1
00:00:00,000 --> 00:00:00,930


2
00:00:00,930 --> 00:00:05,670
PROFESSOR: So let's
forget about this example

3
00:00:05,670 --> 00:00:10,910
and review what we learned
in 11.3, chapter 11.

4
00:00:10,910 --> 00:00:17,160
Chapter 11, again, was
functions of several variables.

5
00:00:17,160 --> 00:00:21,028
In our case, I'll say
functions of two variables.

6
00:00:21,028 --> 00:00:27,010


7
00:00:27,010 --> 00:00:30,400
11.3 taught you, what?

8
00:00:30,400 --> 00:00:32,100
Taught you some
beautiful things.

9
00:00:32,100 --> 00:00:34,980
Practically, if you
understand this picture,

10
00:00:34,980 --> 00:00:37,502
you will remember everything.

11
00:00:37,502 --> 00:00:43,630
This picture is going to
try and [INAUDIBLE] a graph

12
00:00:43,630 --> 00:00:45,540
that's sitting
above here somewhere

13
00:00:45,540 --> 00:00:50,870
in Euclidean free space,
dimensional space.

14
00:00:50,870 --> 00:00:53,180
You have the origin.

15
00:00:53,180 --> 00:00:57,154
And you say I want markers.

16
00:00:57,154 --> 00:00:59,118
No, you don't say
I want markers.

17
00:00:59,118 --> 00:01:02,740
I say I want markers.

18
00:01:02,740 --> 00:01:09,380
We want to fix a point
x0, y0 on the surface,

19
00:01:09,380 --> 00:01:11,160
assuming the surface is smooth.

20
00:01:11,160 --> 00:01:14,030


21
00:01:14,030 --> 00:01:17,260
That x0 of mine
should be projected.

22
00:01:17,260 --> 00:01:20,250
I'm going to try to draw
better than I did last time.

23
00:01:20,250 --> 00:01:23,410
X0, y0 corresponds
to a certain altitude

24
00:01:23,410 --> 00:01:26,565
z0 that is projected like that.

25
00:01:26,565 --> 00:01:29,050
And this is my
[INAUDIBLE] 0 here.

26
00:01:29,050 --> 00:01:31,310
But I don't care much
about that right now.

27
00:01:31,310 --> 00:01:34,850
I care about the
fact that locally, I

28
00:01:34,850 --> 00:01:40,760
represent the function
as a graph-- z of f-- f

29
00:01:40,760 --> 00:01:43,650
of x and y defined
over a domain.

30
00:01:43,650 --> 00:01:47,630
I have a domain
that is an open set.

31
00:01:47,630 --> 00:01:50,992
And you connect to-- that's
more than you need to know.

32
00:01:50,992 --> 00:01:52,045
Could be anything.

33
00:01:52,045 --> 00:01:56,340
Could be a square, could be
a-- this could be something,

34
00:01:56,340 --> 00:01:58,343
a nice patch of them like.

35
00:01:58,343 --> 00:02:01,040


36
00:02:01,040 --> 00:02:05,350
So the projection of my
point here is x0, y0.

37
00:02:05,350 --> 00:02:08,149
I'm going to draw these
parallels as well as I can.

38
00:02:08,149 --> 00:02:10,660
But I cannot draw very well.

39
00:02:10,660 --> 00:02:12,130
But I'm trying.

40
00:02:12,130 --> 00:02:18,220
X0 and y0-- and
remember from last time.

41
00:02:18,220 --> 00:02:19,890
What did we say?

42
00:02:19,890 --> 00:02:25,125
I'm going to draw a plane
of equation x equals x0.

43
00:02:25,125 --> 00:02:27,730


44
00:02:27,730 --> 00:02:30,202
All right, I'll try.

45
00:02:30,202 --> 00:02:33,746
I'll try and do a good job--
x equals x0 is this plane.

46
00:02:33,746 --> 00:02:35,245
STUDENT: Don't you
have the x amount

47
00:02:35,245 --> 00:02:36,712
and the y amounts backward?

48
00:02:36,712 --> 00:02:38,882
Or [INAUDIBLE]

49
00:02:38,882 --> 00:02:39,465
PROFESSOR: No.

50
00:02:39,465 --> 00:02:40,430
STUDENT: [INAUDIBLE]

51
00:02:40,430 --> 00:02:44,280
PROFESSOR: x is this 1
coming towards you like that.

52
00:02:44,280 --> 00:02:47,280
And I also think about
that always, Ryan.

53
00:02:47,280 --> 00:02:48,730
Do I have them backward?

54
00:02:48,730 --> 00:02:49,930
This time, I was lucky.

55
00:02:49,930 --> 00:02:51,370
I didn't have them backward.

56
00:02:51,370 --> 00:02:52,910
So y goes this way.

57
00:02:52,910 --> 00:02:57,920
For y0, let me pick another
color, a more beautiful color.

58
00:02:57,920 --> 00:03:01,175


59
00:03:01,175 --> 00:03:07,180
For y0, my video is not
going to see the y0.

60
00:03:07,180 --> 00:03:10,763
But hopefully, it's going to
see it, this beautiful line.

61
00:03:10,763 --> 00:03:12,470
Spring is coming.

62
00:03:12,470 --> 00:03:15,486
So this is going
to be the plane.

63
00:03:15,486 --> 00:03:19,860


64
00:03:19,860 --> 00:03:24,160
Label it [INAUDIBLE]
y equals 0y.

65
00:03:24,160 --> 00:03:30,690
Now, the green plane cuts the
surface into a plane curve,

66
00:03:30,690 --> 00:03:33,030
of course, because
teasing the plane

67
00:03:33,030 --> 00:03:35,730
that I drew with the line.

68
00:03:35,730 --> 00:03:41,080
And in the plane that I drew
with red-- was it red or pink?

69
00:03:41,080 --> 00:03:42,430
It's Valentine's Day.

70
00:03:42,430 --> 00:03:43,040
It's pink.

71
00:03:43,040 --> 00:03:47,160
OK, so I have it like that.

72
00:03:47,160 --> 00:03:48,630
So what is the pink curve?

73
00:03:48,630 --> 00:03:53,680
The pink curve is the
intersection between z

74
00:03:53,680 --> 00:03:58,080
equals-- x equals x0
plane with my surface.

75
00:03:58,080 --> 00:04:00,110
My surface is black.

