude and longitude

634
00:37:02,240 --> 00:37:03,320
sort of.

635
00:37:03,320 --> 00:37:12,324
So we have x of uv i plus
y of uv j plus z of uv k.

636
00:37:12,324 --> 00:37:20,620


637
00:37:20,620 --> 00:37:23,030
You can do that.

638
00:37:23,030 --> 00:37:26,010
And you say, but can you give
us an example, because this

639
00:37:26,010 --> 00:37:28,210
looks so abstract for god sake.

640
00:37:28,210 --> 00:37:31,830
If you give me the graph
the way you gave it to me

641
00:37:31,830 --> 00:37:37,307
before z equals f of xy,
please parametrize this for me.

642
00:37:37,307 --> 00:37:41,880


643
00:37:41,880 --> 00:37:44,640
Parametrize it for
me because I'm lost.

644
00:37:44,640 --> 00:37:45,610
You are not lost.

645
00:37:45,610 --> 00:37:47,530
We can do this together.

646
00:37:47,530 --> 00:37:51,480
Now what's the simplest
way to parametrize

647
00:37:51,480 --> 00:37:57,260
a graph of the type
z equals f of xy?

648
00:37:57,260 --> 00:38:01,970
Take the xy to be
u and v. Take x

649
00:38:01,970 --> 00:38:05,360
and y to be your
independent variables

650
00:38:05,360 --> 00:38:07,850
and take z to be the
dependent variable.

651
00:38:07,850 --> 00:38:12,700


652
00:38:12,700 --> 00:38:16,930
I'm again expressing these
things in terms of variables

653
00:38:16,930 --> 00:38:18,340
like I did last time.

654
00:38:18,340 --> 00:38:23,370
Then I say, let's take this kind
of parametrization. [INAUDIBLE]

655
00:38:23,370 --> 00:38:24,380
vu, right.

656
00:38:24,380 --> 00:38:33,080
y would be v. Then I'm
going to write r of x and y

657
00:38:33,080 --> 00:38:36,710
just like that guy will
be [INAUDIBLE] of xn.

658
00:38:36,710 --> 00:38:38,770
[? y ?] will say, wait a minute.

659
00:38:38,770 --> 00:38:42,884
I will have to re-denote
everybody with capitals.

660
00:38:42,884 --> 00:38:46,300
Then my life will become
better because you

661
00:38:46,300 --> 00:38:47,300
don't have to erase.

662
00:38:47,300 --> 00:38:50,670
You just make little
x big, little y bigs,

663
00:38:50,670 --> 00:38:53,890
bigs, big, capitalized XYZ.

664
00:38:53,890 --> 00:39:02,150
And then I'll say OK, XYZ
will be my setting here in 3D.

665
00:39:02,150 --> 00:39:07,020


666
00:39:07,020 --> 00:39:07,560
All right.

667
00:39:07,560 --> 00:39:10,290
So how am I going
to re-parametrize

668
00:39:10,290 --> 00:39:12,576
the whole surface?

669
00:39:12,576 --> 00:39:22,220
Whole surface will be r of
xy equals in this case, well,

670
00:39:22,220 --> 00:39:23,280
let's think about it.

671
00:39:23,280 --> 00:39:29,020
In this case, I'm
going to have xy.

672
00:39:29,020 --> 00:39:31,350
And where's the little f?

673
00:39:31,350 --> 00:39:32,550
I just erased it.

674
00:39:32,550 --> 00:39:35,085
I was smart, right,
that I erased f of xy.

675
00:39:35,085 --> 00:39:37,830


676
00:39:37,830 --> 00:39:46,010
So I have x, y, and
z, which is f of xy.

677
00:39:46,010 --> 00:39:53,240


678
00:39:53,240 --> 00:40:01,430
And this is the generic point
p of coordinates xy f of xy.

679
00:40:01,430 --> 00:40:04,980


680
00:40:04,980 --> 00:40:07,580
So I say, OK, what does it mean?

681
00:40:07,580 --> 00:40:10,100
I will project this point.

682
00:40:10,100 --> 00:40:13,175
And this is the point
when big x becomes little

683
00:40:13,175 --> 00:40:17,860
x, when big y becomes--
where is my y-axis?

684
00:40:17,860 --> 00:40:20,090
Somebody ate my y axis.

685
00:40:20,090 --> 00:40:22,190
[INAUDIBLE]

686
00:40:22,190 --> 00:40:28,400
So when big Y becomes
little y, little y

687
00:40:28,400 --> 00:40:33,830
is just an instance of big Y.
And big Z will take what value?

688
00:40:33,830 --> 00:40:35,630
Well, I need to project that.

689
00:40:35,630 --> 00:40:39,120
How do you project from
a point to the z-axis?

690
00:40:39,120 --> 00:40:42,680
You have to take the
parallel from the point

691
00:40:42,680 --> 00:40:47,630
to the horizontal
plane until you

692
00:40:47,630 --> 00:40:52,940
hit the-- [INAUDIBLE] the whole
plane parallel to the floor

693
00:40:52,940 --> 00:40:54,210
through the point p.

694
00:40:54,210 --> 00:40:55,450
And what do I get here?

695
00:40:55,450 --> 00:40:56,410
STUDENT: [INAUDIBLE].

696
00:40:56,410 --> 00:40:58,670
PROFESSOR TODA: Not
z0, but it's little z

697
00:40:58,670 --> 00:41:03,120
equals f of xy, which is an
instance of the variable xz.

698
00:41:03,120 --> 00:41:06,460
For you programmers, you know
that big z will be a variable

699
00:41:06,460 --> 00:41:11,640
and little z will be
[INAUDIBLE] a variable.

700
00:41:11,640 --> 00:41:12,140
OK.

701
00:41:12,140 --> 00:41:16,610
So I parametrized my graph
in a more general way,

702
00:41:16,610 --> 00:41:18,578
general parametrization
for a graph.

703
00:41:18,578 --> 00:41:25,960


704
00:41:25,960 --> 00:41:33,420
And now, what are-- what's the
meaning of r sub x and r sub y?

705
00:41:33,420 --> 00:41:34,489
What are they?

706
00:41:34,489 --> 00:41:35,364
STUDENT: [INAUDIBLE].

707
00:41:35,364 --> 00:41:38,180


708
00:41:38,180 --> 00:41:41,660
PROFESSOR TODA: Now, we
don't say that in the book.

709
00:41:41,660 --> 00:41:42,990
Shame on us.

710
00:41:42,990 --> 00:41:43,630
Shame on us.

711
00:41:43,630 --> 00:41:47,480
We should have because I was
browsing through the projects

712
00:41:47,480 --> 00:41:49,900
about a year and a half ago.

713
00:41:49,900 --> 00:41:52,970
The senior projects of
a few of my students

714
00:41:52,970 --> 00:41:56,340
who are-- two of them were
in mechanical engineering.

715
00:41:56,340 --> 00:42:00,660
One of them was in
petroleum engineering.

716
00:42:00,660 --> 00:42:03,965
And he actually showed me
that they were doing this.

717
00:42:03,965 --> 00:42:07,830
They were taking vectors
that depend on parameters--

718
00:42:07,830 --> 00:42:11,250
this is a vector [INAUDIBLE]--
and differentiated them with

719
00:42:11,250 --> 00:42:13,720
respect to those parameters.

720
00:42:13,720 --> 00:42:17,215
And I was thinking OK, did we
do the partial derivatives r sub

721
00:42:17,215 --> 00:42:17,960
x, r sub y?

722
00:42:17,960 --> 00:42:19,340
Not so much.

723
00:42:19,340 --> 00:42:22,380
But now I want to do it
because I think that prepares

724
00:42:22,380 --> 00:42:24,640
you better as engineers.

725
00:42:24,640 --> 00:42:29,070
So what is r sub x
and what is r sub y?

726
00:42:29,070 --> 00:42:31,250
And you say, well,
OK. [INAUDIBLE],

727
00:42:31,250 --> 00:42:34,860
I think I know how to do
that in my sleep, right.

728
00:42:34,860 --> 00:42:36,780
If you want me to do
that theoretically

729
00:42:36,780 --> 00:42:39,720
from this formula,
but on the picture,

730
00:42:39,720 --> 00:42:42,450
I really don't know what it is.

731
00:42:42,450 --> 00:42:45,590
So I'm asking you what
I'm going to have in terms

732
00:42:45,590 --> 00:42:47,240
of r sub x and r sub y.

733
00:42:47,240 --> 00:42:48,950
They will be vectors.

734
00:42:48,950 --> 00:42:51,880
This should be a
vector as well, right.

735
00:42:51,880 --> 00:42:56,620
And for me, vector triple
means the identification

736
00:42:56,620 --> 00:42:59,930
between the three coordinates
and the physical vector.

737
00:42:59,930 --> 00:43:01,960
So this is the physical vector.

738
00:43:01,960 --> 00:43:06,032
Go ahead and write x prime
with respect to x is 1.

739
00:43:06,032 --> 00:43:08,684


740
00:43:08,684 --> 00:43:13,776
y prime with respect to x is 0.

741
00:43:13,776 --> 00:43:15,970
The third [INAUDIBLE]
prime with respect

742
00:43:15,970 --> 00:43:20,190
to x is just whatever
this little f is,

743
00:43:20,190 --> 00:43:21,984
it's not any of my business.

744
00:43:21,984 --> 00:43:24,786
It's a [INAUDIBLE]
function f sub x.

745
00:43:24,786 --> 00:43:28,290


746
00:43:28,290 --> 00:43:30,590
Well, what is the second vector?

747
00:43:30,590 --> 00:43:32,285
STUDENT: 0, 1, f sub y.

748
00:43:32,285 --> 00:43:34,810
PROFESSOR TODA: 0, 1, f sub y.

749
00:43:34,810 --> 00:43:36,596
Now, are they slopes?

750
00:43:36,596 --> 00:43:37,096
No.

751
00:43:37,096 --> 00:43:38,010
These are slopes.

752
00:43:38,010 --> 00:43:40,770
That's a slope and
that's a slope.

753
00:43:40,770 --> 00:43:44,950
And we learned
about those in 11.3,

754
00:43:44,950 --> 00:43:49,530
and we understood that those
are ski slopes, they were.

755
00:43:49,530 --> 00:43:52,312
In the direction of x
and the direction of y,

756
00:43:52,312 --> 00:44:00,030
the slopes of the tangents
to the coordinate lines.

757
00:44:00,030 --> 00:44:04,980
But this looks like I have
a direction of a line,

758
00:44:04,980 --> 00:44:08,610
and this would be the lope, and
that's the direction of a line,

759
00:44:08,610 --> 00:44:10,310
and that would be the slope.

760
00:44:10,310 --> 00:44:12,700
What are those lines?

761
00:44:12,700 --> 00:44:16,314
STUDENT: [INAUDIBLE] to
the function [INAUDIBLE].

762
00:44:16,314 --> 00:44:17,480
PROFESSOR TODA: Let me draw.

763
00:44:17,480 --> 00:44:19,440
Then shall I erase
the whole thing?

764
00:44:19,440 --> 00:44:20,160
No.

765
00:44:20,160 --> 00:44:23,948
I'm just going to keep--
I'll erase the tangent.

766
00:44:23,948 --> 00:44:27,470
Don't erase anything
on your notebooks.

767
00:44:27,470 --> 00:44:28,922
So this is the point p.

768
00:44:28,922 --> 00:44:29,630
It's still there.

769
00:44:29,630 --> 00:44:30,570
This is the surface.

770
00:44:30,570 --> 00:44:33,060
It's still there.

771
00:44:33,060 --> 00:44:38,200
So my surface will be x,
slices of x, [? S ?] constant

772
00:44:38,200 --> 00:44:39,590
are coming towards you.

773
00:44:39,590 --> 00:44:45,800
They are these [? walls ?]
like that, like this, yes.

774
00:44:45,800 --> 00:44:47,606
It's like the CT scan.

775
00:44:47,606 --> 00:44:52,190
I think that when they
slice up your body,

776
00:44:52,190 --> 00:44:54,260
tch tch tch tch tch
tch, take pictures

777
00:44:54,260 --> 00:44:57,590
of the slices of your body,
that's the same kind of thing.

778
00:44:57,590 --> 00:44:59,508
So x0, x0, x0, x0.

779
00:44:59,508 --> 00:45:05,414
I'm going to [INAUDIBLE]
planes and I had x equals x0.

780
00:45:05,414 --> 00:45:12,402
And in the other direction, I
cut and I get, what do I get?

781
00:45:12,402 --> 00:45:18,400


782
00:45:18,400 --> 00:45:20,226
Well, I started bad.

783
00:45:20,226 --> 00:45:23,650


784
00:45:23,650 --> 00:45:25,195
Great, Magdalena, this is--

785
00:45:25,195 --> 00:45:27,226
What is this pink?

786
00:45:27,226 --> 00:45:32,350
It's not Valentine's Day
anymore. y equals [INAUDIBLE].

787
00:45:32,350 --> 00:45:34,810
And this is the point.

788
00:45:34,810 --> 00:45:39,320
So, as Alex was
trying to tell you,

789
00:45:39,320 --> 00:45:44,980
our sub x would represent the
vector, the physical vector

790
00:45:44,980 --> 00:45:52,260
in 3D, that is originating
at p and tangent to which

791
00:45:52,260 --> 00:45:55,760
of the two, to the purple
one or to the red one?

792
00:45:55,760 --> 00:45:57,185
STUDENT: Red.

793
00:45:57,185 --> 00:45:58,135
Uh, purple.

794
00:45:58,135 --> 00:45:59,560
PROFESSOR TODA:
Make up your mind.

795
00:45:59,560 --> 00:46:01,494
STUDENT: The purple one.

796
00:46:01,494 --> 00:46:03,660
PROFESSOR TODA: [INAUDIBLE]
constant and [INAUDIBLE]

797
00:46:03,660 --> 00:46:06,770
constant in the red
one, y equals y0, right?

798
00:46:06,770 --> 00:46:08,915
So, this depends on x.

799
00:46:08,915 --> 00:46:11,010
So this has r sub x.

800
00:46:11,010 --> 00:46:14,800


801
00:46:14,800 --> 00:46:18,830
This is the velocity with
respect to the variable x.

802
00:46:18,830 --> 00:46:23,200
And the other one, the
blue one, x equals x0,

803
00:46:23,200 --> 00:46:27,640
means x0 is held fixed
and y is the variable.

804
00:46:27,640 --> 00:46:30,505
So I have to do r sub y,
and what am I gonna get?

805
00:46:30,505 --> 00:46:32,696
I'm gonna get the blue vector.

806
00:46:32,696 --> 00:46:34,880
What's the property
of the blue vector?

807
00:46:34,880 --> 00:46:37,830
It's tangent to the purple line.

808
00:46:37,830 --> 00:46:44,160
So r sub y has to be
tangent to the curve.

809
00:46:44,160 --> 00:46:47,440


810
00:46:47,440 --> 00:46:55,310
x0, y, f of x0 and
y is the curve.

811
00:46:55,310 --> 00:46:59,770
And r sub x is tangent
to which curve?

812
00:46:59,770 --> 00:47:02,400
Who is telling me which curve?

813
00:47:02,400 --> 00:47:12,020
x, y0 sub constant,
f of x and y0.

814
00:47:12,020 --> 00:47:14,489
So that's a curve that
depends only on y,

815
00:47:14,489 --> 00:47:16,854
y is the time in this case.

816
00:47:16,854 --> 00:47:19,000
And that's the curve
that depends only on x.

817
00:47:19,000 --> 00:47:21,210
x is the time in this case.

818
00:47:21,210 --> 00:47:24,580
r sub x and r sub y are
the tangent vectors.

819
00:47:24,580 --> 00:47:26,830
What's magical about them?

820
00:47:26,830 --> 00:47:30,540
If I shape this
triangle between them,

821
00:47:30,540 --> 00:47:32,172
that will be the tangent plane.

822
00:47:32,172 --> 00:47:35,950


823
00:47:35,950 --> 00:47:39,170
And I make a smile because I
discovered the tangent plane

824
00:47:39,170 --> 00:47:43,230
in a different way than
we did it last time.

825
00:47:43,230 --> 00:47:51,005
So the tangent plane represents
the plane of the vector r sub

826
00:47:51,005 --> 00:47:54,532
x and r sub y.

827
00:47:54,532 --> 00:48:02,290
The tangent plane
represents the plane

828
00:48:02,290 --> 00:48:13,080
given by vectors r sub x and
r sub y with what conditions?

829
00:48:13,080 --> 00:48:14,025
It's a conditional.

830
00:48:14,025 --> 00:48:17,010


831
00:48:17,010 --> 00:48:20,630
r sub x and r sub
y shouldn't be 0.

832
00:48:20,630 --> 00:48:24,850
r sub x different from 0,
r sub y different from 0,

833
00:48:24,850 --> 00:48:27,455
and r sub x and r sub
y are not collinear.

834
00:48:27,455 --> 00:48:32,160


835
00:48:32,160 --> 00:48:35,050
What's gonna happen
if they are collinear?

836
00:48:35,050 --> 00:48:36,880
Well, they're gonna
collapse; they are not

837
00:48:36,880 --> 00:48:38,190
gonna determine a plane.

838
00:48:38,190 --> 00:48:40,770
So there will be
no tangent planes.

839
00:48:40,770 --> 00:48:43,720
So they have to be
linearly independent.

840
00:48:43,720 --> 00:48:47,940
For the people who are taking
now linear algebra, I'm saying.

841
00:48:47,940 --> 00:48:50,940
So we have no other
choice, we have

842
00:48:50,940 --> 00:48:54,820
to assume that these vectors,
called partial velocities,

843
00:48:54,820 --> 00:49:04,120
by the way, for the
motion across the surface.

844
00:49:04,120 --> 00:49:04,620
OK?