76
00:04:00,110 --> 00:04:03,800
I'm going to say s on surface.

77
00:04:03,800 --> 00:04:05,770
And then I have a pink curve.

78
00:04:05,770 --> 00:04:06,850
Let's call it c1.

79
00:04:06,850 --> 00:04:11,100
Because you cannot see
pink on your notes.

80
00:04:11,100 --> 00:04:14,590
You only can imagine that
it's not the same thing.

81
00:04:14,590 --> 00:04:24,398
C2 is y equals y0 plane
intersected with s.

82
00:04:24,398 --> 00:04:26,382
And what have we
learned last time?

83
00:04:26,382 --> 00:04:32,720
Last time, we learned that
we introduce some derivatives

84
00:04:32,720 --> 00:04:37,240
at the point at 0, y is
0, so that they represent

85
00:04:37,240 --> 00:04:40,400
those partial derivatives of
the function z with respect

86
00:04:40,400 --> 00:04:42,010
to x and y.

87
00:04:42,010 --> 00:04:56,550
So we have the partial z sub x
at x0, y0 and the partial and z

88
00:04:56,550 --> 00:05:00,955
sub y at x0, y0.

89
00:05:00,955 --> 00:05:04,360
Do we have a more
elegant definition?

90
00:05:04,360 --> 00:05:06,860
That's elegant enough for
me, thank you very much.

91
00:05:06,860 --> 00:05:09,810
But if I wanted to give the
original definition, what

92
00:05:09,810 --> 00:05:11,180
was that?

93
00:05:11,180 --> 00:05:19,741
That is d of bx at x0, y0, which
is a limit of the difference

94
00:05:19,741 --> 00:05:20,240
quotient.

95
00:05:20,240 --> 00:05:23,000
And this time, we're going to--
not going to do the x of y.

96
00:05:23,000 --> 00:05:25,020
I I'm different today.

97
00:05:25,020 --> 00:05:27,470
So I do h goes to 0.

98
00:05:27,470 --> 00:05:31,040
H is my smallest
displacement of [INAUDIBLE].

99
00:05:31,040 --> 00:05:34,330
Here, I have f of-- now,
who is the variable?

100
00:05:34,330 --> 00:05:34,850
X.

101
00:05:34,850 --> 00:05:38,940
So who is going to say fixed?

102
00:05:38,940 --> 00:05:47,730
Y. So I'm going to say I'm
displacing mister x0 with an h.

103
00:05:47,730 --> 00:05:55,320
And y0 will be fixed minus
f of x0, y0, all over h.

104
00:05:55,320 --> 00:06:02,310
So again, instead of-- instead
of a delta x, I call the h.

105
00:06:02,310 --> 00:06:06,010
And the derivative
with respect to y

106
00:06:06,010 --> 00:06:11,320
will assume that
x0 is a constant.

107
00:06:11,320 --> 00:06:13,750
I saw how well
[INAUDIBLE] explained that

108
00:06:13,750 --> 00:06:15,880
and I'm ambitious.

109
00:06:15,880 --> 00:06:18,040
I want to do an even better
job than [INAUDIBLE].

110
00:06:18,040 --> 00:06:20,370
Hopefully, I might manage.

111
00:06:20,370 --> 00:06:32,820
D of ty equals [INAUDIBLE] h
going to 0 of that CF of-- now,

112
00:06:32,820 --> 00:06:34,910
who's telling me what we have?

113
00:06:34,910 --> 00:06:37,970
Of course, mister
x, y and y, yy.

114
00:06:37,970 --> 00:06:43,620
F of x0, y0 is their constant
waiting for his turn.

115
00:06:43,620 --> 00:06:46,660
H is your parameter.

116
00:06:46,660 --> 00:06:50,940
And then you'll have, what?

117
00:06:50,940 --> 00:06:51,990
H0 is fixed, right?

118
00:06:51,990 --> 00:06:53,280
STUDENT: So h0 is--

119
00:06:53,280 --> 00:06:54,280
PROFESSOR: --fixed.

120
00:06:54,280 --> 00:06:56,260
Y is the variable.

121
00:06:56,260 --> 00:07:01,910
So I go into the direction
of y starting from y0.

122
00:07:01,910 --> 00:07:05,650
And I displace that with
a small quantity, right?

123
00:07:05,650 --> 00:07:09,010
So these are my
partial velocity--

124
00:07:09,010 --> 00:07:11,890
my partial derivatives, I'm
sorry, not partial velocities.

125
00:07:11,890 --> 00:07:12,765
Forget what I said.

126
00:07:12,765 --> 00:07:17,656
I said something that
you will learn later.

127
00:07:17,656 --> 00:07:18,910
What are those?

128
00:07:18,910 --> 00:07:29,700
Those are the slopes at x0, y0
of the tangents at the point

129
00:07:29,700 --> 00:07:31,330
here, OK?

130
00:07:31,330 --> 00:07:36,870
The tangents to the two curves,
the pink one-- the pink one

131
00:07:36,870 --> 00:07:42,700
and the green one, all right?

132
00:07:42,700 --> 00:07:48,670
For the pink one, for the pink
curve, what is the variable?

133
00:07:48,670 --> 00:07:51,110
The variable is the y, right?

134
00:07:51,110 --> 00:07:56,880
So this is c1 is a
curve that depends on y.

135
00:07:56,880 --> 00:07:59,670
And c2 is a curve
that depends on x.

136
00:07:59,670 --> 00:08:03,530
So this comes with x0 fixed.

137
00:08:03,530 --> 00:08:05,560
I better write it like that.

138
00:08:05,560 --> 00:08:08,390
F of x is 0.

139
00:08:08,390 --> 00:08:15,200
Y, instead of c2 of x, I'll
say f of y-- f of x and yz.

140
00:08:15,200 --> 00:08:17,970


141
00:08:17,970 --> 00:08:22,297
So, which slope is which?

142
00:08:22,297 --> 00:08:26,280


143
00:08:26,280 --> 00:08:29,890
The d of dy at this point
is the slope to this one.

144
00:08:29,890 --> 00:08:31,335
Are you guys with me?

145
00:08:31,335 --> 00:08:33,610
The slope of that tangent.

146
00:08:33,610 --> 00:08:36,760
Considered in the
plane where it is.

147
00:08:36,760 --> 00:08:38,740
How about the other one?

148
00:08:38,740 --> 00:08:47,630
S of x will be the slope of this
line in the green plane, OK?