845
00:49:04,620 --> 00:49:06,970
These are the partial
velocities, or partial velocity

846
00:49:06,970 --> 00:49:08,630
vectors.

847
00:49:08,630 --> 00:49:12,860
Partial velocity vectors
have to determine a plane,

848
00:49:12,860 --> 00:49:16,560
so I have to assume
that they are non-zero,

849
00:49:16,560 --> 00:49:20,120
they never become 0, and
they are not collinear.

850
00:49:20,120 --> 00:49:23,270
If they are collinear,
life is over for you.

851
00:49:23,270 --> 00:49:24,140
OK?

852
00:49:24,140 --> 00:49:29,390
So I have to assume that I
throw away all the points where

853
00:49:29,390 --> 00:49:35,100
the velocities become 0, and
all the points where--those are

854
00:49:35,100 --> 00:49:39,710
singularity points--where
my velocity vectors are 0.

855
00:49:39,710 --> 00:49:43,710


856
00:49:43,710 --> 00:49:45,820
Have you ever studied design?

857
00:49:45,820 --> 00:49:47,350
Any kind of experimental design.

858
00:49:47,350 --> 00:49:52,310
Like, how do you design a car,
the coordinate lines on a car?

859
00:49:52,310 --> 00:49:53,280
I'm just dreaming.

860
00:49:53,280 --> 00:50:00,200
You have a car, a beautiful
car, and then you have-- Well,

861
00:50:00,200 --> 00:50:04,790
I cannot draw really
well, but anyway.

862
00:50:04,790 --> 00:50:08,730
I have these coordinate
lines on this car.

863
00:50:08,730 --> 00:50:12,060
It's a mesh what I have there.

864
00:50:12,060 --> 00:50:15,510
Actually, we do that in
animation all the time.

865
00:50:15,510 --> 00:50:21,030
We have meshes over the
models we have in animation.

866
00:50:21,030 --> 00:50:22,660
Think Avatar.

867
00:50:22,660 --> 00:50:27,210
Now, those are all
coordinate lines.

868
00:50:27,210 --> 00:50:33,650
Those coordinate lines would be,
even your singularities, where?

869
00:50:33,650 --> 00:50:38,510
For example, if you take a body
in a mesh like that, in a net,

870
00:50:38,510 --> 00:50:43,190
in, like, a fishnet, then
you pull from the fishnet,

871
00:50:43,190 --> 00:50:52,980
all the coordinate lines
will come together,

872
00:50:52,980 --> 00:50:55,310
and this would be a singularity.

873
00:50:55,310 --> 00:50:57,890
We avoid this kind
of singularity.

874
00:50:57,890 --> 00:51:00,430
So these are points where
something bad happened.

875
00:51:00,430 --> 00:51:05,380
Either the velocity
vectors become collinear.

876
00:51:05,380 --> 00:51:07,430
You see what I'm talking about?

877
00:51:07,430 --> 00:51:11,260
Or the velocity
vectors shrank to 0.

878
00:51:11,260 --> 00:51:14,190
So that's a bad point;
that's a singularity point.

879
00:51:14,190 --> 00:51:16,870
They have this
problem when meshing.

880
00:51:16,870 --> 00:51:20,670
So when they make
these models that

881
00:51:20,670 --> 00:51:26,850
involve two-dimensional meshing
and three-dimensional ambient

882
00:51:26,850 --> 00:51:31,490
space, like it is in
animation, the mesh

883
00:51:31,490 --> 00:51:34,630
is called regular
if we don't have

884
00:51:34,630 --> 00:51:39,770
this kind of singularity, where
the velocity vectors become 0,

885
00:51:39,770 --> 00:51:42,000
or collinear.

886
00:51:42,000 --> 00:51:45,795
It's very important for a
person who programs in animation

887
00:51:45,795 --> 00:51:47,220
to know mathematics.

888
00:51:47,220 --> 00:51:50,100
If they don't understand
these things, it's over.

889
00:51:50,100 --> 00:51:55,910
Because you write the matrix,
and you will know the vectors

890
00:51:55,910 --> 00:51:59,954
will become collinear when the
two vectors--let's say two rows

891
00:51:59,954 --> 00:52:00,495
of a matrix--

892
00:52:00,495 --> 00:52:00,810
STUDENT: Parallel.

893
00:52:00,810 --> 00:52:01,880
PROFESSOR TODA:
Are proportional.

894
00:52:01,880 --> 00:52:02,510
Or parallel.

895
00:52:02,510 --> 00:52:03,870
Or proportional.

896
00:52:03,870 --> 00:52:07,550
So, everything is numerical
in terms of those matrices,

897
00:52:07,550 --> 00:52:12,890
but it's just a discretization
of a continuous phenomenon,

898
00:52:12,890 --> 00:52:14,010
which is this one.

899
00:52:14,010 --> 00:52:17,690


900
00:52:17,690 --> 00:52:19,970
Do you remember Toy Story?

901
00:52:19,970 --> 00:52:20,815
OK.

902
00:52:20,815 --> 00:52:24,300
The Toy Story people,
the renderers,

903
00:52:24,300 --> 00:52:27,010
the ones who did the rendering
techniques for Toy Story,

904
00:52:27,010 --> 00:52:30,410
both have their
master's in mathematics.

905
00:52:30,410 --> 00:52:33,970
And you realize why
now to do that you

906
00:52:33,970 --> 00:52:38,860
have to know calc I, calc
II, calc III, linear algebra,

907
00:52:38,860 --> 00:52:41,120
be able to deal with matrices.

908
00:52:41,120 --> 00:52:45,610
Have a programming course
or two; that's essential.

909
00:52:45,610 --> 00:52:50,042
They took advanced calculus
because some people

910
00:52:50,042 --> 00:52:55,420
don't cover thi-- I was about to
skip it right now in calc III.

911
00:52:55,420 --> 00:53:00,110
But they teach that in
advanced calculus 4350, 4351.

912
00:53:00,110 --> 00:53:02,672
So that's about as
far as you can get,

913
00:53:02,672 --> 00:53:05,870
and differential equation's
also very important.

914
00:53:05,870 --> 00:53:09,510
So, if you master those and
you go into something else,

915
00:53:09,510 --> 00:53:12,320
like programming,
electrical engineering,

916
00:53:12,320 --> 00:53:14,250
you're ready for animation.

917
00:53:14,250 --> 00:53:16,970
[INAUDIBLE] If you went
I want to be a rendering

918
00:53:16,970 --> 00:53:20,140
guy for the next movie,
then they'll say no,

919
00:53:20,140 --> 00:53:21,600
we won't take you.

920
00:53:21,600 --> 00:53:23,920
I have a friend who
works for Disney.

921
00:53:23,920 --> 00:53:26,780
She wanted to get a PhD.

922
00:53:26,780 --> 00:53:29,384
At some point, she
changed her mind

923
00:53:29,384 --> 00:53:31,967
and ended up just with a
master's in mathematics

924
00:53:31,967 --> 00:53:33,800
while I was in Kansas,
University of Kansas,

925
00:53:33,800 --> 00:53:36,565
and she said, "You know what?

926
00:53:36,565 --> 00:53:41,620
Disney's just giving me
$65,000 as an intern."

927
00:53:41,620 --> 00:53:45,582
And I was like OK and probably
asked [INAUDIBLE] $40,000 as

928
00:53:45,582 --> 00:53:46,676
a postdoc.

929
00:53:46,676 --> 00:53:48,050
And she said,
"Good luck to you."

930
00:53:48,050 --> 00:53:49,420
Good luck to you, too.

931
00:53:49,420 --> 00:53:52,520
But we stayed in touch,
and right now she's

932
00:53:52,520 --> 00:53:57,460
making twice as much as
I'm making, for Disney.

933
00:53:57,460 --> 00:53:58,676
Is she happy?

934
00:53:58,676 --> 00:53:59,568
Yeah.

935
00:53:59,568 --> 00:54:00,460
Would I be happy?

936
00:54:00,460 --> 00:54:01,406
No.

937
00:54:01,406 --> 00:54:05,764
Because she works
for 11 hours a day.

938
00:54:05,764 --> 00:54:08,120
11 hours a day, on a chair.

939
00:54:08,120 --> 00:54:09,090
That would kill me.

940
00:54:09,090 --> 00:54:15,070
I mean, I spend about six hours
sitting on a chair every day

941
00:54:15,070 --> 00:54:19,160
of the week, but
it's still too much.

942
00:54:19,160 --> 00:54:20,800
She's a hard worker, though.

943
00:54:20,800 --> 00:54:22,820
She loves what she's doing.

944
00:54:22,820 --> 00:54:24,060
The problem is your eyes.

945
00:54:24,060 --> 00:54:27,420
After a while, your
eyes are going bad.

946
00:54:27,420 --> 00:54:33,600
So, what is the normal for
the plane in this case?

947
00:54:33,600 --> 00:54:37,298
I'll try my best
ability to draw normal.

948
00:54:37,298 --> 00:54:38,714
The normal has to
be perpendicular

949
00:54:38,714 --> 00:54:41,950
to the tangent space, right?

950
00:54:41,950 --> 00:54:43,700
Tangent plane.

951
00:54:43,700 --> 00:54:46,230
So, n has to be
perpendicular to our sub

952
00:54:46,230 --> 00:54:49,790
x and has to be
perpendicular to our sub y.

953
00:54:49,790 --> 00:54:53,045


954
00:54:53,045 --> 00:54:56,240
So, can you have any
guess how in the world

955
00:54:56,240 --> 00:54:59,470
I'm gonna get n vector?

956
00:54:59,470 --> 00:55:01,454
STUDENT: [INAUDIBLE]

957
00:55:01,454 --> 00:55:02,870
PROFESSOR TODA:
[INAUDIBLE] That's

958
00:55:02,870 --> 00:55:05,070
why you need to
know linear algebra

959
00:55:05,070 --> 00:55:09,040
sort of at the same time, but
you guys are making it fine.

960
00:55:09,040 --> 00:55:10,460
It's not a big deal.

961
00:55:10,460 --> 00:55:16,450
You have a matrix, i, j, k
in the front row vectors,

962
00:55:16,450 --> 00:55:21,570
and then you have r sub x that
you gave me, and I erased it.

963
00:55:21,570 --> 00:55:23,605
1, 0, f sub x.

964
00:55:23,605 --> 00:55:26,590


965
00:55:26,590 --> 00:55:29,150
0, 1, f sub y.

966
00:55:29,150 --> 00:55:40,604
And you have exactly 18
seconds to compute this vector.

967
00:55:40,604 --> 00:55:47,627


968
00:55:47,627 --> 00:55:48,460
STUDENT: [INAUDIBLE]

969
00:55:48,460 --> 00:55:52,890


970
00:55:52,890 --> 00:55:55,690
PROFESSOR TODA: You want k, but
I want to leave k at the end

971
00:55:55,690 --> 00:55:58,540
because I always
order my vectors.

972
00:55:58,540 --> 00:56:02,137
Something i plus something
j plus something k.

973
00:56:02,137 --> 00:56:02,970
[INTERPOSING VOICES]

974
00:56:02,970 --> 00:56:05,276


975
00:56:05,276 --> 00:56:06,400
PROFESSOR TODA: Am I right?

976
00:56:06,400 --> 00:56:07,025
Minus f sub x--

977
00:56:07,025 --> 00:56:09,910
STUDENT: Minus f of x plus k.

978
00:56:09,910 --> 00:56:11,892
PROFESSOR TODA: --times i.

979
00:56:11,892 --> 00:56:14,262
For j, do I have to change sign?

980
00:56:14,262 --> 00:56:18,370
Yeah, because 1 plus 2 is odd.

981
00:56:18,370 --> 00:56:21,273
So I go minus 1.

982
00:56:21,273 --> 00:56:22,600
And do it slowly.

983
00:56:22,600 --> 00:56:25,740
You're not gonna make fun of
me; I gotta make fun of you, OK?

984
00:56:25,740 --> 00:56:28,100
And minus 1 times--

985
00:56:28,100 --> 00:56:29,440
STUDENT: Did you forget f y?

986
00:56:29,440 --> 00:56:37,150
PROFESSOR TODA: --f sub y--I go
like that--sub y times j plus

987
00:56:37,150 --> 00:56:39,228
k.

988
00:56:39,228 --> 00:56:42,120
As you said very well
in the most elegant way

989
00:56:42,120 --> 00:56:45,750
without being like yours,
but I say it like this.

990
00:56:45,750 --> 00:56:49,870
So you have minus f
sub x, minus f sub y,

991
00:56:49,870 --> 00:56:54,580
and 1 as a triple with angular
brackets--You love that.

992
00:56:54,580 --> 00:57:00,250
I don't; I like it parentheses
[INAUDIBLE]--equals n.

993
00:57:00,250 --> 00:57:03,485
But n is non-unitary,
but I don't care.

994
00:57:03,485 --> 00:57:04,730
Why don't I care?

995
00:57:04,730 --> 00:57:08,270
I can write the
tangent plane very well

996
00:57:08,270 --> 00:57:13,216
without that n being
unitary, right?

997
00:57:13,216 --> 00:57:14,540
It doesn't matter in the end.

998
00:57:14,540 --> 00:57:17,680
These would be my a, b, c.

999
00:57:17,680 --> 00:57:18,860
Now I know my ABC.

1000
00:57:18,860 --> 00:57:20,400
I know my ABC.

1001
00:57:20,400 --> 00:57:26,315
So, the tangent plane
is your next guess.

1002
00:57:26,315 --> 00:57:30,140
The tangent plane would
be perpendicular to n.

1003
00:57:30,140 --> 00:57:32,150
So this is n.

1004
00:57:32,150 --> 00:57:35,515
The tangent plane passes
through the point p

1005
00:57:35,515 --> 00:57:37,350
and is perpendicular to n.

1006
00:57:37,350 --> 00:57:43,147
So, what is the equation
of the tangent plane?

1007
00:57:43,147 --> 00:57:44,730
STUDENT: Do you want
scalar equations?

1008
00:57:44,730 --> 00:57:49,160
PROFESSOR TODA: A by x minus 0.

1009
00:57:49,160 --> 00:57:50,220
Very good.

1010
00:57:50,220 --> 00:57:56,330
That's exactly what I
wanted you to write.

1011
00:57:56,330 --> 00:58:01,390
All right, so, does
it look familiar?

1012
00:58:01,390 --> 00:58:01,920
Not yet.

1013
00:58:01,920 --> 00:58:02,387
[STUDENT SNEEZES]

1014
00:58:02,387 --> 00:58:02,854
STUDENT: Bless you.

1015
00:58:02,854 --> 00:58:03,757
STUDENT: Bless you.

1016
00:58:03,757 --> 00:58:04,840
PROFESSOR TODA: Bless you.

1017
00:58:04,840 --> 00:58:05,994
Who sneezed?

1018
00:58:05,994 --> 00:58:08,800
OK.

1019
00:58:08,800 --> 00:58:10,370
Am I almost done?

1020
00:58:10,370 --> 00:58:11,700
Well, I am almost done.

1021
00:58:11,700 --> 00:58:14,930
I have to go backwards,
and whatever I get

1022
00:58:14,930 --> 00:58:17,760
I'll put it big here in
a big formula on top.

1023
00:58:17,760 --> 00:58:22,370
I'm gonna say oh, my God.

1024
00:58:22,370 --> 00:58:24,020
No, that's not
what I'm gonna say.

1025
00:58:24,020 --> 00:58:33,330
I'm gonna say minus f sub x at
my point p--that is a, right?

1026
00:58:33,330 --> 00:58:37,089
Times x minus x0.

1027
00:58:37,089 --> 00:58:45,956
Plus minus f sub y at
the point p; that's b.

1028
00:58:45,956 --> 00:58:54,660
y minus y0 plus--c is 1, right?

1029
00:58:54,660 --> 00:58:55,360
c is 1.

1030
00:58:55,360 --> 00:58:58,020
I'm not gonna write
it because if I write

1031
00:58:58,020 --> 00:59:03,990
it you'll want to make fun
of me. z minus z0 equals 0.

1032
00:59:03,990 --> 00:59:08,560
Now it starts looking like
something familiar, finally.

1033
00:59:08,560 --> 00:59:14,961
Now we discovered
that the tangent plane

1034
00:59:14,961 --> 00:59:20,630
can be written as z minus z0.

1035
00:59:20,630 --> 00:59:24,630
I'm keeping the guys z minus
z0 on the left-hand side.

1036
00:59:24,630 --> 00:59:28,630
And these guys are gonna
move to the right-hand side.

1037
00:59:28,630 --> 00:59:33,570
So, I'm gonna have
again, my friend,

1038
00:59:33,570 --> 00:59:45,460
the equation of the tangent
plane for the graph z equals f

1039
00:59:45,460 --> 00:59:46,180
of x,y.

1040
00:59:46,180 --> 00:59:51,940


1041
00:59:51,940 --> 00:59:54,870
But you will say
OK, I think by now

1042
00:59:54,870 --> 00:59:57,340
we've learned these
by heart, we know

1043
00:59:57,340 --> 01:00:00,480
the equation of the tangent
plane, and now we're asleep.

1044
01:00:00,480 --> 01:00:06,160
But what if your surface
would be implicit the way

1045
01:00:06,160 --> 01:00:08,760
you gave it to us at first.

1046
01:00:08,760 --> 01:00:11,840
Maybe you remember the sphere
that was an implicit equation,

1047
01:00:11,840 --> 01:00:14,720
x squared plus x squared
plus x squared equals--

1048
01:00:14,720 --> 01:00:16,030
What do you want it to be?

1049
01:00:16,030 --> 01:00:16,777
STUDENT: 16.

1050
01:00:16,777 --> 01:00:17,610
PROFESSOR TODA: Huh?

1051
01:00:17,610 --> 01:00:18,800
STUDENT: 16.

1052
01:00:18,800 --> 01:00:20,920
PROFESSOR TODA: 16.

1053
01:00:20,920 --> 01:00:22,385
So, radius should be 4.

1054
01:00:22,385 --> 01:00:26,798


1055
01:00:26,798 --> 01:00:31,060
And in such a case, the equation
is of the type f of x, y, z

1056
01:00:31,060 --> 01:00:33,190
equals constant.

1057
01:00:33,190 --> 01:00:35,740
Can we write again the
equation [INAUDIBLE]?

1058
01:00:35,740 --> 01:00:39,770


1059
01:00:39,770 --> 01:00:42,240
Well, you say well,
you just taught

1060
01:00:42,240 --> 01:00:51,240
us some theory that says I have
to think of u and v, but not x

1061
01:00:51,240 --> 01:00:51,850
and y.

1062
01:00:51,850 --> 01:00:55,190
Because if I think of x
and y, what would they be?

1063
01:00:55,190 --> 01:00:57,960
I think the sphere
as being an apple.

1064
01:00:57,960 --> 01:01:01,880
Not an apple, something
you can cut easily.

1065
01:01:01,880 --> 01:01:05,480
Well, an apple, an
orange, something.

1066
01:01:05,480 --> 01:01:07,080
A round piece of soft cheese.

1067
01:01:07,080 --> 01:01:09,510
I started being hungry,
and I'm dreaming.

1068
01:01:09,510 --> 01:01:14,190
So, this is a huge something
you're gonna slice up.

1069
01:01:14,190 --> 01:01:19,100
If you are gonna
do it with x and y,

1070
01:01:19,100 --> 01:01:21,580
the slices would be like this.

1071
01:01:21,580 --> 01:01:24,630
Like that and like this, right?

1072
01:01:24,630 --> 01:01:27,120
And in that case,
your coordinate curves

1073
01:01:27,120 --> 01:01:30,540
are sort of weird.

1074
01:01:30,540 --> 01:01:33,610
If you want to do it in
different coordinates,

1075
01:01:33,610 --> 01:01:35,080
so we want to
change coordinates,

1076
01:01:35,080 --> 01:01:39,810
and those coordinates should
be plotted to the longitude,

1077
01:01:39,810 --> 01:01:43,628
then we cannot use x and y.

1078
01:01:43,628 --> 01:01:44,990
Am I right?

1079
01:01:44,990 --> 01:01:46,590
We cannot use x and y.

1080
01:01:46,590 --> 01:01:50,630
So those u and v will be
different coordinates,

1081
01:01:50,630 --> 01:01:55,160
and then we can do it
like that, latitude.

1082
01:01:55,160 --> 01:01:57,790


1083
01:01:57,790 --> 01:02:00,010
[INAUDIBLE] minus [INAUDIBLE].

1084
01:02:00,010 --> 01:02:00,805
And longitude.

1085
01:02:00,805 --> 01:02:03,080
We are gonna talk about
spherical coordinates

1086
01:02:03,080 --> 01:02:05,202
later, not today.

1087
01:02:05,202 --> 01:02:06,160
Latitude and longitude.

1088
01:02:06,160 --> 01:02:10,340


1089
01:02:10,340 --> 01:02:12,890
1 point extra credit,
because eventually we

1090
01:02:12,890 --> 01:02:16,884
are gonna get
there, chapter 12.7.

1091
01:02:16,884 --> 01:02:20,650
12.7 comes way
after spring break.

1092
01:02:20,650 --> 01:02:27,390
But before we get there, who
is in mechanical engineering

1093
01:02:27,390 --> 01:02:28,830
again?

1094
01:02:28,830 --> 01:02:32,710
You know about Euler's
angles, and stuff like that.

1095
01:02:32,710 --> 01:02:33,550
OK.

1096
01:02:33,550 --> 01:02:40,330
Can you write me
the equations of x

1097
01:02:40,330 --> 01:02:47,850
and y and z of the sphere
with respect to u and v,

1098
01:02:47,850 --> 01:02:51,200
u being latitude and
v being longitude?

1099
01:02:51,200 --> 01:02:53,980


1100
01:02:53,980 --> 01:02:58,641
These have to be
trigonometric functions.

1101
01:02:58,641 --> 01:03:03,860


1102
01:03:03,860 --> 01:03:10,770
In terms of u and v, when u is
latitude and v is longitude.

1103
01:03:10,770 --> 01:03:15,310
1 point extra credit
until a week from today.

1104
01:03:15,310 --> 01:03:16,280
How about that?

1105
01:03:16,280 --> 01:03:20,650


1106
01:03:20,650 --> 01:03:23,850
U and v are latitude
and longitude.

1107
01:03:23,850 --> 01:03:33,800
And express the xyz point in
the ambient space on the sphere.

1108
01:03:33,800 --> 01:03:36,460
x squared plus x squared
plus x squared would be 16.

1109
01:03:36,460 --> 01:03:40,020
So you'll have lots of
cosines and sines [INAUDIBLE]

1110
01:03:40,020 --> 01:03:46,024
of those angles, the latitude
angle and the longitude angle.

1111
01:03:46,024 --> 01:03:49,800
And I would suggest to you that
you take--for the extra credit

1112
01:03:49,800 --> 01:03:54,910
thing--you take the longitude
angle to be from 0 to 2pi,

1113
01:03:54,910 --> 01:04:00,150
from the Greenwich 0 meridian
going back to himself,

1114
01:04:00,150 --> 01:04:07,725
and--well, there are two ways
we do this in mathematics

1115
01:04:07,725 --> 01:04:09,810
because mathematicians
are so diverse.

1116
01:04:09,810 --> 01:04:14,850
Some of us, say, for me,
I measure the latitude

1117
01:04:14,850 --> 01:04:17,100
starting from the North Pole.

1118
01:04:17,100 --> 01:04:20,270
I think that's because we all
believe in Santa or something.

1119
01:04:20,270 --> 01:04:23,440
So, we start measuring
always from the North Pole

1120
01:04:23,440 --> 01:04:27,030
because that's the most
important place on Earth.

1121
01:04:27,030 --> 01:04:35,633
They go 0, pi over 2, and then--
what is our lat--shame on me.

1122
01:04:35,633 --> 01:04:36,480
STUDENT: It's 33.

1123
01:04:36,480 --> 01:04:37,271
PROFESSOR TODA: 33?

1124
01:04:37,271 --> 01:04:39,220
OK.

1125
01:04:39,220 --> 01:04:44,060
Then pi would be the
equator, and then pi

1126
01:04:44,060 --> 01:04:45,834
would be the South Pole.

1127
01:04:45,834 --> 01:04:50,625
But some other mathematicians,
especially biologists

1128
01:04:50,625 --> 01:04:54,530
and differential geometry
people, I'm one of them,

1129
01:04:54,530 --> 01:04:56,090
we go like that.

1130
01:04:56,090 --> 01:05:01,620
Minus pi over 2, South Pole
0, pi over 2 North Pole.

1131
01:05:01,620 --> 01:05:06,820
So we shift that
kind of interval.

1132
01:05:06,820 --> 01:05:10,280
Then for us, the trigonometric
functions of these angles

1133
01:05:10,280 --> 01:05:12,020
would be a little
bit different when we

1134
01:05:12,020 --> 01:05:14,395
do the spherical coordinates.

1135
01:05:14,395 --> 01:05:16,335
OK, that's just extra credit.

1136
01:05:16,335 --> 01:05:19,070
It has nothing to do with
what I'm gonna do right now.

1137
01:05:19,070 --> 01:05:22,960
What I'm gonna do right now
is to pick a point on Earth.

1138
01:05:22,960 --> 01:05:26,000
We have to find Lubbock.

1139
01:05:26,000 --> 01:05:27,210
STUDENT: It's on the left.

1140
01:05:27,210 --> 01:05:28,740
PROFESSOR TODA: Here?

1141
01:05:28,740 --> 01:05:29,870
Is that a good point?

1142
01:05:29,870 --> 01:05:32,400


1143
01:05:32,400 --> 01:05:34,486
This is LBB.

1144
01:05:34,486 --> 01:05:38,430
That's Lubbock
International Airport.

1145
01:05:38,430 --> 01:05:47,530
So, for Lubbock--let's call it
p as well--draw the r sub u,

1146
01:05:47,530 --> 01:05:52,550
r sub v. So, u was latitude.

1147
01:05:52,550 --> 01:05:55,750
So if I fix the latitude,
that means I fix

1148
01:05:55,750 --> 01:05:58,650
the 33 point whatever you said.

1149
01:05:58,650 --> 01:06:00,060
u equals u0.

1150
01:06:00,060 --> 01:06:09,630
It is fixed, so I have u
fixed, and v equals v0 is that.

1151
01:06:09,630 --> 01:06:14,345
I fixed the meridian
where we are.

1152
01:06:14,345 --> 01:06:15,990
What is this tangent vector?

1153
01:06:15,990 --> 01:06:20,518


1154
01:06:20,518 --> 01:06:22,950
To the pink parallel,
the tangent vector

1155
01:06:22,950 --> 01:06:25,660
would be r sub what?

1156
01:06:25,660 --> 01:06:26,160
STUDENT: v.

1157
01:06:26,160 --> 01:06:27,785
PROFESSOR TODA: r
sub v. You are right.

1158
01:06:27,785 --> 01:06:28,920
You've got the idea.

1159
01:06:28,920 --> 01:06:33,370
And the blue vector would
be the partial velocity.

1160
01:06:33,370 --> 01:06:39,466
That's the tangent vector
to the blue meridian,

1161
01:06:39,466 --> 01:06:43,920
which is r sub u.

1162
01:06:43,920 --> 01:06:48,675
And what is n gonna be? n's
gonna be r sub u [INAUDIBLE].

1163
01:06:48,675 --> 01:06:53,370
But is there any other way
to do it in a simpler way

1164
01:06:53,370 --> 01:06:55,515
without you guys going oh, man.

1165
01:06:55,515 --> 01:06:58,055
Suppose some of you don't
wanna do the extra credit

1166
01:06:58,055 --> 01:07:00,332
and then say the
heck with it; I don't

1167
01:07:00,332 --> 01:07:03,610
care about her stinking extra
credit until chapter 12,

1168
01:07:03,610 --> 01:07:07,700
when I have to study the
spherical coordinates,

1169
01:07:07,700 --> 01:07:11,170
and is there another
way to get n.

1170
01:07:11,170 --> 01:07:13,408
I told you another way to get n.

1171
01:07:13,408 --> 01:07:15,384
Well, we are getting there.

1172
01:07:15,384 --> 01:07:21,750
n was the gradient of f
over the length of that.

1173
01:07:21,750 --> 01:07:26,490
And if we want it unitary,
the length of f was what?

1174
01:07:26,490 --> 01:07:31,720
f sub x, f sub y, f
sub z vector, where

1175
01:07:31,720 --> 01:07:36,530
the implicit equation of
the surface was f of x, y, z

1176
01:07:36,530 --> 01:07:38,400
equals c.

1177
01:07:38,400 --> 01:07:40,240
So now we've done this before.

1178
01:07:40,240 --> 01:07:42,470
You say Magdalena, you're
repeating yourself.

1179
01:07:42,470 --> 01:07:47,210
I know I'm repeating myself, but
I want you to learn this twice

1180
01:07:47,210 --> 01:07:49,260
so you can remember it.

1181
01:07:49,260 --> 01:07:52,410
What is f of x, y, z?

1182
01:07:52,410 --> 01:07:56,700
In my case, it's x squared
plus y squared plus z squared

1183
01:07:56,700 --> 01:07:59,930
minus 16, or even nothing.

1184
01:07:59,930 --> 01:08:01,850
Because the constant
doesn't matter anyway

1185
01:08:01,850 --> 01:08:04,434
when I do the gradient.

1186
01:08:04,434 --> 01:08:05,600
You guys are doing homework.

1187
01:08:05,600 --> 01:08:08,210
You saw how the gradient goes.

1188
01:08:08,210 --> 01:08:13,730
So gradient of f would
be 2x times-- and that's

1189
01:08:13,730 --> 01:08:19,384
the partial derivative times i
plus 2y times j plus 2z times

1190
01:08:19,384 --> 01:08:22,964
k-- that's very important.

1191
01:08:22,964 --> 01:08:28,270
[? Lovett ?] has some
coordinates we plug in.

1192
01:08:28,270 --> 01:08:33,500
Now, can we write-- two things.

1193
01:08:33,500 --> 01:08:35,620
I want two things from you.

1194
01:08:35,620 --> 01:08:41,340
Write me a total
differential b tangent plane

1195
01:08:41,340 --> 01:08:46,140
at the point-- so, a, write
the total differential.

1196
01:08:46,140 --> 01:08:50,970


1197
01:08:50,970 --> 01:08:53,670
I'm not going to ask you you
to do a linear approximation.

1198
01:08:53,670 --> 01:08:55,810
I could.

1199
01:08:55,810 --> 01:09:23,660
B, write the tangent plane
to the sphere at the point

1200
01:09:23,660 --> 01:09:25,189
that-- I don't know.

1201
01:09:25,189 --> 01:09:26,870
I don't want one that's trivial.

1202
01:09:26,870 --> 01:09:30,040


1203
01:09:30,040 --> 01:09:37,770
Let's take this 0, square root
of 8, and square root of 8.

1204
01:09:37,770 --> 01:09:39,640
I just have to make
sure that I don't

1205
01:09:39,640 --> 01:09:41,700
come with some
nonsensical point that's

1206
01:09:41,700 --> 01:09:43,290
not going to be on the sphere.

1207
01:09:43,290 --> 01:09:45,863
This will be because I
plugged it in in my mind.

1208
01:09:45,863 --> 01:09:50,229
I get 8 plus 8 is 16 last
time I checked, right?

1209
01:09:50,229 --> 01:09:54,980
So after we do this
we take a break.

1210
01:09:54,980 --> 01:09:58,282
Suppose that this is a
problem on your midterm,

1211
01:09:58,282 --> 01:10:00,742
or on your final or
on your homework,

1212
01:10:00,742 --> 01:10:04,326
or on somebody [? YouTubed it ?]
for a lot of money,

1213
01:10:04,326 --> 01:10:10,010
you asked them, $25 an hour
for me to work that problem.

1214
01:10:10,010 --> 01:10:10,570
That's good.

1215
01:10:10,570 --> 01:10:16,730
I mean-- it's-- it's a
class that you're taking

1216
01:10:16,730 --> 01:10:20,030
for your general requirement
because your school wants you

1217
01:10:20,030 --> 01:10:22,470
to take calc 3.

1218
01:10:22,470 --> 01:10:25,570
But it gives you-- and
I know from experience,

1219
01:10:25,570 --> 01:10:27,670
some of my students came
back to me and said,

1220
01:10:27,670 --> 01:10:30,160
after I took calc
3, I understood it

1221
01:10:30,160 --> 01:10:33,380
so well that I was able to
tutor calc 1, calc 2, calc 3,

1222
01:10:33,380 --> 01:10:35,840
so I got a double job.

1223
01:10:35,840 --> 01:10:38,060
Several hours a week,
the tutoring center,

1224
01:10:38,060 --> 01:10:39,557
math department,
and several hours

1225
01:10:39,557 --> 01:10:40,640
at the [INAUDIBLE] center.

1226
01:10:40,640 --> 01:10:42,670
You know what I'm talking about?

1227
01:10:42,670 --> 01:10:46,220
So I've had students who did
well and ended up liking this,

1228
01:10:46,220 --> 01:10:49,276
and said I can tutor
this in my sleep.

1229
01:10:49,276 --> 01:10:53,760
So-- and also private tutoring
is always a possibility.

1230
01:10:53,760 --> 01:10:55,200
OK.

1231
01:10:55,200 --> 01:10:58,670
Write total differential.

1232
01:10:58,670 --> 01:11:04,260
df equals, and now
I'll say at any point.

1233
01:11:04,260 --> 01:11:06,965
So I don't care what
the value will be.

1234
01:11:06,965 --> 01:11:08,820
I didn't say at what point.

1235
01:11:08,820 --> 01:11:09,923
It means in general.

1236
01:11:09,923 --> 01:11:12,010
Why is that?

1237
01:11:12,010 --> 01:11:14,900
You tell me, you
know that by now.

1238
01:11:14,900 --> 01:11:18,410
2x times what?

1239
01:11:18,410 --> 01:11:20,305
Now, you learned
your lesson, you're

1240
01:11:20,305 --> 01:11:21,980
never gonna make mistakes.

1241
01:11:21,980 --> 01:11:25,490
2y plus 2z dz.

1242
01:11:25,490 --> 01:11:26,450
That is very good.

1243
01:11:26,450 --> 01:11:28,010
That's the total differential.

1244
01:11:28,010 --> 01:11:33,960
Now, what is the equation
of the tangent plane?

1245
01:11:33,960 --> 01:11:37,040
It's not gonna be that.

1246
01:11:37,040 --> 01:11:40,670
Because I'm not
considering a graph.

1247
01:11:40,670 --> 01:11:44,590
I'm considering an
implicitly given surface

1248
01:11:44,590 --> 01:11:52,720
by this implicit equation f of
x, y, z, equals c, your friend.

1249
01:11:52,720 --> 01:11:57,734
So what was, in that case,
the equation of the plane

1250
01:11:57,734 --> 01:11:59,630
written as?

1251
01:11:59,630 --> 01:12:02,480
STUDENT: [INAUDIBLE]

1252
01:12:02,480 --> 01:12:05,130
PROFESSOR TODA: I'm--
yeah, you guys are smart.

1253
01:12:05,130 --> 01:12:06,440
I mean, you are fast.

1254
01:12:06,440 --> 01:12:07,790
Let's do it in general.

1255
01:12:07,790 --> 01:12:11,635
F sub x-- we did that last
time, [INAUDIBLE] times--

1256
01:12:11,635 --> 01:12:14,260
do you guys remember?