149
00:08:47,630 --> 00:08:52,180
That is considered as a
plane of axis of coordinates.

150
00:08:52,180 --> 00:08:57,260
Good, good-- so we
know what they are.

151
00:08:57,260 --> 00:09:04,660
A quick example to review--
I've given you some really ugly,

152
00:09:04,660 --> 00:09:06,450
nasty functions today.

153
00:09:06,450 --> 00:09:08,900
The last time, you
did a good job.

154
00:09:08,900 --> 00:09:11,730
So today, I'm not
challenging you anymore.

155
00:09:11,730 --> 00:09:16,165
I'm just going to give
you one simple example.

156
00:09:16,165 --> 00:09:19,370
And I'm asked you, what
does this guy look like

157
00:09:19,370 --> 00:09:25,280
and what will the meanings
of z sub x and z sub y be?

158
00:09:25,280 --> 00:09:26,520
What will they be at?

159
00:09:26,520 --> 00:09:33,130
Let's say I think I know what I
want to take at the point 0, 0.

160
00:09:33,130 --> 00:09:40,400
And maybe you're going to
tell me what else it will be.

161
00:09:40,400 --> 00:09:42,832
And eventually at
another point like z

162
00:09:42,832 --> 00:09:55,438
sub a, so coordinates, 1
over square root of 2 and 1

163
00:09:55,438 --> 00:09:58,390
over square root of 2.

164
00:09:58,390 --> 00:10:02,818
And v sub y is same-- 1
over square root of 2,

165
00:10:02,818 --> 00:10:05,000
1 over square root of 2.

166
00:10:05,000 --> 00:10:10,150
Can one draw them and have
a geometric explanation

167
00:10:10,150 --> 00:10:13,580
of what's going on?

168
00:10:13,580 --> 00:10:16,590
Well, I don't want you to
forget the definitions,

169
00:10:16,590 --> 00:10:19,990
but since you absorbed them
with your mind hopefully

170
00:10:19,990 --> 00:10:23,760
and with your eyes, you're not
going to need them anymore.

171
00:10:23,760 --> 00:10:27,050
We should be able to draw
this quadric that you love.

172
00:10:27,050 --> 00:10:29,250
I'm sure you love it.

173
00:10:29,250 --> 00:10:31,630
When it's-- what
does it look like?

174
00:10:31,630 --> 00:10:35,995


175
00:10:35,995 --> 00:10:37,450
STUDENT: [INAUDIBLE]

176
00:10:37,450 --> 00:10:40,660
PROFESSOR: Wait a minute, you're
not awake or I'm not awake.

177
00:10:40,660 --> 00:10:44,150
So if you do x squared
plus y squared,

178
00:10:44,150 --> 00:10:46,880
don't write it down please.

179
00:10:46,880 --> 00:10:49,180
It would be that.

180
00:10:49,180 --> 00:10:50,685
And what is this?

181
00:10:50,685 --> 00:10:51,960
STUDENT: That's a [INAUDIBLE].

182
00:10:51,960 --> 00:10:54,830
PROFESSOR: A circular
paraboloid-- you are correct.

183
00:10:54,830 --> 00:10:57,420
We've done that before.

184
00:10:57,420 --> 00:10:59,660
I'd say it looks
like an egg shell,

185
00:10:59,660 --> 00:11:02,515
but it's actually--
this is a parabola

186
00:11:02,515 --> 00:11:04,815
if it's going to infinity.

187
00:11:04,815 --> 00:11:06,270
And you said a bunch of circles.

188
00:11:06,270 --> 00:11:06,770
Yes, sir.

189
00:11:06,770 --> 00:11:08,770
STUDENT: So is it an
upside down graph?

190
00:11:08,770 --> 00:11:10,827
PROFESSOR: It's an
upside down paraboloid.

191
00:11:10,827 --> 00:11:11,660
STUDENT: [INAUDIBLE]

192
00:11:11,660 --> 00:11:13,580
PROFESSOR: So, very
good-- how do we do that?

193
00:11:13,580 --> 00:11:16,286
We make this guy
look in the mirror.

194
00:11:16,286 --> 00:11:17,265
This is the lake.

195
00:11:17,265 --> 00:11:18,455
The lake is xy plane.

196
00:11:18,455 --> 00:11:21,470
So this guy is
looking in the mirror.

197
00:11:21,470 --> 00:11:25,160
Take his image and
shift it just like he

198
00:11:25,160 --> 00:11:29,270
said-- shift it one unit up.

199
00:11:29,270 --> 00:11:30,750
This is one.

200
00:11:30,750 --> 00:11:34,024
You're going to have
another paraboloid.

201
00:11:34,024 --> 00:11:39,370
So from this construction,
I'm going to draw.

202
00:11:39,370 --> 00:11:42,045
And he's going to look
like you took a cup

203
00:11:42,045 --> 00:11:44,160
and you put it upside down.

204
00:11:44,160 --> 00:11:45,770
But it's more like
an eggshell, right?

205
00:11:45,770 --> 00:11:48,950
It's not a cup because
a cup is supposed

206
00:11:48,950 --> 00:11:50,770
to have a flat bottom, right?

207
00:11:50,770 --> 00:11:54,690
But this is like an eggshell.

208
00:11:54,690 --> 00:11:58,090
And I'll draw.

209
00:11:58,090 --> 00:12:01,800
And for this fellow, we
have a beautiful picture

210
00:12:01,800 --> 00:12:09,945
that looks like this hopefully
But I'm going to try and draw.

211
00:12:09,945 --> 00:12:12,560
STUDENT: Are you looking
from a top to bottom?

212
00:12:12,560 --> 00:12:16,470
PROFESSOR: We can look it
whatever you want to look.

213
00:12:16,470 --> 00:12:19,300
That's a very good thing.

214
00:12:19,300 --> 00:12:23,052
You're getting too close
to what I wanted to go.

215
00:12:23,052 --> 00:12:25,380
We'll discuss in one minute.

216
00:12:25,380 --> 00:12:29,800
So you can imagine this
is a hill full of snow.

217
00:12:29,800 --> 00:12:33,272
Although in two days,
we have Valentine's Day

218
00:12:33,272 --> 00:12:35,310
and there is no snow.