1257
01:12:14,260 --> 01:12:16,470
x minus x0.

1258
01:12:16,470 --> 01:12:20,660
And this is at the point plus
big F sub y at the point times

1259
01:12:20,660 --> 01:12:25,590
y minus y0 plus big F sub
z at the point z minus z0.

1260
01:12:25,590 --> 01:12:26,810
This is just review.

1261
01:12:26,810 --> 01:12:27,990
Equals 0.

1262
01:12:27,990 --> 01:12:28,490
Stop.

1263
01:12:28,490 --> 01:12:31,466
Where do these guys come from?

1264
01:12:31,466 --> 01:12:32,954
From the gradient.

1265
01:12:32,954 --> 01:12:34,830
From the gradient.

1266
01:12:34,830 --> 01:12:40,150
Which are the a,b,c, now I
know my ABCs, from the normal.

1267
01:12:40,150 --> 01:12:41,925
My ABCs from the normal.

1268
01:12:41,925 --> 01:12:46,636
So in this case-- I
don't want to erase

1269
01:12:46,636 --> 01:12:49,016
this beautiful picture.

1270
01:12:49,016 --> 01:12:54,910
The last thing I have to do
before the break is-- you

1271
01:12:54,910 --> 01:12:56,900
said 0.

1272
01:12:56,900 --> 01:12:58,960
I'm a lazy person by definition.

1273
01:12:58,960 --> 01:13:02,990
Can you tell me why
you said 0 times?

1274
01:13:02,990 --> 01:13:05,000
STUDENT: Because the
x value is [INAUDIBLE]

1275
01:13:05,000 --> 01:13:07,430
PROFESSOR TODA: You said
2x, plug in and x equals 0

1276
01:13:07,430 --> 01:13:10,050
from your point,
Magdalena, so you don't

1277
01:13:10,050 --> 01:13:12,410
have to write down everything.

1278
01:13:12,410 --> 01:13:19,670
But I'm gonna write down 0
times x minus 0 plus-- what's

1279
01:13:19,670 --> 01:13:20,613
next for me?

1280
01:13:20,613 --> 01:13:21,710
STUDENT: 2 square root 8.

1281
01:13:21,710 --> 01:13:23,630
PROFESSOR TODA: 2y, 2 root 8.

1282
01:13:23,630 --> 01:13:26,187
Is root 8 beautiful?

1283
01:13:26,187 --> 01:13:28,095
It looks like heck.

1284
01:13:28,095 --> 01:13:32,870
At the end I'm gonna
brush it up a little bit.

1285
01:13:32,870 --> 01:13:39,120
This is the partial-- f sub y of
t times y minus-- who is y, z?

1286
01:13:39,120 --> 01:13:40,782
Root 8.

1287
01:13:40,782 --> 01:13:41,734
Do I like it?

1288
01:13:41,734 --> 01:13:43,638
I hate it, but it
doesn't matter.

1289
01:13:43,638 --> 01:13:45,542
Because I'm gonna simplify.

1290
01:13:45,542 --> 01:13:52,470
Plus again, 2 root 8, thank you.

1291
01:13:52,470 --> 01:13:56,710
This is my c guy.

1292
01:13:56,710 --> 01:14:02,440
Times z minus root 8 equals 0.

1293
01:14:02,440 --> 01:14:05,430
I picked another example
from the one from the book,

1294
01:14:05,430 --> 01:14:08,830
because you are gonna
read the book anyway.

1295
01:14:08,830 --> 01:14:11,960
I'm gonna erase that.

1296
01:14:11,960 --> 01:14:14,830
And I'm gonna brush
this up because it

1297
01:14:14,830 --> 01:14:17,490
looks horrible to me.

1298
01:14:17,490 --> 01:14:19,890
Thank God this goes away.

1299
01:14:19,890 --> 01:14:21,980
So the plane will
simply be a combination

1300
01:14:21,980 --> 01:14:24,250
of my y and z in a constant.

1301
01:14:24,250 --> 01:14:28,000
And if I want to
make my life easier,

1302
01:14:28,000 --> 01:14:30,466
I'm gonna divide by what?

1303
01:14:30,466 --> 01:14:32,280
By this.

1304
01:14:32,280 --> 01:14:34,390
So in the end, it
doesn't matter.

1305
01:14:34,390 --> 01:14:35,916
Come on.

1306
01:14:35,916 --> 01:14:42,300
I'll get y minus root 8 plus
c minus root 8 equals 0.

1307
01:14:42,300 --> 01:14:44,020
Do I like it?

1308
01:14:44,020 --> 01:14:44,770
I hate it.

1309
01:14:44,770 --> 01:14:46,660
No, you know, I don't like it.

1310
01:14:46,660 --> 01:14:49,040
Why don't I like it?

1311
01:14:49,040 --> 01:14:50,430
It's not simplified.

1312
01:14:50,430 --> 01:14:56,000
So in any case, if this
were multiple choice,

1313
01:14:56,000 --> 01:14:59,450
it would not be written
like that, right?

1314
01:14:59,450 --> 01:15:03,990
So what would be the
simplified claim in this case?

1315
01:15:03,990 --> 01:15:09,270
The way I would write
it-- a y plus a z minus--

1316
01:15:09,270 --> 01:15:11,486
think, what is root 8?

1317
01:15:11,486 --> 01:15:12,530
STUDENT: 2 root 2.

1318
01:15:12,530 --> 01:15:13,738
PROFESSOR TODA: And 2 root 2.

1319
01:15:13,738 --> 01:15:20,990
And 2 root 2, how
much-- minus 4 root 2.

1320
01:15:20,990 --> 01:15:28,840
And this is how you are expected
to leave this answer boxed.

1321
01:15:28,840 --> 01:15:37,812
This is that tangent
plane at the point.

1322
01:15:37,812 --> 01:15:41,200


1323
01:15:41,200 --> 01:15:42,652
To the sphere.

1324
01:15:42,652 --> 01:15:45,570


1325
01:15:45,570 --> 01:15:48,570
There are programs--
one time I was teaching

1326
01:15:48,570 --> 01:15:53,970
advance geometry, 4331, and one
thing I gave my students to do,

1327
01:15:53,970 --> 01:15:58,920
which was a lot of fun--
using a parametrization,

1328
01:15:58,920 --> 01:16:02,710
plot the entire
sphere with MathLab.

1329
01:16:02,710 --> 01:16:04,060
We did it with MathLab.

1330
01:16:04,060 --> 01:16:06,930
Some people said they know
[INAUDIBLE] I didn't care.

1331
01:16:06,930 --> 01:16:09,326
So MathLab for me
was easier, so we

1332
01:16:09,326 --> 01:16:11,800
plotted the sphere in MathLab.

1333
01:16:11,800 --> 01:16:14,940
We picked a point,
and we drew-- well,

1334
01:16:14,940 --> 01:16:21,960
we drew-- with MathLab we
drew the tangent plane that

1335
01:16:21,960 --> 01:16:25,950
was tangent to the
sphere at that point.

1336
01:16:25,950 --> 01:16:27,220
And they liked it.

1337
01:16:27,220 --> 01:16:29,550
It was-- you know
what this class is,

1338
01:16:29,550 --> 01:16:31,750
is-- if you're math
majors you take it.

1339
01:16:31,750 --> 01:16:34,250
It's called advanced geometries.

1340
01:16:34,250 --> 01:16:35,630
Mainly it's theoretical.

1341
01:16:35,630 --> 01:16:38,540
It teaches you Euclidian
axioms and stuff,

1342
01:16:38,540 --> 01:16:41,540
and then some
non-Euclidian geometries.

1343
01:16:41,540 --> 01:16:45,830
But I thought that I would
do it into an honors class.

1344
01:16:45,830 --> 01:16:49,270
And I put one third of that
last class visualization

1345
01:16:49,270 --> 01:16:50,850
with MathLab of geometry.

1346
01:16:50,850 --> 01:16:54,020
And I think that was what
they liked the most, not so

1347
01:16:54,020 --> 01:16:56,070
much the axiomatic
part and the proofs,

1348
01:16:56,070 --> 01:17:03,270
but the hands-on computation
and visualization in the lab.

1349
01:17:03,270 --> 01:17:04,980
We have this lab, 113.

1350
01:17:04,980 --> 01:17:07,340
We used to have two labs,
but now we are poor,

1351
01:17:07,340 --> 01:17:09,090
we only have one.

1352
01:17:09,090 --> 01:17:10,510
No, we lost the lab.

1353
01:17:10,510 --> 01:17:13,660
The undergraduate
lab-- 009, next to you,

1354
01:17:13,660 --> 01:17:18,560
is lost because-- I used
to each calc 3 there.

1355
01:17:18,560 --> 01:17:21,536
Not because-- that's
not why we lost it.

1356
01:17:21,536 --> 01:17:24,830
We lost it because we-- we
put some 20 graduate students

1357
01:17:24,830 --> 01:17:25,330
there.

1358
01:17:25,330 --> 01:17:26,686
We have no space.

1359
01:17:26,686 --> 01:17:30,810
And we have 130 graduate
students in mathematics.

1360
01:17:30,810 --> 01:17:32,430
Where do you put them?

1361
01:17:32,430 --> 01:17:34,165
We just cram them into cubicles.

1362
01:17:34,165 --> 01:17:37,590
So they made 20 cubicles
here, and they put some,

1363
01:17:37,590 --> 01:17:40,010
so we lost the lab.

1364
01:17:40,010 --> 01:17:41,860
It's sad.

1365
01:17:41,860 --> 01:17:42,730
All right.

1366
01:17:42,730 --> 01:17:45,090
So that's it for now.

1367
01:17:45,090 --> 01:17:47,540
We are gonna take a
short break, and we

1368
01:17:47,540 --> 01:17:52,120
will continue for one more hour,
which is mostly application.

1369
01:17:52,120 --> 01:17:54,660
I'm sort of done with 11.4.

1370
01:17:54,660 --> 01:17:57,838
I'll jump into 11.5 next.

1371
01:17:57,838 --> 01:18:00,832
Take a short break.

1372
01:18:00,832 --> 01:18:02,828
Thanks for the attendance.

1373
01:18:02,828 --> 01:18:04,824
Oh, and you did the calculus.

1374
01:18:04,824 --> 01:18:05,822
Very good.

1375
01:18:05,822 --> 01:19:51,584


1376
01:19:51,584 --> 01:19:55,077
Did this homework give you
a lot of headaches, troubles

1377
01:19:55,077 --> 01:19:56,075
or anything, or not?

1378
01:19:56,075 --> 01:19:57,572
Not too much?

1379
01:19:57,572 --> 01:19:59,069
It's a long homework.

1380
01:19:59,069 --> 01:20:00,566
49 problems-- 42 problems.

1381
01:20:00,566 --> 01:20:05,556


1382
01:20:05,556 --> 01:20:07,053
It wasn't bad?

1383
01:20:07,053 --> 01:22:39,080


1384
01:22:39,080 --> 01:22:45,831
OK, questions from the-- what
was it, the first part-- mainly

1385
01:22:45,831 --> 01:22:47,620
the first part of chapter 11.

1386
01:22:47,620 --> 01:22:49,510
This is where we are.

1387
01:22:49,510 --> 01:22:56,690
Right now we hit the
half point because 11.8

1388
01:22:56,690 --> 01:22:59,250
is the last section.

1389
01:22:59,250 --> 01:23:03,310
And we will do that, that's
Lagrange multipliers.

1390
01:23:03,310 --> 01:23:06,900
So, let's do a little
bit of a review.

1391
01:23:06,900 --> 01:23:08,836
Questions about homework.

1392
01:23:08,836 --> 01:23:11,160
Do you have them?

1393
01:23:11,160 --> 01:23:13,990
Imagine this would
be office hour.

1394
01:23:13,990 --> 01:23:15,116
What would you ask?

1395
01:23:15,116 --> 01:23:17,804


1396
01:23:17,804 --> 01:23:19,510
STUDENT: I know it's
a stupid question,

1397
01:23:19,510 --> 01:23:22,210
but my visualization [INAUDIBLE]
coming along, and question

1398
01:23:22,210 --> 01:23:26,910
three about the sphere passing
the plane and passing the line.

1399
01:23:26,910 --> 01:23:31,590
So you have a 3, 5,
and 4 x, y, and z,

1400
01:23:31,590 --> 01:23:34,335
and you have a radius of 5.

1401
01:23:34,335 --> 01:23:36,275
Is it passing the x, y plane?

1402
01:23:36,275 --> 01:23:40,710
Is it passing [INAUDIBLE]
x plane and [INAUDIBLE]

1403
01:23:40,710 --> 01:23:42,220
passing the other plane.

1404
01:23:42,220 --> 01:23:44,010
PROFESSOR TODA: So-- say again.

1405
01:23:44,010 --> 01:23:46,050
So you have 3 and 4 and 5--

1406
01:23:46,050 --> 01:23:47,520
STUDENT: x minus-- yes.

1407
01:23:47,520 --> 01:23:49,380
PROFESSOR TODA: What
are the coordinates?

1408
01:23:49,380 --> 01:23:50,691
STUDENT: 3, 4, and 5.

1409
01:23:50,691 --> 01:23:53,410
PROFESSOR TODA: 3, 4, and
5, just as you said them.

1410
01:23:53,410 --> 01:23:54,190
You can--

1411
01:23:54,190 --> 01:23:55,730
STUDENT: And the radius is 5.

1412
01:23:55,730 --> 01:23:56,840
PROFESSOR TODA: Radius of?

1413
01:23:56,840 --> 01:23:57,340
STUDENT: 5.

1414
01:23:57,340 --> 01:23:59,640
Radius is equal to 5.

1415
01:23:59,640 --> 01:24:00,600
[INAUDIBLE]

1416
01:24:00,600 --> 01:24:02,110
PROFESSOR TODA: Yeah, well, OK.

1417
01:24:02,110 --> 01:24:07,779
So assume you have a
sphere of radius 5, which

1418
01:24:07,779 --> 01:24:09,270
means you have 25.

1419
01:24:09,270 --> 01:24:14,710
If you do the 3 squared plus
4 squared plus 5 squared,

1420
01:24:14,710 --> 01:24:16,465
what is that?

1421
01:24:16,465 --> 01:24:17,090
For this point.

1422
01:24:17,090 --> 01:24:18,750
You have two separate points.

1423
01:24:18,750 --> 01:24:22,919
For this point you
have 25 plus 25.

1424
01:24:22,919 --> 01:24:24,875
Are you guys with me?

1425
01:24:24,875 --> 01:24:30,254
So you have the
specific x0, y0, z0.

1426
01:24:30,254 --> 01:24:39,060
You do the sum of the
squares, and you get 50.

1427
01:24:39,060 --> 01:24:43,900
My question is, is this point
outside, inside the sphere

1428
01:24:43,900 --> 01:24:45,370
or on the sphere?

1429
01:24:45,370 --> 01:24:47,010
On the sphere,
obviously, it's not,

1430
01:24:47,010 --> 01:24:54,140
because it does not verify the
equation of the sphere, right?

1431
01:24:54,140 --> 01:24:59,140
STUDENT: [INAUDIBLE] those the
location of the center point.

1432
01:24:59,140 --> 01:25:01,307
STUDENT: Where's the
center of the sphere?

1433
01:25:01,307 --> 01:25:02,140
STUDENT: [INAUDIBLE]

1434
01:25:02,140 --> 01:25:05,640


1435
01:25:05,640 --> 01:25:09,140
PROFESSOR TODA: The center
of the sphere would be at 0.

1436
01:25:09,140 --> 01:25:11,640
STUDENT: [INAUDIBLE]

1437
01:25:11,640 --> 01:25:13,520
PROFESSOR TODA: We are
making up a question.

1438
01:25:13,520 --> 01:25:14,730
So, right?

1439
01:25:14,730 --> 01:25:16,784
So practically, I am
making up a question.

1440
01:25:16,784 --> 01:25:17,450
STUDENT: Oh, OK.

1441
01:25:17,450 --> 01:25:22,930
PROFESSOR TODA: So I'm saying if
you have a sphere of radius 5,

1442
01:25:22,930 --> 01:25:27,175
and somebody gives you this
point of coordinates 3, 4,

1443
01:25:27,175 --> 01:25:29,240
and 5, where is the point?

1444
01:25:29,240 --> 01:25:34,934
Is it inside the sphere, outside
the sphere or on the sphere?

1445
01:25:34,934 --> 01:25:37,100
On the sphere it cannot be
because it doesn't verify

1446
01:25:37,100 --> 01:25:39,776
the sphere.

1447
01:25:39,776 --> 01:25:44,580
Ah, it looks like a Mr. Egg.

1448
01:25:44,580 --> 01:25:47,280
I don't like it.

1449
01:25:47,280 --> 01:25:50,610
I'm sorry, it's a sphere.

1450
01:25:50,610 --> 01:25:54,880
So a point on a sphere that
will have-- that's a hint.

1451
01:25:54,880 --> 01:25:58,470
A point on a sphere that
will have coordinates 3 and 4

1452
01:25:58,470 --> 01:26:02,490
would be exactly 3, 4, and 0.

1453
01:26:02,490 --> 01:26:05,960
So it would be where?

1454
01:26:05,960 --> 01:26:07,760
STUDENT: 16, 4.

1455
01:26:07,760 --> 01:26:11,480
PROFESSOR TODA: 3 squared plus
4 squared is 5 squared, right?

1456
01:26:11,480 --> 01:26:13,300
So those are
Pythagorean numbers.

1457
01:26:13,300 --> 01:26:15,162
That's the beauty of them.

1458
01:26:15,162 --> 01:26:22,602


1459
01:26:22,602 --> 01:26:27,570
I'm trying to draw well.

1460
01:26:27,570 --> 01:26:28,410
Right.

1461
01:26:28,410 --> 01:26:29,980
This is the point a.

1462
01:26:29,980 --> 01:26:33,235


1463
01:26:33,235 --> 01:26:36,620
You go up how many?

1464
01:26:36,620 --> 01:26:38,940
You shift by 5.

1465
01:26:38,940 --> 01:26:41,329
So are you inside or outside?

1466
01:26:41,329 --> 01:26:42,307
STUDENT: Outside.

1467
01:26:42,307 --> 01:26:43,182
PROFESSOR TODA: Yeah.

1468
01:26:43,182 --> 01:26:50,131


1469
01:26:50,131 --> 01:26:55,020
STUDENT: Are you outside
or are you exactly on-- oh.

1470
01:26:55,020 --> 01:26:55,770
Sorry, I thought--

1471
01:26:55,770 --> 01:26:56,400
PROFESSOR TODA: You go--

1472
01:26:56,400 --> 01:26:58,286
STUDENT: I thought you
were saying point a.

1473
01:26:58,286 --> 01:26:59,960
Point a is like
exactly-- [INAUDIBLE]

1474
01:26:59,960 --> 01:27:00,810
PROFESSOR TODA: You
are on the equator,

1475
01:27:00,810 --> 01:27:02,260
and from the Equator
of the Earth,

1476
01:27:02,260 --> 01:27:05,750
you're going parallel to the
z-axis, then you stay outside.

1477
01:27:05,750 --> 01:27:08,570
But the question is
more subtle than that.

1478
01:27:08,570 --> 01:27:12,000
This is pretty--
you figured it out.

1479
01:27:12,000 --> 01:27:15,310
1 point-- 0.5 extra credit.

1480
01:27:15,310 --> 01:27:18,580
That we don't have--
I wish we had-- maybe

1481
01:27:18,580 --> 01:27:19,900
we'll find some time.

1482
01:27:19,900 --> 01:27:23,030
When I-- when we rewrite the
book, maybe we should do that.

1483
01:27:23,030 --> 01:27:38,636
So express the points outside
the sphere, inside the sphere,

1484
01:27:38,636 --> 01:27:50,210
and on the sphere
using exclusively

1485
01:27:50,210 --> 01:27:51,636
equalities and inequalities.

1486
01:27:51,636 --> 01:27:57,900


1487
01:27:57,900 --> 01:27:58,900
And that's extra credit.

1488
01:27:58,900 --> 01:28:01,000
So, of course, the
[INAUDIBLE] is obvious.

1489
01:28:01,000 --> 01:28:06,800
The sphere is the set of
the triples x, y, z in R3.

1490
01:28:06,800 --> 01:28:09,680


1491
01:28:09,680 --> 01:28:13,480
OK, I'm teaching you a little
bit of mathematical language.

1492
01:28:13,480 --> 01:28:19,560
x, y, z belongs to R3,
R3 being the free space,

1493
01:28:19,560 --> 01:28:23,810
with the property that x squared
plus y squared plus z squared

1494
01:28:23,810 --> 01:28:26,720
equals given a squared.

1495
01:28:26,720 --> 01:28:29,840
What if you have less than,
what if you have greater than?

1496
01:28:29,840 --> 01:28:31,836
Ah, shut up, Magdalena.

1497
01:28:31,836 --> 01:28:33,400
This is all up to you.

1498
01:28:33,400 --> 01:28:35,910
You will figure
out how the points

1499
01:28:35,910 --> 01:28:40,920
on the outside and the points
on the inside are characterized.

1500
01:28:40,920 --> 01:28:47,060
And unfortunately we don't
emphasize that in the textbook.

1501
01:28:47,060 --> 01:28:49,510
I'll erase.

1502
01:28:49,510 --> 01:28:51,682
You figured it out.

1503
01:28:51,682 --> 01:28:53,265
And now I want to
move on to something

1504
01:28:53,265 --> 01:28:57,056
a little bit challenging,
but not very challenging.

1505
01:28:57,056 --> 01:29:11,203


1506
01:29:11,203 --> 01:29:12,494
STUDENT: Professor, [INAUDIBLE]

1507
01:29:12,494 --> 01:29:19,830


1508
01:29:19,830 --> 01:29:21,330
PROFESSOR TODA: The
last requirement

1509
01:29:21,330 --> 01:29:22,630
on the extra credit?

1510
01:29:22,630 --> 01:29:26,660
So I said the sphere
represents the set of all

1511
01:29:26,660 --> 01:29:29,532
triples x, y, z in
R3 with the property

1512
01:29:29,532 --> 01:29:31,990
that x squared plus y squared
plus y squared plus z squared

1513
01:29:31,990 --> 01:29:33,880
equals a squared.

1514
01:29:33,880 --> 01:29:36,840
With the equality sign.

1515
01:29:36,840 --> 01:29:40,020
Represent the points on
the inside of the sphere

1516
01:29:40,020 --> 01:29:44,560
and the outside of the sphere
using just inequalities.

1517
01:29:44,560 --> 01:29:45,280
Mathematics.

1518
01:29:45,280 --> 01:29:48,710
No writing, no words,
just mathematics.

1519
01:29:48,710 --> 01:29:50,150
In set theory symbols.

1520
01:29:50,150 --> 01:29:54,824
Like, the set of points
with braces like that.

1521
01:29:54,824 --> 01:29:57,680
OK.

1522
01:29:57,680 --> 01:30:02,620
I'll help you review a little
bit of stuff from the chain

1523
01:30:02,620 --> 01:30:12,150
rule in-- in chapter--
I don't know, guys,

1524
01:30:12,150 --> 01:30:14,770
it was a long time ago.

1525
01:30:14,770 --> 01:30:15,730
Shame on me.

1526
01:30:15,730 --> 01:30:19,320
Chapter 3, calc 1.

1527
01:30:19,320 --> 01:30:38,180
Versus chain rule rules in
calc in-- chapter 5 calc 3.

1528
01:30:38,180 --> 01:30:40,550
This is a little
bit of a warmup.

1529
01:30:40,550 --> 01:30:42,325
I don't want to
[INAUDIBLE] again

1530
01:30:42,325 --> 01:30:44,330
next time when we
meet on Thursday.

1531
01:30:44,330 --> 01:30:45,990
Bless you.

1532
01:30:45,990 --> 01:30:48,924
The bless you was
out of the context.

1533
01:30:48,924 --> 01:30:51,580
What was the chain rule?

1534
01:30:51,580 --> 01:30:53,670
We did compositions
of functions,

1535
01:30:53,670 --> 01:31:01,090
and we had a diagram that we
don't show you, but we should.

1536
01:31:01,090 --> 01:31:05,050
There is practically a function
that comes from a set A

1537
01:31:05,050 --> 01:31:08,490
to a set B to a set
C. These are the sets.

1538
01:31:08,490 --> 01:31:12,760
And we have g and an f.

1539
01:31:12,760 --> 01:31:17,480
And we have g of f of t.

1540
01:31:17,480 --> 01:31:22,450
t is your favorite letter here.

1541
01:31:22,450 --> 01:31:26,790
How do you do the
derivative with respect

1542
01:31:26,790 --> 01:31:28,940
to g composed with f?

1543
01:31:28,940 --> 01:31:32,920


1544
01:31:32,920 --> 01:31:36,850
I asked the same question to
my Calc 1 and Calc 2 students,

1545
01:31:36,850 --> 01:31:42,470
and they really had a hard
time expressing themselves,

1546
01:31:42,470 --> 01:31:44,710
expressing the chain rule.

1547
01:31:44,710 --> 01:31:46,530
And when I gave them
an example, they

1548
01:31:46,530 --> 01:31:49,600
said, oh, I know how to
do it on the example.

1549
01:31:49,600 --> 01:31:55,090
I just don't know how to do it
on the-- I like the numbers,

1550
01:31:55,090 --> 01:31:57,510
but I don't like them letters.

1551
01:31:57,510 --> 01:32:02,345
So how do we do
it in an example?

1552
01:32:02,345 --> 01:32:05,340


1553
01:32:05,340 --> 01:32:09,140
I chose natural log,
which you find everywhere.

1554
01:32:09,140 --> 01:32:14,442
So how do you do d
dt of this animal?

1555
01:32:14,442 --> 01:32:15,888
It's an animal.

1556
01:32:15,888 --> 01:32:18,310
STUDENT: [INAUDIBLE]

1557
01:32:18,310 --> 01:32:21,490
PROFESSOR TODA: So the idea
is you go from the outside

1558
01:32:21,490 --> 01:32:23,140
to the inside, one at a time.

1559
01:32:23,140 --> 01:32:24,680
My students know that.

1560
01:32:24,680 --> 01:32:27,480
You prime the function,
the outer function,

1561
01:32:27,480 --> 01:32:30,572
the last one you applied,
to the function inside.

1562
01:32:30,572 --> 01:32:33,770
And you prime that with
respect to the argument.

1563
01:32:33,770 --> 01:32:37,040
This is called the
argument in that case.

1564
01:32:37,040 --> 01:32:40,694
Derivative of natural
log is 1 over what?

1565
01:32:40,694 --> 01:32:43,610
The argument.

1566
01:32:43,610 --> 01:32:46,315
And you cover up natural
log with your hand,

1567
01:32:46,315 --> 01:32:47,170
and you keep going.

1568
01:32:47,170 --> 01:32:51,868
And you say, next I go,
times the derivative

1569
01:32:51,868 --> 01:32:55,700
of this square, plus 1,
prime with respect to t.

1570
01:32:55,700 --> 01:32:58,460
So I go times 2t.

1571
01:32:58,460 --> 01:33:01,290
And that's what we have.

1572
01:33:01,290 --> 01:33:04,620
And they say, when you explain
it like that, they said to me,

1573
01:33:04,620 --> 01:33:06,350
I can understand it.

1574
01:33:06,350 --> 01:33:09,050
But I'm having a
problem understanding it

1575
01:33:09,050 --> 01:33:12,960
when you express this diagram--
that it throws me off.

1576
01:33:12,960 --> 01:33:19,242
So in order to avoid that kind
of theoretical misconception,

1577
01:33:19,242 --> 01:33:24,990
I'm saying, let us see
what the heck this is.

1578
01:33:24,990 --> 01:33:32,805
d dt of g of f of t, because
this is what you're doing,

1579
01:33:32,805 --> 01:33:34,680
has to have some understanding.

1580
01:33:34,680 --> 01:33:38,615
The problem is that Mister
f of t, that lives here,

1581
01:33:38,615 --> 01:33:40,370
has a different argument.

1582
01:33:40,370 --> 01:33:45,410
The letter in B should
be, let's say, u.

1583
01:33:45,410 --> 01:33:48,510


1584
01:33:48,510 --> 01:33:51,684
That doesn't say
anything practically.

1585
01:33:51,684 --> 01:33:54,000
How do you differentiate
with respect to what?

1586
01:33:54,000 --> 01:33:56,240
You cannot say d dt here.

1587
01:33:56,240 --> 01:34:00,940
So you have to call f
of t something generic.

1588
01:34:00,940 --> 01:34:05,210
You have to have a
generic variable for that.

1589
01:34:05,210 --> 01:34:13,690
So you have then dg du, at
what specific value of u?

1590
01:34:13,690 --> 01:34:18,050
At the specific value of
u that we have as f of t.

1591
01:34:18,050 --> 01:34:21,570
Do you understand the
specificity of this?

1592
01:34:21,570 --> 01:34:26,700
Times-- that's the chain
rule, the product coming

1593
01:34:26,700 --> 01:34:31,764
from the chain rule-- df pt.

1594
01:34:31,764 --> 01:34:33,890
You take du dt or d of dt.

1595
01:34:33,890 --> 01:34:34,940
It is the same thing.

1596
01:34:34,940 --> 01:34:36,884
Say it again, df dt.

1597
01:34:36,884 --> 01:34:41,260


1598
01:34:41,260 --> 01:34:43,540
I had a student ask me,
what if I put du dt?

1599
01:34:43,540 --> 01:34:44,790
Would it be wrong?

1600
01:34:44,790 --> 01:34:50,079
No, as long as you understand
that u is a-something,

1601
01:34:50,079 --> 01:34:54,867
as the image of this t.

1602
01:34:54,867 --> 01:34:55,950
Do you know what he liked?

1603
01:34:55,950 --> 01:34:58,935


1604
01:34:58,935 --> 01:35:01,790
He said, do you know
what I like about that?

1605
01:35:01,790 --> 01:35:07,165
I like that I can imagine
that these are two cowboys-- I

1606
01:35:07,165 --> 01:35:09,450
told the same thing to my son.

1607
01:35:09,450 --> 01:35:12,510
He was so excited,
not about that,

1608
01:35:12,510 --> 01:35:14,596
but about these two cowboys.

1609
01:35:14,596 --> 01:35:17,329
Of course, he is 10.

1610
01:35:17,329 --> 01:35:18,245
These are the cowboys.

1611
01:35:18,245 --> 01:35:20,420
They are across.

1612
01:35:20,420 --> 01:35:22,640
One is on top of
the building there,

1613
01:35:22,640 --> 01:35:24,850
shooting at this
guy, who is here

1614
01:35:24,850 --> 01:35:28,480
across the street on the bottom.

1615
01:35:28,480 --> 01:35:31,390
So they are
annihilating each other.

1616
01:35:31,390 --> 01:35:33,290
They shoot and they die.

1617
01:35:33,290 --> 01:35:37,080
And they die, and
you're left with 1/3.

1618
01:35:37,080 --> 01:35:41,780
The same idea is that, actually,
these guys do not simplify.

1619
01:35:41,780 --> 01:35:46,100
du and-- [? du, ?] they're not
cowboys who shoot at each other

1620
01:35:46,100 --> 01:35:48,580
at the same time and both
die at the same time.

1621
01:35:48,580 --> 01:35:53,220
It is not so romantic.

1622
01:35:53,220 --> 01:35:59,640
But the idea of remembering
this formula is the same.

1623
01:35:59,640 --> 01:36:03,700
Because practically, if you want
to annihilate the two cowboys

1624
01:36:03,700 --> 01:36:06,330
and put your hands over them
so you don't see them anymore,

1625
01:36:06,330 --> 01:36:10,580
du dt, you would
have to remember, oh,

1626
01:36:10,580 --> 01:36:12,430
so that was the
derivative with respect

1627
01:36:12,430 --> 01:36:15,930
to t that I initially
have of the guy on top,

1628
01:36:15,930 --> 01:36:19,110
which was g of f of
the composed function.

1629
01:36:19,110 --> 01:36:22,850
So if you view g of f of t
as the composed function,

1630
01:36:22,850 --> 01:36:23,838
who is that?

1631
01:36:23,838 --> 01:36:28,980
The composition g
composed with f of t

1632
01:36:28,980 --> 01:36:31,900
is the function g of f of t.

1633
01:36:31,900 --> 01:36:34,940
This is the function that
you want to differentiate

1634
01:36:34,940 --> 01:36:37,370
with respect to time, t.

1635
01:36:37,370 --> 01:36:40,980
This is this, prime
with respect to t.

1636
01:36:40,980 --> 01:36:46,024
It's like they would be killing
each other, and you would die.

1637
01:36:46,024 --> 01:36:48,200
And I liked this
idea, and I said,

1638
01:36:48,200 --> 01:36:50,390
I should tell that to my
students and to my son.

1639
01:36:50,390 --> 01:36:52,925
And, of course, my son
started jumping around

1640
01:36:52,925 --> 01:36:56,485
and said that he understands
multiplication of fractions

1641
01:36:56,485 --> 01:36:57,890
better now.

1642
01:36:57,890 --> 01:37:01,390
They don't learn about
simplifications-- I don't

1643
01:37:01,390 --> 01:37:03,042
know how they teach these kids.

1644
01:37:03,042 --> 01:37:06,320


1645
01:37:06,320 --> 01:37:07,830
It became so complicated.

1646
01:37:07,830 --> 01:37:10,900
It's as if mathematics--
mathematics is the same.

1647
01:37:10,900 --> 01:37:12,110
It hasn't changed.

1648
01:37:12,110 --> 01:37:14,310
It's the people
who make the rules

1649
01:37:14,310 --> 01:37:17,415
on how to teach it that change.

1650
01:37:17,415 --> 01:37:21,530
So he simply doesn't see
that this simplifies.

1651
01:37:21,530 --> 01:37:24,690
And when I tell him simplify,
he's like, what is simplify?

1652
01:37:24,690 --> 01:37:25,850
What is this word simplify?

1653
01:37:25,850 --> 01:37:27,235
My teacher doesn't use it.

1654
01:37:27,235 --> 01:37:31,705
So I feel like sometimes
I want to shoot myself.

1655
01:37:31,705 --> 01:37:35,380
But he went over that and
he understood about the idea

1656
01:37:35,380 --> 01:37:37,420
of simplification.

1657
01:37:37,420 --> 01:37:39,370
[? He ?] composing
something on top

1658
01:37:39,370 --> 01:37:43,190
and the bottom finding the
common factors up and down,

1659
01:37:43,190 --> 01:37:44,820
crossing them out, and so on.

1660
01:37:44,820 --> 01:37:47,260
And so now he knows
what it means.

1661
01:37:47,260 --> 01:37:50,576
But imagine going to
college without having

1662
01:37:50,576 --> 01:37:51,450
this early knowledge.

1663
01:37:51,450 --> 01:37:55,132
You come to college,
you were good in school,

1664
01:37:55,132 --> 01:37:57,090
and you've never learned
enough simplification.

1665
01:37:57,090 --> 01:38:00,220
And then somebody like me,
and tells you simplification.

1666
01:38:00,220 --> 01:38:03,010
You say, she is a foreigner.

1667
01:38:03,010 --> 01:38:07,770
She has a language barrier
that is [INAUDIBLE] she has

1668
01:38:07,770 --> 01:38:10,050
that I've never heard before.