219
00:12:35,310 --> 00:12:38,610
But assume that we
go to New Mexico

220
00:12:38,610 --> 00:12:41,380
and we find a hill
that more or less looks

221
00:12:41,380 --> 00:12:44,510
like a perfect hill like that.

222
00:12:44,510 --> 00:12:50,430
And we start thinking
of skiing down the hill.

223
00:12:50,430 --> 00:12:52,570
Where am I at 0, 0?

224
00:12:52,570 --> 00:12:55,087
I am on top of the hill.

225
00:12:55,087 --> 00:12:59,420
I'm on top of the
hill and I decide

226
00:12:59,420 --> 00:13:03,740
to analyze the slope
of the tangents

227
00:13:03,740 --> 00:13:08,100
to the surface in the
direction of-- who is this?

228
00:13:08,100 --> 00:13:10,290
Like now, and you
make me nervous.

229
00:13:10,290 --> 00:13:13,550
So in the direction of
y, I have one slope.

230
00:13:13,550 --> 00:13:17,620
In the direction of x, I have
another slope in general.

231
00:13:17,620 --> 00:13:21,050
Only in this case, they
are the same slope.

232
00:13:21,050 --> 00:13:25,490
And what is that same slope if
I'm here on top of the hill?

233
00:13:25,490 --> 00:13:30,180
This is me-- well, I don't
know, one of you guys.

234
00:13:30,180 --> 00:13:34,004


235
00:13:34,004 --> 00:13:37,282
That looks horrible.

236
00:13:37,282 --> 00:13:38,240
What's going to happen?

237
00:13:38,240 --> 00:13:39,820
We don't want to think about it.

238
00:13:39,820 --> 00:13:42,350
But it definitely is too steep.

239
00:13:42,350 --> 00:13:46,030
So this will be the slope
of the line in the direction

240
00:13:46,030 --> 00:13:46,950
with respect to y.

241
00:13:46,950 --> 00:13:50,080
So I'm going to think
f sub y and f sub

242
00:13:50,080 --> 00:13:55,930
x if I change my skis go
this direction and I go down.

243
00:13:55,930 --> 00:14:01,726
So I could go down this
way and break my neck.

244
00:14:01,726 --> 00:14:07,530
Or I could go down this way
and break my neck as well.

245
00:14:07,530 --> 00:14:15,670
OK, it has to go like-- right?

246
00:14:15,670 --> 00:14:17,937
Can you tell me what
these guys will be?

247
00:14:17,937 --> 00:14:20,270
I'm going to put them in pink
because they're beautiful.

248
00:14:20,270 --> 00:14:21,120
STUDENT: 0 [INAUDIBLE].

249
00:14:21,120 --> 00:14:22,828
PROFESSOR: Thank God,
they are beautiful.

250
00:14:22,828 --> 00:14:23,900
Larry, what does it mean?

251
00:14:23,900 --> 00:14:28,070
That means that the two
tangents, the tangents

252
00:14:28,070 --> 00:14:33,710
to the curves, are horizontal.

253
00:14:33,710 --> 00:14:38,100
And if I were to draw the plane
between those two tangents--

254
00:14:38,100 --> 00:14:44,566
one tangent is in pink
pen, our is in green.

255
00:14:44,566 --> 00:14:46,112
Today, I'm all about colors.

256
00:14:46,112 --> 00:14:47,090
I'm in a good mood.

257
00:14:47,090 --> 00:14:50,690


258
00:14:50,690 --> 00:14:54,970
And that's going to be the
so-called tangent plane--

259
00:14:54,970 --> 00:15:06,980
tangent plane to the surface
at x0, y0, which is the origin.

260
00:15:06,980 --> 00:15:09,290
That was a nice point.

261
00:15:09,290 --> 00:15:10,680
That is a nice point.

262
00:15:10,680 --> 00:15:12,550
Not all the points
will be [INAUDIBLE]

263
00:15:12,550 --> 00:15:14,680
and nice but beautiful.

264
00:15:14,680 --> 00:15:18,304
[INAUDIBLE] I take the
nice-- well, not so nice,

265
00:15:18,304 --> 00:15:19,150
I don't know.

266
00:15:19,150 --> 00:15:21,534
You'll have to figure it out.

267
00:15:21,534 --> 00:15:26,530
How do I get-- well, first
of all, where is this point?

268
00:15:26,530 --> 00:15:31,720
If I take x to be 1 over square
2 and y to be 1 over square 2

269
00:15:31,720 --> 00:15:33,870
and I plug them in,
what's z going to be?

270
00:15:33,870 --> 00:15:35,010
STUDENT: 0.

271
00:15:35,010 --> 00:15:37,641
PROFESSOR: 0, and I
did that on purpose.

272
00:15:37,641 --> 00:15:41,020
Because in that case, I'm
going to be on flat line again.

273
00:15:41,020 --> 00:15:42,250
This look like [INAUDIBLE].

274
00:15:42,250 --> 00:15:45,090
Except [INAUDIBLE]
is not z equal 0.

275
00:15:45,090 --> 00:15:47,490
What is [INAUDIBLE] like?

276
00:15:47,490 --> 00:15:48,390
z equals--

277
00:15:48,390 --> 00:15:49,322
STUDENT: [INAUDIBLE]

278
00:15:49,322 --> 00:15:49,946
PROFESSOR: Huh?

279
00:15:49,946 --> 00:15:51,485
STUDENT: I don't know.

280
00:15:51,485 --> 00:15:54,080
PROFESSOR: Do you want to
go in meters or in feet?

281
00:15:54,080 --> 00:15:56,280
STUDENT: [INAUDIBLE].

282
00:15:56,280 --> 00:15:57,600
It's about a mile.

283
00:15:57,600 --> 00:15:59,080
PROFESSOR: Yes, I don't know.

284
00:15:59,080 --> 00:16:02,840
I thought it's about one
kilometer, 1,000 something

285
00:16:02,840 --> 00:16:03,820
meters.

286
00:16:03,820 --> 00:16:05,870
But somebody said it's more so.

287
00:16:05,870 --> 00:16:10,040
It's flat land, and I'd say
about a mile above the sea

288
00:16:10,040 --> 00:16:11,090
level.

289
00:16:11,090 --> 00:16:20,410
All right, now, I am going to
be in flat land right here,

290
00:16:20,410 --> 00:16:24,770
1 over a root 2, 1
over a root 2, and 0.

291
00:16:24,770 --> 00:16:26,060
What happened here?