1669
01:38:10,050 --> 01:38:15,250
So I wish the people who
really re-conceive, re-write

1670
01:38:15,250 --> 01:38:18,820
the curriculum for K12
would be a little bit

1671
01:38:18,820 --> 01:38:21,730
more respectful of the history.

1672
01:38:21,730 --> 01:38:25,590
Imagine that I
would teach calculus

1673
01:38:25,590 --> 01:38:28,813
without ever telling you
anything about Leibniz, who

1674
01:38:28,813 --> 01:38:31,200
was Leibniz, he doesn't exist.

1675
01:38:31,200 --> 01:38:34,100
Or Euler, or one
of these fathers.

1676
01:38:34,100 --> 01:38:37,660
They are the ones who
created these notations.

1677
01:38:37,660 --> 01:38:42,630
And if we never tell you
about them, that I guess,

1678
01:38:42,630 --> 01:38:47,400
wherever they are, it is an
injustice that we are doing.

1679
01:38:47,400 --> 01:38:48,180
All right.

1680
01:38:48,180 --> 01:38:53,520
Chain rule in
Chapter 5 of Calc 3.

1681
01:38:53,520 --> 01:38:56,310
This is a little bit
more complicated,

1682
01:38:56,310 --> 01:38:59,690
but I'm going to teach it
to you because I like it.

1683
01:38:59,690 --> 01:39:05,582
Imagine that you have z equals
x squared plus y squared.

1684
01:39:05,582 --> 01:39:06,570
What is that?

1685
01:39:06,570 --> 01:39:08,052
It's an example of a graph.

1686
01:39:08,052 --> 01:39:10,614
And I just taught
you what a graph is.

1687
01:39:10,614 --> 01:39:13,490


1688
01:39:13,490 --> 01:39:22,939
But imagine that
xy follow a curve.

1689
01:39:22,939 --> 01:39:25,900


1690
01:39:25,900 --> 01:39:27,884
[INAUDIBLE] with
respect to time.

1691
01:39:27,884 --> 01:39:37,780


1692
01:39:37,780 --> 01:39:41,330
And you will say, Magdalena,
can you draw that?

1693
01:39:41,330 --> 01:39:45,670
What in the world do you mean
that x and y follow a curve?

1694
01:39:45,670 --> 01:39:46,770
I'll try to draw it.

1695
01:39:46,770 --> 01:39:48,640
First of all, you are on a walk.

1696
01:39:48,640 --> 01:39:50,480
You are in a beautiful valley.

1697
01:39:50,480 --> 01:39:51,480
It's not a vase.

1698
01:39:51,480 --> 01:39:57,088
It's a circular
paraboloid, as an example.

1699
01:39:57,088 --> 01:40:00,944


1700
01:40:00,944 --> 01:40:01,908
It's like an egg shell.

1701
01:40:01,908 --> 01:40:05,290


1702
01:40:05,290 --> 01:40:07,050
You have a curve on that.

1703
01:40:07,050 --> 01:40:08,050
You draw that.

1704
01:40:08,050 --> 01:40:10,340
You have nothing better
to do than decorating eggs

1705
01:40:10,340 --> 01:40:10,960
for Easter.

1706
01:40:10,960 --> 01:40:12,190
Hey, wait.

1707
01:40:12,190 --> 01:40:14,695
Easter is far, far away.

1708
01:40:14,695 --> 01:40:17,355
But let's say you want to
decorate eggs for Easter.

1709
01:40:17,355 --> 01:40:22,800
You take some color of paint
and put paint on the egg.

1710
01:40:22,800 --> 01:40:28,400
You are actually describing
an arc of a curve.

1711
01:40:28,400 --> 01:40:38,240
And x and y, their
projection on the floor

1712
01:40:38,240 --> 01:40:39,605
will be x of t, y of t.

1713
01:40:39,605 --> 01:40:42,790


1714
01:40:42,790 --> 01:40:45,470
Because you paint in time.

1715
01:40:45,470 --> 01:40:46,220
You paint in time.

1716
01:40:46,220 --> 01:40:48,090
You describe this in time.

1717
01:40:48,090 --> 01:40:54,090
Now, if x of ty of t is
being projected on the floor.

1718
01:40:54,090 --> 01:40:58,820
Of course, you have a curve
here as well, which is what?

1719
01:40:58,820 --> 01:41:05,700
Which it will be x
of t, y of t, z of t.

1720
01:41:05,700 --> 01:41:06,620
Oh, my god.

1721
01:41:06,620 --> 01:41:11,910
Yes, because the altitude also
depends on the motion in time.

1722
01:41:11,910 --> 01:41:13,810
All right.

1723
01:41:13,810 --> 01:41:16,460
So what's missing here?

1724
01:41:16,460 --> 01:41:18,880
It's missing the third
coordinate, duh, that's

1725
01:41:18,880 --> 01:41:21,382
0 because I'm on the floor.

1726
01:41:21,382 --> 01:41:26,500
I'm on the xy plane, which
is the floor z equals z.

1727
01:41:26,500 --> 01:41:28,560
But now let's
suppose that I want

1728
01:41:28,560 --> 01:41:36,570
to say this is f of x and y,
and I want to differentiate

1729
01:41:36,570 --> 01:41:39,400
f with respect to t.

1730
01:41:39,400 --> 01:41:40,730
And you go, say what?

1731
01:41:40,730 --> 01:41:41,440
Oh, my god.

1732
01:41:41,440 --> 01:41:42,480
What is that?

1733
01:41:42,480 --> 01:41:45,840
I differentiate f
with respect to t.

1734
01:41:45,840 --> 01:41:48,730
By differentiating
f with respect to t,

1735
01:41:48,730 --> 01:41:54,780
I mean that I have f of
x and y differentiated

1736
01:41:54,780 --> 01:41:56,145
with respect to t.

1737
01:41:56,145 --> 01:41:58,070
And you say, wait, Magdalena.

1738
01:41:58,070 --> 01:41:59,820
This doesn't make any sense.

1739
01:41:59,820 --> 01:42:03,640
And you would be right to say
it doesn't make any sense.

1740
01:42:03,640 --> 01:42:07,280
Can somebody tell me why
it doesn't make any sense?

1741
01:42:07,280 --> 01:42:13,580
It's not clear where in the
world the variable t is inside.

1742
01:42:13,580 --> 01:42:17,220
So I'm going to say, OK,
x are themselves functions

1743
01:42:17,220 --> 01:42:19,500
of t, functions of that.

1744
01:42:19,500 --> 01:42:21,342
x of t, y of t.

1745
01:42:21,342 --> 01:42:23,950
If I don't do that,
it's not clear.

1746
01:42:23,950 --> 01:42:27,722
So this is a composed
function just like this one.

1747
01:42:27,722 --> 01:42:28,680
Look at the similarity.

1748
01:42:28,680 --> 01:42:31,100
It's really beautiful.

1749
01:42:31,100 --> 01:42:35,670
This is a function of
a function, g of f.

1750
01:42:35,670 --> 01:42:38,520
This is a function
of two functions.

1751
01:42:38,520 --> 01:42:43,304
Say it again, f is a function
of two functions, x and y.

1752
01:42:43,304 --> 01:42:45,232
This was a function
of a function of t.

1753
01:42:45,232 --> 01:42:47,642
This was a function
of two functions of t.

1754
01:42:47,642 --> 01:42:48,606
Oh, my God.

1755
01:42:48,606 --> 01:42:52,470


1756
01:42:52,470 --> 01:42:55,080
How do we compute this?

1757
01:42:55,080 --> 01:42:56,851
There is a rule.

1758
01:42:56,851 --> 01:42:58,204
It can be proved.

1759
01:42:58,204 --> 01:43:01,950
We will look a little bit into
the theoretical justification

1760
01:43:01,950 --> 01:43:03,403
of this proof later.

1761
01:43:03,403 --> 01:43:05,720
But practically what
you do, you say,

1762
01:43:05,720 --> 01:43:07,955
I have to have some
order in my life.

1763
01:43:07,955 --> 01:43:09,090
OK.?

1764
01:43:09,090 --> 01:43:12,880
So the way we do that,
we differentiate first

1765
01:43:12,880 --> 01:43:17,150
with respect to the first
location, which is x.

1766
01:43:17,150 --> 01:43:21,515
I go there, but I cannot write
df dx because f is a mother

1767
01:43:21,515 --> 01:43:23,110
of two babies.

1768
01:43:23,110 --> 01:43:26,520
f is a function of two
variables, x and y.

1769
01:43:26,520 --> 01:43:28,800
She has to be a mother
to both of them;

1770
01:43:28,800 --> 01:43:31,620
otherwise, they get
jealous of one another.

1771
01:43:31,620 --> 01:43:37,630
So I have to say, partial
of f with respect to x,

1772
01:43:37,630 --> 01:43:38,860
I cannot use d.

1773
01:43:38,860 --> 01:43:43,510
Like Leibniz, I have
to use del, d of dx.

1774
01:43:43,510 --> 01:43:49,030
At the point x of dy of t,
this is the location I have.

1775
01:43:49,030 --> 01:43:50,630
Times what?

1776
01:43:50,630 --> 01:43:51,970
I keep derivation.

1777
01:43:51,970 --> 01:43:55,640
I keep derivating, like
don't drink and derive.

1778
01:43:55,640 --> 01:43:56,630
What is that?

1779
01:43:56,630 --> 01:43:58,981
The chain rule.

1780
01:43:58,981 --> 01:44:05,430
Prime again, this guy x
with respect to t, dx dt.

1781
01:44:05,430 --> 01:44:09,320
And then you go,
plus because she has

1782
01:44:09,320 --> 01:44:11,570
to be a mother to both kids.

1783
01:44:11,570 --> 01:44:14,670
The same thing for
the second child.

1784
01:44:14,670 --> 01:44:17,690
So you go, the derivative
of f with respect

1785
01:44:17,690 --> 01:44:26,990
to y, add x of ty
of t times dy dt.

1786
01:44:26,990 --> 01:44:30,230


1787
01:44:30,230 --> 01:44:35,440
So you see on the surface, x and
y are moving according to time.

1788
01:44:35,440 --> 01:44:39,000
And somehow we want to
measure the derivative

1789
01:44:39,000 --> 01:44:42,792
of the resulting function,
or composition function,

1790
01:44:42,792 --> 01:44:44,610
with respect to time.

1791
01:44:44,610 --> 01:44:46,330
This is a very
important chain rule

1792
01:44:46,330 --> 01:44:50,020
that I would like
you to memorize.

1793
01:44:50,020 --> 01:44:53,430
A chain rule.

1794
01:44:53,430 --> 01:44:54,060
Chain Rule No.

1795
01:44:54,060 --> 01:44:54,560
1.

1796
01:44:54,560 --> 01:44:58,720


1797
01:44:58,720 --> 01:44:59,920
Is it hard?

1798
01:44:59,920 --> 01:45:01,490
No, but for me it was.

1799
01:45:01,490 --> 01:45:04,690
When I was 21 and I saw
that-- and, of course,

1800
01:45:04,690 --> 01:45:06,020
my teacher was good.

1801
01:45:06,020 --> 01:45:10,350
And he told me, Magdalena,
imagine that instead of del you

1802
01:45:10,350 --> 01:45:13,530
would have d's.

1803
01:45:13,530 --> 01:45:16,680
So you have d and d and d and d.

1804
01:45:16,680 --> 01:45:21,300
The dx dx here, dy dy here,
they should be in your mind.

1805
01:45:21,300 --> 01:45:22,720
They are facing each other.

1806
01:45:22,720 --> 01:45:25,850
They are across on a diagonal.

1807
01:45:25,850 --> 01:45:29,140
And then, of course, I didn't
tell my teacher my idea

1808
01:45:29,140 --> 01:45:31,770
with the cowboys,
but it was funny.

1809
01:45:31,770 --> 01:45:38,810
So this is the chain rule
that re-makes, or generalizes

1810
01:45:38,810 --> 01:45:42,870
this idea to two variables.

1811
01:45:42,870 --> 01:45:47,970
Let's finish the example
because we didn't do it.

1812
01:45:47,970 --> 01:45:53,310
What is the derivative
of f in our case?

1813
01:45:53,310 --> 01:46:01,660
df dt will be-- oh, my god--
at any point p, how arbitary,

1814
01:46:01,660 --> 01:46:03,848
would be what?

1815
01:46:03,848 --> 01:46:07,640
First, you write
with respect to x.

1816
01:46:07,640 --> 01:46:10,501
2x, right?

1817
01:46:10,501 --> 01:46:11,000
2x.

1818
01:46:11,000 --> 01:46:16,900
But then you have to compute
this dx, add the pair you give.

1819
01:46:16,900 --> 01:46:19,650
And the pair they
gave you has a t.

1820
01:46:19,650 --> 01:46:23,450
So 2x is add x of
ty-- if you're going

1821
01:46:23,450 --> 01:46:25,335
to write it first
like that, you're

1822
01:46:25,335 --> 01:46:29,730
going to find it weird-- times,
I'm done with the first guy.

1823
01:46:29,730 --> 01:46:32,795
Then I'm going to take
the second guy in red,

1824
01:46:32,795 --> 01:46:35,310
and I'll put it here.

1825
01:46:35,310 --> 01:46:39,278
dx dt, but dx dt
everybody knows.

1826
01:46:39,278 --> 01:46:45,080
[INAUDIBLE] Let me
write it like this.

1827
01:46:45,080 --> 01:46:52,186
Plus [INAUDIBLE] that
guy again with green-- dy

1828
01:46:52,186 --> 01:46:59,146
computed at the pair x
of dy of [? t ?] times,

1829
01:46:59,146 --> 01:47:01,511
again, in red, dy dt.

1830
01:47:01,511 --> 01:47:06,730


1831
01:47:06,730 --> 01:47:08,772
So how do we write
the whole thing?

1832
01:47:08,772 --> 01:47:10,990
Could I have written it
from the beginning better?

1833
01:47:10,990 --> 01:47:11,490
Yeah.

1834
01:47:11,490 --> 01:47:20,630
2x of t, dx dt plus 2y of t dy.

1835
01:47:20,630 --> 01:47:21,610
Is it hard?

1836
01:47:21,610 --> 01:47:25,150
No, this is the idea.

1837
01:47:25,150 --> 01:47:28,070
Let's have something
more specific.

1838
01:47:28,070 --> 01:47:30,230
I'm going to erase
the whole thing.

1839
01:47:30,230 --> 01:47:36,230


1840
01:47:36,230 --> 01:47:40,205
I'll give you a problem
that we gave on the final

1841
01:47:40,205 --> 01:47:41,630
a few years ago.

1842
01:47:41,630 --> 01:47:44,890
And I'll show you how my
students cheated on that.

1843
01:47:44,890 --> 01:47:53,386
And I let them cheat, in
a way, because in the end

1844
01:47:53,386 --> 01:47:54,060
they were smart.

1845
01:47:54,060 --> 01:47:59,350
It didn't matter how they did
the problem, as long as they

1846
01:47:59,350 --> 01:48:01,790
got the correct answer.

1847
01:48:01,790 --> 01:48:03,330
So the problem was like that.

1848
01:48:03,330 --> 01:48:10,155
And my colleague did that many
years ago, several years ago,

1849
01:48:10,155 --> 01:48:11,980
did that several times.

1850
01:48:11,980 --> 01:48:19,700
So he said, let's do f of
t, dt squared and g of t.

1851
01:48:19,700 --> 01:48:27,000
I'll I'll do this
one, dq plus 1.

1852
01:48:27,000 --> 01:48:42,980
And then let's
[INAUDIBLE] the w of u

1853
01:48:42,980 --> 01:48:54,100
and B, exactly the same thing I
gave you before, [INAUDIBLE] I

1854
01:48:54,100 --> 01:48:56,040
remember that.

1855
01:48:56,040 --> 01:49:05,708
And he said, compute the
derivative of w of f of t,

1856
01:49:05,708 --> 01:49:10,280
and g of t with respect to t.

1857
01:49:10,280 --> 01:49:12,250
And you will ask,
wait a minute here.

1858
01:49:12,250 --> 01:49:14,560
Why do you put d and not del?

1859
01:49:14,560 --> 01:49:17,850
Because this is a composed
function that in the end

1860
01:49:17,850 --> 01:49:20,580
is a function of t only.

1861
01:49:20,580 --> 01:49:22,680
So if you do it as
a composed function,

1862
01:49:22,680 --> 01:49:26,040
because this goes like this.

1863
01:49:26,040 --> 01:49:31,560
t goes to two
functions, f of t and u.

1864
01:49:31,560 --> 01:49:34,454


1865
01:49:34,454 --> 01:49:40,850
And there is a function w
that takes both of them, that

1866
01:49:40,850 --> 01:49:42,870
is a function of both of them.

1867
01:49:42,870 --> 01:49:46,835
In the end, this composition
that's straight from here

1868
01:49:46,835 --> 01:49:50,826
to here, is a function
of one variable only.

1869
01:49:50,826 --> 01:49:54,850


1870
01:49:54,850 --> 01:49:58,280
So my students then-- it was in
the beginning of the examine,

1871
01:49:58,280 --> 01:49:59,020
I remember.

1872
01:49:59,020 --> 01:50:02,300
And they said, well,
I forgot, they said.

1873
01:50:02,300 --> 01:50:03,860
I stayed up almost all night.

1874
01:50:03,860 --> 01:50:05,432
Don't do that.

1875
01:50:05,432 --> 01:50:06,390
Don't do what they did.

1876
01:50:06,390 --> 01:50:08,330
Many of my students
stay up all night

1877
01:50:08,330 --> 01:50:11,090
before the final because
I think I scare people,

1878
01:50:11,090 --> 01:50:12,700
and that's not what I mean.

1879
01:50:12,700 --> 01:50:15,480
I just want you to study.

1880
01:50:15,480 --> 01:50:18,670
But they stay up before
the final and the next day,

1881
01:50:18,670 --> 01:50:19,380
I'm a vegetable.

1882
01:50:19,380 --> 01:50:21,410
I don't even remember
the chain rule.

1883
01:50:21,410 --> 01:50:23,450
So they did not
remember the chain rule

1884
01:50:23,450 --> 01:50:25,000
that I've just wrote.

1885
01:50:25,000 --> 01:50:28,490
And they said, oh, but I
think I know how to do it.

1886
01:50:28,490 --> 01:50:29,990
And I said, shh.

1887
01:50:29,990 --> 01:50:31,980
Just don't say anything.

1888
01:50:31,980 --> 01:50:34,840
Let me show you how the
course coordinator wanted

1889
01:50:34,840 --> 01:50:37,170
that done several years ago.

1890
01:50:37,170 --> 01:50:40,165
So he wanted it done
by the chain rule.

1891
01:50:40,165 --> 01:50:41,550
He didn't say how you do it.

1892
01:50:41,550 --> 01:50:42,050
OK?

1893
01:50:42,050 --> 01:50:44,160
He said just get to
the right answer.

1894
01:50:44,160 --> 01:50:45,614
It doesn't matter.

1895
01:50:45,614 --> 01:50:46,780
He wanted it done like that.

1896
01:50:46,780 --> 01:50:55,700
He said, dw of f of tg
of p with respect to t,

1897
01:50:55,700 --> 01:51:06,738
would be dw du, instead
of u you have f of t.

1898
01:51:06,738 --> 01:51:16,730
f of tg of t times df
dt plus dw with respect

1899
01:51:16,730 --> 01:51:18,940
to the second variable.

1900
01:51:18,940 --> 01:51:25,137
So this would be u, and
this would be v with respect

1901
01:51:25,137 --> 01:51:27,300
to the variable v,
the second variable

1902
01:51:27,300 --> 01:51:30,726
where [? measure ?]
that f of dg of t.

1903
01:51:30,726 --> 01:51:39,410
Evaluate it there times dg dt.

1904
01:51:39,410 --> 01:51:46,136
So it's like dv dt, which is dg
dt. [INAUDIBLE] So he did that,

1905
01:51:46,136 --> 01:51:48,045
and he expected
people to do what?

1906
01:51:48,045 --> 01:51:51,192
He expected people to take
a u squared the same 2 times

1907
01:51:51,192 --> 01:51:54,251
u, just like you
did before, 2 times.

1908
01:51:54,251 --> 01:51:57,715
And instead of u, since u is
f of t to [INAUDIBLE] puts

1909
01:51:57,715 --> 01:52:13,171
2f of t, this is the first
squiggly thing, times v of dt.

1910
01:52:13,171 --> 01:52:19,932
2t is this smiley face.

1911
01:52:19,932 --> 01:52:31,200
This is 2t plus--
what is the f dv?

1912
01:52:31,200 --> 01:52:37,600
Dw with respect to dv is
going to be 2v 2 time gf t.

1913
01:52:37,600 --> 01:52:46,794
When I evaluate add gf
t, this funny fellow

1914
01:52:46,794 --> 01:52:57,580
with this funny fellow, times qg
d, which, with your permission

1915
01:52:57,580 --> 01:53:00,890
I'm going to erase
and write 3p squared.

1916
01:53:00,890 --> 01:53:04,030


1917
01:53:04,030 --> 01:53:07,340
And the last row he expected
my students to write

1918
01:53:07,340 --> 01:53:22,135
was 2t squared times 2t plus
2pq plus 1, times 3t squared.

1919
01:53:22,135 --> 01:53:27,580


1920
01:53:27,580 --> 01:53:31,540
Are you guys with me?

1921
01:53:31,540 --> 01:53:43,210
So [INAUDIBLE] 2t 2x
2t squared, correct.

1922
01:53:43,210 --> 01:53:49,730
I forgot to identify
this as that.

1923
01:53:49,730 --> 01:53:50,230
All right.

1924
01:53:50,230 --> 01:53:52,650
So in the end, the answer
is a simplified answer.

1925
01:53:52,650 --> 01:53:53,930
Can you tell me what it is?

1926
01:53:53,930 --> 01:53:55,250
I'm too lazy to write it down.

1927
01:53:55,250 --> 01:53:56,935
You compute it.

1928
01:53:56,935 --> 01:53:58,930
How much is it simplified?

1929
01:53:58,930 --> 01:54:00,421
Find it as a polynomial.

1930
01:54:00,421 --> 01:54:01,415
STUDENT: [INAUDIBLE].

1931
01:54:01,415 --> 01:54:04,397


1932
01:54:04,397 --> 01:54:08,870
PROFESSOR TODA:
So you have 6, 6--

1933
01:54:08,870 --> 01:54:10,150
STUDENT: 16 cubed plus 3--

1934
01:54:10,150 --> 01:54:15,300
PROFESSOR TODA: T
to the 5th plus--

1935
01:54:15,300 --> 01:54:17,240
STUDENT: [INAUDIBLE].

1936
01:54:17,240 --> 01:54:19,350
PROFESSOR TODA: In
order, in order.

1937
01:54:19,350 --> 01:54:20,420
What's the next guy?

1938
01:54:20,420 --> 01:54:21,636
STUDENT: [INAUDIBLE].

1939
01:54:21,636 --> 01:54:22,830
PROFESSOR TODA: 4t cubed.

1940
01:54:22,830 --> 01:54:23,975
And the last guy--

1941
01:54:23,975 --> 01:54:24,925
STUDENT: 6t squared.

1942
01:54:24,925 --> 01:54:26,050
PROFESSOR TODA: 6t squared.

1943
01:54:26,050 --> 01:54:31,220


1944
01:54:31,220 --> 01:54:31,720
Yes?

1945
01:54:31,720 --> 01:54:33,320
Did you get the same thing?

1946
01:54:33,320 --> 01:54:34,220
OK.

1947
01:54:34,220 --> 01:54:37,145
Now, how did my students do it?

1948
01:54:37,145 --> 01:54:37,644
[INAUDIBLE]

1949
01:54:37,644 --> 01:54:40,171


1950
01:54:40,171 --> 01:54:41,420
Did they apply the chain rule?

1951
01:54:41,420 --> 01:54:41,920
No.

1952
01:54:41,920 --> 01:54:44,100
They said OK, this
is how it goes.

1953
01:54:44,100 --> 01:54:46,960


1954
01:54:46,960 --> 01:54:58,210
W of U of T and V of T is U is
F. So this guy is T squared,

1955
01:54:58,210 --> 01:55:01,886
T squared squared,
plus this guy is T

1956
01:55:01,886 --> 01:55:08,960
cubed plus 1 taken and
shaken and squared.

1957
01:55:08,960 --> 01:55:13,710
And then when I do the
whole thing, derivative

1958
01:55:13,710 --> 01:55:22,640
of this with respect
to T, I get--

1959
01:55:22,640 --> 01:55:27,570
I'm too lazy-- T to the
4 prime is 40 cubed.

1960
01:55:27,570 --> 01:55:28,870
I'm not going to do on the map.

1961
01:55:28,870 --> 01:55:37,380
2 out T cubed plus 1 times
chain rule, 3t squared.

1962
01:55:37,380 --> 01:55:49,610
40 cubed plus 16 to the 5 plus--
[INAUDIBLE] 2 and 6t squared.

1963
01:55:49,610 --> 01:55:56,450
So you realize that I
have to give them 100%.

1964
01:55:56,450 --> 01:55:59,465
Although they were very
honest and said, we blanked.

1965
01:55:59,465 --> 01:56:01,130
We don't remember
the chain rule.

1966
01:56:01,130 --> 01:56:02,926
We don't remember the formula.

1967
01:56:02,926 --> 01:56:03,550
So that's fine.

1968
01:56:03,550 --> 01:56:05,080
Do whatever you can.

1969
01:56:05,080 --> 01:56:06,920
So I gave them 100% for that.

1970
01:56:06,920 --> 01:56:11,280
But realize that the
author of the problem

1971
01:56:11,280 --> 01:56:14,080
was a little bit naive.

1972
01:56:14,080 --> 01:56:16,520
Because you could have
done this differently.

1973
01:56:16,520 --> 01:56:22,190
I mean if you wanted to
actually test the whole thing,

1974
01:56:22,190 --> 01:56:26,170
you wouldn't have given-- let's
say you wouldn't have given

1975
01:56:26,170 --> 01:56:32,400
the actual-- yeah, you wouldn't
have given the actual functions

1976
01:56:32,400 --> 01:56:37,530
and say write the chain
formula symbolically

1977
01:56:37,530 --> 01:56:44,875
for this function applied
for F of T and G of T.

1978
01:56:44,875 --> 01:56:49,340
So it was-- they
were just lucky.

1979
01:56:49,340 --> 01:56:52,470
Remember that you need
to know this chain rule.

1980
01:56:52,470 --> 01:56:53,970
It's going to be
one of the problems

1981
01:56:53,970 --> 01:56:56,900
to be emphasized in the exams.

1982
01:56:56,900 --> 01:57:02,408
Maybe one of the top 15 or
16 most important topics.

1983
01:57:02,408 --> 01:57:07,170


1984
01:57:07,170 --> 01:57:07,900
Is that OK?

1985
01:57:07,900 --> 01:57:09,396
Can I erase the whole thing?

1986
01:57:09,396 --> 01:57:09,896
OK.

1987
01:57:09,896 --> 01:57:11,393
Let me erase the whole thing.

1988
01:57:11,393 --> 01:57:44,326


1989
01:57:44,326 --> 01:57:44,826
OK.

1990
01:57:44,826 --> 01:57:45,824
Any other questions?

1991
01:57:45,824 --> 01:58:02,291


1992
01:58:02,291 --> 01:58:03,661
No?

1993
01:58:03,661 --> 01:58:05,285
I'm not going to let
you go right away,

1994
01:58:05,285 --> 01:58:07,780
we're going to work one
more problem or two more

1995
01:58:07,780 --> 01:58:08,778
simple problems.

1996
01:58:08,778 --> 01:58:10,773
And then we are going to go.

1997
01:58:10,773 --> 01:58:11,273
OK?

1998
01:58:11,273 --> 01:58:22,750


1999
01:58:22,750 --> 01:58:26,480
So question.

2000
01:58:26,480 --> 01:58:27,986
A question.

2001
01:58:27,986 --> 01:58:32,946


2002
01:58:32,946 --> 01:58:39,890
What do you think the
gradient is good at?

2003
01:58:39,890 --> 01:58:49,314


2004
01:58:49,314 --> 01:58:50,980
Two reasons, right.

2005
01:58:50,980 --> 01:58:54,340
Review number one.

2006
01:58:54,340 --> 01:58:59,120
If you have an increasingly
defined function,

2007
01:58:59,120 --> 01:59:02,860
then the gradient of F was what?

2008
01:59:02,860 --> 01:59:21,904
Equals direction of the
normal to the surface S--

2009
01:59:21,904 --> 01:59:26,395
let's say S is given
increasingly at the point

2010
01:59:26,395 --> 01:59:27,393
with [INAUDIBLE].

2011
01:59:27,393 --> 01:59:31,884


2012
01:59:31,884 --> 01:59:33,381
But any other reason?

2013
01:59:33,381 --> 02:00:00,327


2014
02:00:00,327 --> 02:00:01,824
Let's take that again.

2015
02:00:01,824 --> 02:00:05,816
Z equals x squared
plus y squared.

2016
02:00:05,816 --> 02:00:07,812
Let's compute a few
partial derivatives.

2017
02:00:07,812 --> 02:00:09,309
Let's compute the gradient.

2018
02:00:09,309 --> 02:00:21,120
The gradient is Fs of x, Fs
of y, where this is F of xy

2019
02:00:21,120 --> 02:00:24,888
or Fs of xi plus Fs of yj.

2020
02:00:24,888 --> 02:00:28,880


2021
02:00:28,880 --> 02:00:31,375
[INAUDIBLE]

2022
02:00:31,375 --> 02:00:34,369
And we drew it.

2023
02:00:34,369 --> 02:00:42,353
I drew this case, and we also
drew another related example,

2024
02:00:42,353 --> 02:00:45,846
where we took Z equals 1 minus
x squared minus y squared.

2025
02:00:45,846 --> 02:00:46,844
And we went skiing.

2026
02:00:46,844 --> 02:00:52,333
And we were so happy last week
to go skiing, because we still

2027
02:00:52,333 --> 02:00:57,650
had snow in New
Mexico, and we-- and we

2028
02:00:57,650 --> 02:01:02,546
said now we computed the
Z to be minus 2x minus 2y.

2029
02:01:02,546 --> 02:01:06,018


2030
02:01:06,018 --> 02:01:09,597
And we said, I'm
looking at the slopes.

2031
02:01:09,597 --> 02:01:12,936
This is the x duration
and the y duration.

2032
02:01:12,936 --> 02:01:18,670
And I'm looking at the slopes of
the lines of these two curves.

2033
02:01:18,670 --> 02:01:23,630
So one that goes
down, like that.

2034
02:01:23,630 --> 02:01:25,010
So this was for what?

2035
02:01:25,010 --> 02:01:27,540
For y equals 0.

2036
02:01:27,540 --> 02:01:32,190
And this was for x equals 0.

2037
02:01:32,190 --> 02:01:36,645


2038
02:01:36,645 --> 02:01:39,580
Curve, x equals
0 curve in plane.

2039
02:01:39,580 --> 02:01:40,360
Right?

2040
02:01:40,360 --> 02:01:42,742
We just cross-section
our surface,

2041
02:01:42,742 --> 02:01:43,950
and we have this [INAUDIBLE].

2042
02:01:43,950 --> 02:01:51,594
And then we have the two
tangents, two slopes.

2043
02:01:51,594 --> 02:01:54,064
And we computed them everywhere.

2044
02:01:54,064 --> 02:02:00,486


2045
02:02:00,486 --> 02:02:01,968
At every point.

2046
02:02:01,968 --> 02:02:06,910


2047
02:02:06,910 --> 02:02:10,845
But realize that to go
up or down these hills,

2048
02:02:10,845 --> 02:02:15,095
I can go on a curve
like that, or I

2049
02:02:15,095 --> 02:02:17,950
can go-- remember the
train of Mickey Mouse going

2050
02:02:17,950 --> 02:02:20,182
on the hilly point on the hill?

2051
02:02:20,182 --> 02:02:22,174
We try to take different paths.

2052
02:02:22,174 --> 02:02:24,166
We are going hiking.

2053
02:02:24,166 --> 02:02:28,648
We are going hiking, and we'll
take hiking through the pass.

2054
02:02:28,648 --> 02:02:38,608


2055
02:02:38,608 --> 02:02:41,098
OK.

2056
02:02:41,098 --> 02:03:01,420
How do we get the maximum
rate of change of the function

2057
02:03:01,420 --> 02:03:03,600
Z equals F of x1?

2058
02:03:03,600 --> 02:03:05,870
So now I'm
anticipating something.

2059
02:03:05,870 --> 02:03:10,680
I'd like to see your intuition,
your inborn sense of I

2060
02:03:10,680 --> 02:03:12,300
know what's going to happen.

2061
02:03:12,300 --> 02:03:14,092
And you know what
that from Mister--

2062
02:03:14,092 --> 02:03:14,842
STUDENT: Heinrich.

2063
02:03:14,842 --> 02:03:17,590
PROFESSOR TODA: [? Heinrich ?]
from high school.

2064
02:03:17,590 --> 02:03:21,280
So I'm asking-- let me
rephrase the question

2065
02:03:21,280 --> 02:03:23,130
like a non-mathematician.

2066
02:03:23,130 --> 02:03:24,230
Let's go hiking.

2067
02:03:24,230 --> 02:03:30,274
This is [INAUDIBLE] we
go to the lighthouse.

2068
02:03:30,274 --> 02:03:33,790
Which path shall I take
on my mountain, my hill,

2069
02:03:33,790 --> 02:03:37,570
my god knows what
geography, in order

2070
02:03:37,570 --> 02:03:40,440
to obtain the maximum
rate of change?

2071
02:03:40,440 --> 02:03:43,515
That means the
highest derivative.

2072
02:03:43,515 --> 02:03:46,470
In what direction do I get
the highest derivative?

2073
02:03:46,470 --> 02:03:49,110
STUDENT: In what direction you
get the highest derivative--

2074
02:03:49,110 --> 02:03:50,735
PROFESSOR TODA: So
in which direction--

2075
02:03:50,735 --> 02:03:53,330
in which direction
on this hill do

2076
02:03:53,330 --> 02:03:55,098
I get the highest derivative?

2077
02:03:55,098 --> 02:03:57,014
The highest rate of change.

2078
02:03:57,014 --> 02:04:03,740
Rate of change means I want to
get the fastest possible way

2079
02:04:03,740 --> 02:04:04,850
somewhere.

2080
02:04:04,850 --> 02:04:08,230
STUDENT: The shortest slope?

2081
02:04:08,230 --> 02:04:09,857
Along just the straight line up.

2082
02:04:09,857 --> 02:04:10,831
PROFESSOR TODA: Along--

2083
02:04:10,831 --> 02:04:12,292
STUDENT: You don't want
to take any [INAUDIBLE].

2084
02:04:12,292 --> 02:04:13,270
PROFESSOR TODA: Right.

2085
02:04:13,270 --> 02:04:13,910
STUDENT: [INAUDIBLE].

2086
02:04:13,910 --> 02:04:15,140
It could be along any axis.

2087
02:04:15,140 --> 02:04:17,710
PROFESSOR TODA: So could
you see which direction

2088
02:04:17,710 --> 02:04:19,070
those are-- very good.

2089
02:04:19,070 --> 02:04:21,230
Actually you were getting
to the same direction.

2090
02:04:21,230 --> 02:04:24,370
So [INAUDIBLE] says
Magdalena, don't be silly.