292
00:16:26,060 --> 00:16:28,986
Here, I just already
broke my neck, you know.

293
00:16:28,986 --> 00:16:32,290
Well, if I came
in this direction,

294
00:16:32,290 --> 00:16:36,616
I would need to draw a
prospective trajectory that

295
00:16:36,616 --> 00:16:38,880
was hopefully not mine.

296
00:16:38,880 --> 00:16:43,699
And the tangent would--
the tangent, the slope

297
00:16:43,699 --> 00:16:45,650
of the tangent, would be funny.

298
00:16:45,650 --> 00:16:47,740
Let's see what you need to do.

299
00:16:47,740 --> 00:16:52,540
You need to say, OK, prime
with respect to x, minus 2x.

300
00:16:52,540 --> 00:16:56,990
And then at the point x
equals 1 over [INAUDIBLE] 2 y

301
00:16:56,990 --> 00:16:59,948
equals 1 over 2,
you just plug in.

302
00:16:59,948 --> 00:17:00,920
And what do you have?

303
00:17:00,920 --> 00:17:02,420
STUDENT: Square
root of [INAUDIBLE].

304
00:17:02,420 --> 00:17:05,000
PROFESSOR: Negative
square root of 2-- my god

305
00:17:05,000 --> 00:17:08,940
that is really bad as a slope.

306
00:17:08,940 --> 00:17:11,002
It's a steep slope.

307
00:17:11,002 --> 00:17:14,010
And this one-- how
about this one?

308
00:17:14,010 --> 00:17:16,098
Same idea, symmetric function.

309
00:17:16,098 --> 00:17:20,339
And it's going to be exactly
the same-- very steep slope.

310
00:17:20,339 --> 00:17:22,819
Why are they negative numbers?

311
00:17:22,819 --> 00:17:27,470
Because the slope is
going down, right?

312
00:17:27,470 --> 00:17:31,990
That's the kind of slope I
have in both directions--

313
00:17:31,990 --> 00:17:38,570
one and-- all right.

314
00:17:38,570 --> 00:17:48,310
If I were to draw
this thing continuing,

315
00:17:48,310 --> 00:17:51,800
how would I represent
those slopes?

316
00:17:51,800 --> 00:17:56,682
This circle-- this circle is
just making my life harder.

317
00:17:56,682 --> 00:17:59,600
But I would need to imagine
those slopes as being

318
00:17:59,600 --> 00:18:02,330
like I'm here, all right?

319
00:18:02,330 --> 00:18:03,700
Are you guys with me?

320
00:18:03,700 --> 00:18:12,360
And I will need to draw
x0-- well, what is that?

321
00:18:12,360 --> 00:18:24,782
1 over root 2 and 1 over root
2 And I would draw two planes.

322
00:18:24,782 --> 00:18:30,084
And I would have two curves.

323
00:18:30,084 --> 00:18:33,220
And when you slice
up, imagine this

324
00:18:33,220 --> 00:18:34,487
would be a piece of cheese.

325
00:18:34,487 --> 00:18:35,320
STUDENT: [INAUDIBLE]

326
00:18:35,320 --> 00:18:35,955
PROFESSOR: And you cut--

327
00:18:35,955 --> 00:18:36,788
STUDENT: [INAUDIBLE]

328
00:18:36,788 --> 00:18:41,782


329
00:18:41,782 --> 00:18:42,490
PROFESSOR: Right?

330
00:18:42,490 --> 00:18:43,073
STUDENT: Yeah.

331
00:18:43,073 --> 00:18:44,970
PROFESSOR: And you cut
in this other side.

332
00:18:44,970 --> 00:18:48,806
Well, this is the one
that's facing you.

333
00:18:48,806 --> 00:18:49,652
You cut like that.

334
00:18:49,652 --> 00:18:51,360
And when you cut like
this, it's facing--

335
00:18:51,360 --> 00:18:52,193
STUDENT: [INAUDIBLE]

336
00:18:52,193 --> 00:18:54,853


337
00:18:54,853 --> 00:18:56,849
PROFESSOR: Hm?

338
00:18:56,849 --> 00:18:59,344
But anyway, let's not
draw the other one.

339
00:18:59,344 --> 00:19:00,841
It's hard, right?

340
00:19:00,841 --> 00:19:04,084
STUDENT: [INAUDIBLE] angle
like this-- just the piece

341
00:19:04,084 --> 00:19:05,622
of the corner of the cheese.

342
00:19:05,622 --> 00:19:06,330
PROFESSOR: Right.

343
00:19:06,330 --> 00:19:08,350
STUDENT: The corner
is facing you.

344
00:19:08,350 --> 00:19:13,248
PROFESSOR: So yeah, so it's--
the corner is facing you.

345
00:19:13,248 --> 00:19:17,000
STUDENT: So basically,
you [INAUDIBLE] this.

346
00:19:17,000 --> 00:19:18,600
PROFESSOR: But-- exactly, but--

347
00:19:18,600 --> 00:19:20,020
STUDENT: Like this.

348
00:19:20,020 --> 00:19:22,890
PROFESSOR: Yeah, well--
yeah, it's hard to draw.

349
00:19:22,890 --> 00:19:25,710
So practically,
this is what you're

350
00:19:25,710 --> 00:19:29,450
looking at it is slope that's
negative in both directions.

351
00:19:29,450 --> 00:19:34,760
So you're going to go
this way or this way.

352
00:19:34,760 --> 00:19:39,140
And it's much steeper than
you imagine [INAUDIBLE].

353
00:19:39,140 --> 00:19:43,000
OK, they are equal.

354
00:19:43,000 --> 00:19:44,525
I'm trying to draw them equal.

355
00:19:44,525 --> 00:19:48,130
I don't know how
equal they can be.

356
00:19:48,130 --> 00:19:55,780
One belongs to one plane
just like you said.

357
00:19:55,780 --> 00:19:58,323
This belongs to this plane.

358
00:19:58,323 --> 00:20:07,476
And the green one belongs to
the plane that's facing you.

359
00:20:07,476 --> 00:20:08,784
So the slope goes this way.

360
00:20:08,784 --> 00:20:11,170
But the two slopes are equal.

361
00:20:11,170 --> 00:20:13,480
You have to have a little
bit of imagination.

362
00:20:13,480 --> 00:20:16,650
We would need some cheese
to make a mountain of cheese

363
00:20:16,650 --> 00:20:18,400
and cut them and slice them.

364
00:20:18,400 --> 00:20:21,640
We'll eat everything
after, yeah.

365
00:20:21,640 --> 00:20:29,030
All right, let's move on to
something more challenging

366
00:20:29,030 --> 00:20:31,780
now that we got to
the tangent plane.

367
00:20:31,780 --> 00:20:34,200
So if somebody would
say, wait a minute,

368
00:20:34,200 --> 00:20:38,230
you said this is the tangent
plane to the surface.

369
00:20:38,230 --> 00:20:40,370
You just introduced
a new notion.

370
00:20:40,370 --> 00:20:41,450
You were fooling us.

371
00:20:41,450 --> 00:20:43,820
I'm fooling you guys.

372
00:20:43,820 --> 00:20:48,590
It's not April 1, but this
kind of a not a neat thing.

373
00:20:48,590 --> 00:20:58,504
I just tried to introduce
you into the section 11.4.

374
00:20:58,504 --> 00:21:02,710
So if you have a piece
of a curve that's smooth

375
00:21:02,710 --> 00:21:08,010
and you have a point
x0, y0, can you

376
00:21:08,010 --> 00:21:12,860
find out the equation
of the tangent plane?

377
00:21:12,860 --> 00:21:16,610
Pi, and this is s form surface.

378
00:21:16,610 --> 00:21:20,820
How can I find the equation
of the tangent plane?

379
00:21:20,820 --> 00:21:27,180


380
00:21:27,180 --> 00:21:33,210
That x0, y0-- 12 is
going to be also z0.

381
00:21:33,210 --> 00:21:44,880
But what I mean that x0,
y0 is in on the floor

382
00:21:44,880 --> 00:21:47,210
as a projection.

383
00:21:47,210 --> 00:21:50,500
So I'm always
looking at the graph.

384
00:21:50,500 --> 00:21:52,496
And that's why.

385
00:21:52,496 --> 00:21:54,299
The moment I stop
looking at the graph,

386
00:21:54,299 --> 00:21:55,340
things will be different.

387
00:21:55,340 --> 00:21:59,770
But I'm looking at the graph
of independent variables x, y.

388
00:21:59,770 --> 00:22:02,505
And that's why those guys
are always on the floor.

389
00:22:02,505 --> 00:22:07,560
A and z would be a function
to keep in the variable.

390
00:22:07,560 --> 00:22:09,300
Now, does anybody know?

391
00:22:09,300 --> 00:22:12,610
Because I know you guys
are reading in advance

392
00:22:12,610 --> 00:22:16,260
and you have better
teachers than me.

393
00:22:16,260 --> 00:22:17,374
You have the internet.

394
00:22:17,374 --> 00:22:18,165
You have the links.

395
00:22:18,165 --> 00:22:18,957
You have YouTube.

396
00:22:18,957 --> 00:22:20,620
You have Khan Academy.

397
00:22:20,620 --> 00:22:24,532
I know from a bunch of you
that you have already gone

398
00:22:24,532 --> 00:22:27,240
over half of the chapter 11.

399
00:22:27,240 --> 00:22:30,840
I just hope that now you
can compare what you learned

400
00:22:30,840 --> 00:22:34,110
with what I'm teaching
you, And I'm not

401
00:22:34,110 --> 00:22:36,840
expecting you to go in
advance, but several of you

402
00:22:36,840 --> 00:22:38,590
already know this formula.

403
00:22:38,590 --> 00:22:43,620
We talked about it in
office hours on yesterday.

404
00:22:43,620 --> 00:22:45,991
Because Tuesday, I
didn't have office hours.

405
00:22:45,991 --> 00:22:49,790
I had a coordinator meeting.

406
00:22:49,790 --> 00:22:55,942
So what equation corresponds
to the tangent plate?

407
00:22:55,942 --> 00:22:56,775
STUDENT: [INAUDIBLE]

408
00:22:56,775 --> 00:23:00,420


409
00:23:00,420 --> 00:23:02,120
PROFESSOR: Several
of you know it.

410
00:23:02,120 --> 00:23:03,970
You know what I hated?

411
00:23:03,970 --> 00:23:05,120
It's fine that you know it.

412
00:23:05,120 --> 00:23:08,250
I'm proud of you guys
and I'll write it.

413
00:23:08,250 --> 00:23:12,320
But when I was a freshman--
or what the heck was I?

414
00:23:12,320 --> 00:23:17,240
A sophomore I think-- no, I was
a freshman when they fed me.

415
00:23:17,240 --> 00:23:19,365
They spoon-fed me this equation.

416
00:23:19,365 --> 00:23:22,380
And I didn't understand
anything at the time.

417
00:23:22,380 --> 00:23:25,910
I hated the fact that
the Professor painted it

418
00:23:25,910 --> 00:23:30,860
on the board just like
that out of the blue.

419
00:23:30,860 --> 00:23:33,700
I want to see a proof.

420
00:23:33,700 --> 00:23:39,630
And he was able to-- I think
he could have done a good job.

421
00:23:39,630 --> 00:23:42,730
But he didn't.

422
00:23:42,730 --> 00:23:46,790
He showed us a bunch
of justifications

423
00:23:46,790 --> 00:23:53,400
like if you generally have
a surface in implicit form,

424
00:23:53,400 --> 00:23:58,100
I told you that
the gradient of F

425
00:23:58,100 --> 00:24:01,530
represents the normal
connection, right?

426
00:24:01,530 --> 00:24:06,510
And he prepared us pretty
good for what could

427
00:24:06,510 --> 00:24:08,850
have been the proof of that.

428
00:24:08,850 --> 00:24:10,195
He said, OK, guys.

429
00:24:10,195 --> 00:24:12,470
You know the duration
of the normal

430
00:24:12,470 --> 00:24:16,000
as even as the gradient over
the next of the gradient,

431
00:24:16,000 --> 00:24:17,954
if you want unit normal.

432
00:24:17,954 --> 00:24:19,340
How did he do that?

433
00:24:19,340 --> 00:24:21,900
Well, he had a
bunch of examples.

434
00:24:21,900 --> 00:24:23,440
He had the sphere.

435
00:24:23,440 --> 00:24:25,805
He showed us that
for the sphere,

436
00:24:25,805 --> 00:24:29,884
you have the normal,
which is the continuation

437
00:24:29,884 --> 00:24:31,400
of the position vector.