2091
02:04:24,370 --> 02:04:28,295
The actual maximum rate of
change for the function Z

2092
02:04:28,295 --> 02:04:31,070
is obviously, because
it is common sense,

2093
02:04:31,070 --> 02:04:36,630
it's obviously happening if
you take the so-called-- what

2094
02:04:36,630 --> 02:04:37,880
are these guys?

2095
02:04:37,880 --> 02:04:40,810
[INAUDIBLE], not meridians.

2096
02:04:40,810 --> 02:04:42,280
STUDENT: Longtitudes?

2097
02:04:42,280 --> 02:04:43,260
PROFESSOR TODA: OK.

2098
02:04:43,260 --> 02:04:44,730
That is-- OK.

2099
02:04:44,730 --> 02:04:47,810
Suppose that we don't hike,
because it's too tiring.

2100
02:04:47,810 --> 02:04:51,170
We go down from the
top of the hill.

2101
02:04:51,170 --> 02:04:53,310
Ah, there's also very good idea.

2102
02:04:53,310 --> 02:04:58,870
So when you let yourself
go down on a sleigh,

2103
02:04:58,870 --> 02:05:02,560
don't think bobsled or
anything-- just a sleigh,

2104
02:05:02,560 --> 02:05:04,110
think of a child's sleigh.

2105
02:05:04,110 --> 02:05:07,680
No, take a plastic bag
and put your butt in it

2106
02:05:07,680 --> 02:05:10,590
and let yourself go.

2107
02:05:10,590 --> 02:05:14,140
What is their
direction actually?

2108
02:05:14,140 --> 02:05:19,925
Your body will find the
fastest way to get down.

2109
02:05:19,925 --> 02:05:23,065
The fastest way to get
down will happen exactly

2110
02:05:23,065 --> 02:05:27,710
in the same
directions going down

2111
02:05:27,710 --> 02:05:29,600
in the directions
of these meridians.

2112
02:05:29,600 --> 02:05:34,100


2113
02:05:34,100 --> 02:05:35,512
OK?

2114
02:05:35,512 --> 02:05:37,000
And now, [INAUDIBLE].

2115
02:05:37,000 --> 02:05:46,424


2116
02:05:46,424 --> 02:05:58,576
The maximum rate of
change will always

2117
02:05:58,576 --> 02:06:07,256
happen in the direction
of the gradient.

2118
02:06:07,256 --> 02:06:14,696


2119
02:06:14,696 --> 02:06:18,950
You can get a little
bit ahead of time

2120
02:06:18,950 --> 02:06:21,860
by just-- I would like this
to [INAUDIBLE] in your heads

2121
02:06:21,860 --> 02:06:23,860
until we get to that section.

2122
02:06:23,860 --> 02:06:26,880
In one section we will be there.

2123
02:06:26,880 --> 02:06:40,430
We also-- it's also reformulated
as the highest, the steepest,

2124
02:06:40,430 --> 02:06:42,079
ascent or descent.

2125
02:06:42,079 --> 02:06:44,574
The steepest.

2126
02:06:44,574 --> 02:06:58,546
The steepest ascent or
the steepest descent

2127
02:06:58,546 --> 02:07:09,524
always happens in the
direction of the gradient.

2128
02:07:09,524 --> 02:07:14,550


2129
02:07:14,550 --> 02:07:17,110
Ascent is when you hike
to the top of the hill.

2130
02:07:17,110 --> 02:07:21,450
Descent is when you let yourself
go in the plastic [INAUDIBLE]

2131
02:07:21,450 --> 02:07:25,270
bag in the snow.

2132
02:07:25,270 --> 02:07:26,080
Right?

2133
02:07:26,080 --> 02:07:30,030
Can you verify this happens
just on this example?

2134
02:07:30,030 --> 02:07:32,540
It's true in general,
for any smooth function.

2135
02:07:32,540 --> 02:07:36,010
Our smooth function is
a really nice function.

2136
02:07:36,010 --> 02:07:39,816
So what is the gradient?

2137
02:07:39,816 --> 02:07:42,720
Well again, it was 2x 2y, right?

2138
02:07:42,720 --> 02:07:45,850


2139
02:07:45,850 --> 02:07:50,508
And that means at a certain
point, x0 y0, whenever you are,

2140
02:07:50,508 --> 02:07:52,300
guys you don't
necessarily have to start

2141
02:07:52,300 --> 02:07:54,750
from the top of the hill.

2142
02:07:54,750 --> 02:07:58,870
You can be-- OK,
this is your cabin.

2143
02:07:58,870 --> 02:08:01,970
And here you are with
friends, or with mom and dad,

2144
02:08:01,970 --> 02:08:05,110
or whoever, on the hill.

2145
02:08:05,110 --> 02:08:09,030
You get out, you take the
sleigh, and you go down.

2146
02:08:09,030 --> 02:08:14,320
So no matter where
you are, there you go.

2147
02:08:14,320 --> 02:08:22,933
You have 2x0 times
i plus 2y0 times j.

2148
02:08:22,933 --> 02:08:31,640
And the direction of the
gradient will be 2x0 2y0.

2149
02:08:31,640 --> 02:08:34,520
Do you like this one?

2150
02:08:34,520 --> 02:08:39,240
Well in this case,
if you were-- suppose

2151
02:08:39,240 --> 02:08:42,300
you were at the
point [INAUDIBLE].

2152
02:08:42,300 --> 02:08:49,230


2153
02:08:49,230 --> 02:08:53,596
You are at the point
of coordinates--

2154
02:08:53,596 --> 02:08:55,130
do you want to be here?

2155
02:08:55,130 --> 02:08:57,000
You want to be here, right?

2156
02:08:57,000 --> 02:08:58,770
So we've done that before.

2157
02:08:58,770 --> 02:09:02,560
I'll take it as 1 over
[? square root of ?]

2158
02:09:02,560 --> 02:09:09,620
2-- I'm trying to be creative
today-- [INAUDIBLE] y equals 0,

2159
02:09:09,620 --> 02:09:14,580
and Z equals-- what's left?

2160
02:09:14,580 --> 02:09:16,460
1/2, right?

2161
02:09:16,460 --> 02:09:17,750
Where am I?

2162
02:09:17,750 --> 02:09:20,390
Guys, do you realize where I am?

2163
02:09:20,390 --> 02:09:21,640
I'll [? take a ?] [INAUDIBLE].

2164
02:09:21,640 --> 02:09:24,340


2165
02:09:24,340 --> 02:09:25,140
y0.

2166
02:09:25,140 --> 02:09:28,656
So I need to be on this
meridian on the red thingy.

2167
02:09:28,656 --> 02:09:33,920


2168
02:09:33,920 --> 02:09:37,356
And somewhere here.

2169
02:09:37,356 --> 02:09:40,250


2170
02:09:40,250 --> 02:09:43,385
What's the duration
of the gradient here?

2171
02:09:43,385 --> 02:09:45,830
Delta z at this p.

2172
02:09:45,830 --> 02:09:56,610


2173
02:09:56,610 --> 02:09:58,760
Then you say ah,
well, I don't get it.

2174
02:09:58,760 --> 02:10:04,040
I have-- the second guy will
become 0, because y0 is 0.

2175
02:10:04,040 --> 02:10:06,980
The first guy will become
1 over square root of 2.

2176
02:10:06,980 --> 02:10:15,280
So I have 2 times 1 over square
root of 2 times i plus 0j.

2177
02:10:15,280 --> 02:10:29,416
It means in the direction of i--
in the direction of i-- from p,

2178
02:10:29,416 --> 02:10:39,201
I have the fastest-- fastest,
Magdalena, fastest-- descent

2179
02:10:39,201 --> 02:10:39,700
possible.

2180
02:10:39,700 --> 02:10:43,010


2181
02:10:43,010 --> 02:10:46,730
But we don't say in
the direction of i

2182
02:10:46,730 --> 02:10:49,640
in our everyday life, right?

2183
02:10:49,640 --> 02:10:53,270
Let's say geographic points.

2184
02:10:53,270 --> 02:10:58,610
We are-- I'm a bug,
and this is north.

2185
02:10:58,610 --> 02:11:00,086
This is south.

2186
02:11:00,086 --> 02:11:05,006


2187
02:11:05,006 --> 02:11:05,990
This is east.

2188
02:11:05,990 --> 02:11:08,960


2189
02:11:08,960 --> 02:11:11,470
And this is west.

2190
02:11:11,470 --> 02:11:18,140
So if I go east, going east
means going in the direction i.

2191
02:11:18,140 --> 02:11:23,040


2192
02:11:23,040 --> 02:11:25,510
Now suppose-- I'm going
to finish with this one.

2193
02:11:25,510 --> 02:11:28,870
Suppose that my house
is not on the prairie

2194
02:11:28,870 --> 02:11:31,710
but my house is here.

2195
02:11:31,710 --> 02:11:34,400
House, h.

2196
02:11:34,400 --> 02:11:37,834
Find me a wood
point to be there.

2197
02:11:37,834 --> 02:11:39,738
STUDENT: Northeast.

2198
02:11:39,738 --> 02:11:41,170
Or to get further down.

2199
02:11:41,170 --> 02:11:45,047
PROFESSOR TODA: Anything, what
would look like why I'm here?

2200
02:11:45,047 --> 02:11:48,041
x0, y0, z0.

2201
02:11:48,041 --> 02:11:50,040
Hm.

2202
02:11:50,040 --> 02:11:57,800
1/2, 1/2, and I
need the minimum.

2203
02:11:57,800 --> 02:12:02,580
So I want to be on the
bisecting plane between the two.

2204
02:12:02,580 --> 02:12:03,420
You understand?

2205
02:12:03,420 --> 02:12:04,400
This is my quarter.

2206
02:12:04,400 --> 02:12:06,870
And I want to be in
this bisecting plane.

2207
02:12:06,870 --> 02:12:10,420
So I'll take 1/2, 1/2, and
what results from here?

2208
02:12:10,420 --> 02:12:11,540
I have to do math.

2209
02:12:11,540 --> 02:12:16,110
1 minus 1/4 minus 1/4 is 1/2.

2210
02:12:16,110 --> 02:12:17,902
Right?

2211
02:12:17,902 --> 02:12:19,740
1/2, 1/2, 1/2.

2212
02:12:19,740 --> 02:12:22,440
This is where my house
is [? and so on. ?]

2213
02:12:22,440 --> 02:12:24,106
And this is full of smoke.

2214
02:12:24,106 --> 02:12:29,940
And what is the
maximum rate of change?

2215
02:12:29,940 --> 02:12:34,500
What is the steepest
descent is the trajectory

2216
02:12:34,500 --> 02:12:37,920
that my body will take
when I let myself go down

2217
02:12:37,920 --> 02:12:39,402
on the sleigh.

2218
02:12:39,402 --> 02:12:40,884
How do I compute that?

2219
02:12:40,884 --> 02:12:43,624
I will just do the same thing.

2220
02:12:43,624 --> 02:12:49,900
Delta z at the point x0
equals 1/2, y0 equals 1/2,

2221
02:12:49,900 --> 02:12:52,100
z0 equals 1/2.

2222
02:12:52,100 --> 02:12:54,060
Well what do I get as direction?

2223
02:12:54,060 --> 02:12:57,490
That will be the
direction of the gradient.

2224
02:12:57,490 --> 02:13:02,920
2 times 1/2-- you
guys with me still?

2225
02:13:02,920 --> 02:13:09,240
i plus 2 times 1/2 with j.

2226
02:13:09,240 --> 02:13:14,311
And there is no Mr.
z0 In the picture.

2227
02:13:14,311 --> 02:13:14,810
Why?

2228
02:13:14,810 --> 02:13:17,100
Because that will
give me the direction

2229
02:13:17,100 --> 02:13:22,000
like on-- in a geographic way.

2230
02:13:22,000 --> 02:13:24,420
North, west, east, south.

2231
02:13:24,420 --> 02:13:26,490
These are the
direction in plane.

2232
02:13:26,490 --> 02:13:28,120
I'm not talking
directions on the hill,

2233
02:13:28,120 --> 02:13:31,496
I'm talking
directions on the map.

2234
02:13:31,496 --> 02:13:33,530
These are directions on the map.

2235
02:13:33,530 --> 02:13:35,930
So what is the direction
i plus j on the map?

2236
02:13:35,930 --> 02:13:39,820
If you show this to a
geography major and say,

2237
02:13:39,820 --> 02:13:43,340
I'm going in the direction
i plus j on the map,

2238
02:13:43,340 --> 02:13:45,700
he will say you are crazy.

2239
02:13:45,700 --> 02:13:47,980
He doesn't understand the thing.

2240
02:13:47,980 --> 02:13:50,280
But you know what you mean.

2241
02:13:50,280 --> 02:13:54,100
East for you is the
direction of i in the x-axis.

2242
02:13:54,100 --> 02:13:56,210
[INAUDIBLE]

2243
02:13:56,210 --> 02:13:58,430
And this is north.

2244
02:13:58,430 --> 02:13:59,630
Are you guys with me?

2245
02:13:59,630 --> 02:14:01,790
The y direction is north.

2246
02:14:01,790 --> 02:14:06,410
So I'm going perfectly
northeast at a 45-degree angle.

2247
02:14:06,410 --> 02:14:08,093
If I tell the
geography major I'm

2248
02:14:08,093 --> 02:14:10,725
going northeast perfectly in
the middle, he will say I know.

2249
02:14:10,725 --> 02:14:13,560
But you will know that
for you, that is i plus j.

2250
02:14:13,560 --> 02:14:15,649
Because you are
the mathematician.

2251
02:14:15,649 --> 02:14:17,100
Right?

2252
02:14:17,100 --> 02:14:18,980
So you go down.

2253
02:14:18,980 --> 02:14:20,780
And this is where you are.

2254
02:14:20,780 --> 02:14:22,500
And you're on the meridian.

2255
02:14:22,500 --> 02:14:25,090
This is the direction i plus j.

2256
02:14:25,090 --> 02:14:29,610
So if I want to project my
trajectory-- I went down

2257
02:14:29,610 --> 02:14:33,260
with the sleigh, all the way
down-- project the trajectory,

2258
02:14:33,260 --> 02:14:36,810
my trajectory is a
body on the snow.

2259
02:14:36,810 --> 02:14:39,320
Projecting it on the
ground is this one.

2260
02:14:39,320 --> 02:14:43,800
So it is exactly the
direction i plus j.

2261
02:14:43,800 --> 02:14:44,320
Right, guys?

2262
02:14:44,320 --> 02:14:48,170
So exactly northeast
perfectly at 45-degree angles.

2263
02:14:48,170 --> 02:14:51,150
Now one caveat.

2264
02:14:51,150 --> 02:14:53,455
One caveat, because
when we get there,

2265
02:14:53,455 --> 02:14:59,060
you should be ready
already, in 11.6 and 11.7.

2266
02:14:59,060 --> 02:15:02,930
When we will say direction,
we are also crazy people.

2267
02:15:02,930 --> 02:15:04,920
I told you, mathematicians
are not normal.

2268
02:15:04,920 --> 02:15:07,108
You have to be a
little bit crazy

2269
02:15:07,108 --> 02:15:11,460
to want to do all the stuff
in your head like that.

2270
02:15:11,460 --> 02:15:16,420
i plus j for us is not a
direction most of the time.

2271
02:15:16,420 --> 02:15:20,015
When we say direction, we mean
we normalize that direction.

2272
02:15:20,015 --> 02:15:23,055
We take the unit
vector, which is unique,

2273
02:15:23,055 --> 02:15:25,940
for responding to i plus j.

2274
02:15:25,940 --> 02:15:28,890
So what is that
unique unit vector?

2275
02:15:28,890 --> 02:15:32,510
You learned in Chapter 9
everything is connected.

2276
02:15:32,510 --> 02:15:33,890
It's a big circle.

2277
02:15:33,890 --> 02:15:35,340
i plus j, very good.

2278
02:15:35,340 --> 02:15:40,190
So direction is a unit vector
for most mathematicians,

2279
02:15:40,190 --> 02:15:45,390
which means you will be i
plus j over square root of 2.

2280
02:15:45,390 --> 02:15:51,966
So in Chapter 5, please
remember, unlike Chapter 9,

2281
02:15:51,966 --> 02:15:55,613
direction is a unit vector.

2282
02:15:55,613 --> 02:15:59,609
In Chapter 9, Chapter 10,
it said direction lmn,

2283
02:15:59,609 --> 02:16:00,650
direction god knows what.

2284
02:16:00,650 --> 02:16:05,900
But in Chapter 11, direction
is a vector in plane,

2285
02:16:05,900 --> 02:16:07,860
like this one, i
plus [INAUDIBLE]

2286
02:16:07,860 --> 02:16:12,015
has to be a unique
normal-- a unique vector.

2287
02:16:12,015 --> 02:16:12,514
OK?

2288
02:16:12,514 --> 02:16:14,318
And we-- keep that in mind.

2289
02:16:14,318 --> 02:16:16,050
Next time, when we
meet on Thursday,

2290
02:16:16,050 --> 02:16:19,956
you will understand why
we need to normalize it.

2291
02:16:19,956 --> 02:16:23,220
Now can we say goodbye to
the snow and everything?

2292
02:16:23,220 --> 02:16:25,810
It's not going to
show up much anymore.

2293
02:16:25,810 --> 02:16:27,555
Remember this example.

2294
02:16:27,555 --> 02:16:30,660
But we will start with
flowers next time.

2295
02:16:30,660 --> 02:16:31,260
OK.

2296
02:16:31,260 --> 02:16:32,760
Have a nice day.

2297
02:16:32,760 --> 02:16:33,959
Yes, sir?

2298
02:16:33,959 --> 02:16:36,410
Let me stop the video.

2299
02:16:36,410 --> 02:16:37,245