438
00:24:31,400 --> 00:24:34,260
Then he said, OK, you
can have approximations

439
00:24:34,260 --> 00:24:39,370
of a surface that is smooth and
round with oscillating spheres

440
00:24:39,370 --> 00:24:45,292
just the way you have for a
curve, a resonating circle,

441
00:24:45,292 --> 00:24:49,560
a resonating circle-- that's
called oscillating circle.

442
00:24:49,560 --> 00:24:53,070
Resonating circle-- in that
case, what will the normal be?

443
00:24:53,070 --> 00:24:55,570
Well, the normal
will have to depend

444
00:24:55,570 --> 00:24:57,120
on the radius of the circle.

445
00:24:57,120 --> 00:25:01,945
So you have a principal normal
or a normal if it's a plane

446
00:25:01,945 --> 00:25:02,790
curve.

447
00:25:02,790 --> 00:25:05,850
And it's easy to
understand that's the same

448
00:25:05,850 --> 00:25:07,210
as the gradient.

449
00:25:07,210 --> 00:25:12,680
So we have enough
justification for the direction

450
00:25:12,680 --> 00:25:16,230
of the gradient of such
a function is always

451
00:25:16,230 --> 00:25:19,867
normal-- normal to the
surface, normal to all

452
00:25:19,867 --> 00:25:22,730
the curves on the surface.

453
00:25:22,730 --> 00:25:26,660
If we want to find
that without swallowing

454
00:25:26,660 --> 00:25:32,410
this like I had to when I
was a student, it's not hard.

455
00:25:32,410 --> 00:25:35,320
And let me show
you how we do it.

456
00:25:35,320 --> 00:25:37,330
We start from the graph, right?

457
00:25:37,330 --> 00:25:40,800
Z equals f of x and y.

458
00:25:40,800 --> 00:25:44,200
And we say, well, Magdalena,
but this is a graph.

459
00:25:44,200 --> 00:25:47,550
It's not an implicit equation.

460
00:25:47,550 --> 00:25:50,320
And I'll say, yes it is.

461
00:25:50,320 --> 00:25:53,980
Let me show you how I make
it an implicit equation.

462
00:25:53,980 --> 00:25:56,302
I move z to the other side.

463
00:25:56,302 --> 00:26:00,630
I put 0 equals f of xy minus z.

464
00:26:00,630 --> 00:26:04,272
Now it is an implicit equation.

465
00:26:04,272 --> 00:26:05,720
So you say you cheated.

466
00:26:05,720 --> 00:26:07,530
Yes, I did.

467
00:26:07,530 --> 00:26:08,190
I have cheated.

468
00:26:08,190 --> 00:26:10,750


469
00:26:10,750 --> 00:26:14,880
It's funny that whenever
somebody gives you a graph,

470
00:26:14,880 --> 00:26:16,920
you can rewrite that
graph immediately

471
00:26:16,920 --> 00:26:18,710
as an implicit equation.

472
00:26:18,710 --> 00:26:23,760
So that implicit equation
is of the form big F of xyz

473
00:26:23,760 --> 00:26:27,470
now equals a
constant, which is 0.

474
00:26:27,470 --> 00:26:32,330
F of xy is your old
friend and minus z.

475
00:26:32,330 --> 00:26:36,340
Now, can you tell me what is
the normal to this surface?

476
00:26:36,340 --> 00:26:40,010
Yeah, give me a splash
in a minute like that.

477
00:26:40,010 --> 00:26:43,192
So what is the gradient of f?

478
00:26:43,192 --> 00:26:44,995
Gradient of f will
be the normal.

479
00:26:44,995 --> 00:26:47,110
I don't care if
it's unit or not.

480
00:26:47,110 --> 00:26:49,460
To heck with the unit or normal.

481
00:26:49,460 --> 00:26:54,180
I'm going to say I wanted
prime with respect to x, y,

482
00:26:54,180 --> 00:26:58,030
and z respectively.

483
00:26:58,030 --> 00:26:59,760
And what is the gradient?

484
00:26:59,760 --> 00:27:01,790
Is the vector.

485
00:27:01,790 --> 00:27:06,920
Big F sub x comma big F
sub y comma big F sub z.

486
00:27:06,920 --> 00:27:08,900
We see that last time.

487
00:27:08,900 --> 00:27:13,620
So the gradient of a
function is the vector

488
00:27:13,620 --> 00:27:17,130
whose coordinates are
the partial velocity--

489
00:27:17,130 --> 00:27:19,560
your friends form last time.

490
00:27:19,560 --> 00:27:22,090
Can we represent this again?

491
00:27:22,090 --> 00:27:22,650
I don't know.

492
00:27:22,650 --> 00:27:24,300
You need to help me.

493
00:27:24,300 --> 00:27:29,339
Who is big F prime
with respect to x?

494
00:27:29,339 --> 00:27:30,130
There is no x here.

495
00:27:30,130 --> 00:27:32,620
Thank God that's
like a constant.

496
00:27:32,620 --> 00:27:36,170
I just have to take this
little one, f, and prime it

497
00:27:36,170 --> 00:27:36,960
with respect to x.

498
00:27:36,960 --> 00:27:41,050
And that's exactly what that's
going to be-- little f sub x.

499
00:27:41,050 --> 00:27:44,800
What is big F with respect to y?

500
00:27:44,800 --> 00:27:46,040
STUDENT: [INAUDIBLE]

501
00:27:46,040 --> 00:27:49,205
PROFESSOR: Little f sub
y prime with respect

502
00:27:49,205 --> 00:27:51,940
to y-- differentiated
with respect to y.

503
00:27:51,940 --> 00:27:56,210
And finally, if I differentiated
with respect to z,

504
00:27:56,210 --> 00:27:57,520
there is no z here, right?

505
00:27:57,520 --> 00:27:58,490
There is no z.

506
00:27:58,490 --> 00:28:00,000
So that's like a constant.

507
00:28:00,000 --> 00:28:04,540
Prime [INAUDIBLE] 0 and minus 1.

508
00:28:04,540 --> 00:28:05,770
So I know the gradient.

509
00:28:05,770 --> 00:28:06,880
I know the normal.

510
00:28:06,880 --> 00:28:09,140
This is the normal.

511
00:28:09,140 --> 00:28:14,180
Now, if somebody gives you
the normal, there you are.

512
00:28:14,180 --> 00:28:20,230
You have the normal to the
surface-- normal to surface.

513
00:28:20,230 --> 00:28:21,282
What does it mean?

514
00:28:21,282 --> 00:28:26,480
Equals normal to the tangent
plane to the surface.

515
00:28:26,480 --> 00:28:30,054
Normal or perpendicular
to the tangent plane-

516
00:28:30,054 --> 00:28:37,520
to the plane-- of the surface.

517
00:28:37,520 --> 00:28:41,470
At that point--
point is the point p.

518
00:28:41,470 --> 00:28:44,500


519
00:28:44,500 --> 00:28:52,530
All right, so if you were to
study a surface that's-- do you

520
00:28:52,530 --> 00:28:53,570
have a [INAUDIBLE]?

521
00:28:53,570 --> 00:28:54,700
STUDENT: Uh, no.

522
00:28:54,700 --> 00:28:55,517
Do you?

523
00:28:55,517 --> 00:28:56,100
PROFESSOR: OK.

524
00:28:56,100 --> 00:28:59,300


525
00:28:59,300 --> 00:29:03,520
OK, I want to study
the tangent plane

526
00:29:03,520 --> 00:29:05,270
at this point to the surface.

527
00:29:05,270 --> 00:29:06,670
Well, that's flat, Magdalena.

528
00:29:06,670 --> 00:29:08,460
You have no imagination.

529
00:29:08,460 --> 00:29:13,770
The tangent plane is this plane,
is the same as the surface.

530
00:29:13,770 --> 00:29:16,490
So, no fun-- no fun.

531
00:29:16,490 --> 00:29:19,880
How about I pick my
favorite plane here

532
00:29:19,880 --> 00:29:24,690
and I take-- what is-- OK.

533
00:29:24,690 --> 00:29:26,765
I have-- this is
Children Internationals.

534
00:29:26,765 --> 00:29:30,300
I have a little girl
abroad that I'm sponsoring.

535
00:29:30,300 --> 00:29:34,500
So you have a point
here and a plane

536
00:29:34,500 --> 00:29:38,780
that passes through that point.

537
00:29:38,780 --> 00:29:40,590
This is the tangent plane.

538
00:29:40,590 --> 00:29:43,610
And my finger is the normal.

539
00:29:43,610 --> 00:29:46,910
And the normal, we call
that normal to the surface

540
00:29:46,910 --> 00:29:49,316
when it's normal to
the tangent plane.

541
00:29:49,316 --> 00:29:52,972
At every point, this
is what the normal is.

542
00:29:52,972 --> 00:29:55,490
All right, can we write
that based on chapter nine?

543
00:29:55,490 --> 00:29:59,179
Now I will see what you remember
from chapter nine if anything

544
00:29:59,179 --> 00:30:00,157
at all.

545
00:30:00,157 --> 00:30:03,580


546
00:30:03,580 --> 00:30:09,430
All right, how do we
write the tangent plane

547
00:30:09,430 --> 00:30:11,826
if we know the normal?

548
00:30:11,826 --> 00:30:21,550
OK, review-- if the normal
vector is ai plus bj plus ck,

549
00:30:21,550 --> 00:30:28,060
that means the plane that
is perpendicular to it

550
00:30:28,060 --> 00:30:30,650
is of what form?

551
00:30:30,650 --> 00:30:37,548
Ax plus by plus cz
plus d equals 0, right?

552
00:30:37,548 --> 00:30:39,540
You've learned that
in chapter nine.

553
00:30:39,540 --> 00:30:43,370
Most of you learned that
last semester in Calculus 2

554
00:30:43,370 --> 00:30:44,590
at the end.

555
00:30:44,590 --> 00:30:51,290
Now, if my normal is f sub
x, f sub y, and minus 1,

556
00:30:51,290 --> 00:30:52,810
those are ABC for God's sake.

557
00:30:52,810 --> 00:30:53,930
Well, good.

558
00:30:53,930 --> 00:30:59,400
Big A, big B, big C
at the given point.

559
00:30:59,400 --> 00:31:09,810
So I'm going to have f sub
x at the given point d times

560
00:31:09,810 --> 00:31:17,760
x plus f sub y at any
given point d times y.

561
00:31:17,760 --> 00:31:18,980
Who is c?

562
00:31:18,980 --> 00:31:20,420
C is minus 1.

563
00:31:20,420 --> 00:31:24,180
Minus 1 times z is--
say you're being silly.

564
00:31:24,180 --> 00:31:26,552
Magdalena, why do
you write minus 1?

565
00:31:26,552 --> 00:31:28,890
Just because I'm having fun.

566
00:31:28,890 --> 00:31:32,780
And plus, d equals 0.

567
00:31:32,780 --> 00:31:34,590
And you say, well,
wait, wait, wait.

568
00:31:34,590 --> 00:31:39,880
This starts looking like that
but it's not the same thing.

569
00:31:39,880 --> 00:31:42,830
All right, what?

570
00:31:42,830 --> 00:31:44,243
How do you get to d?

571
00:31:44,243 --> 00:31:47,700


572
00:31:47,700 --> 00:31:50,510
Now, actually, the
plane perpendicular

573
00:31:50,510 --> 00:31:55,180
to n that passes
through a given point

574
00:31:55,180 --> 00:31:58,920
can be written
much faster, right?

575
00:31:58,920 --> 00:32:06,024
So if a plane is perpendicular
to a certain line,

576
00:32:06,024 --> 00:32:09,535
how do we write if
we know a point?

577
00:32:09,535 --> 00:32:15,190
If we know a point
in the normal ABC--

578
00:32:15,190 --> 00:32:18,920
I have to go backwards to
read it backwards-- then

579
00:32:18,920 --> 00:32:22,990
the plane is going
to be x minus x0

580
00:32:22,990 --> 00:32:29,530
plus b times y times y0 plus
c times z minus c0 equals 0.

581
00:32:29,530 --> 00:32:33,030


582
00:32:33,030 --> 00:32:34,680
So who is the d?

583
00:32:34,680 --> 00:32:39,310
The d is all the constant
that gets out of here.

584
00:32:39,310 --> 00:32:43,671
So the point x0, y0, z0
has to verify the plane.

585
00:32:43,671 --> 00:32:47,080
And that's why when
you plug in x0, y0, z0,

586
00:32:47,080 --> 00:32:50,420
you get 0 plus 0
plus 0 equals 0.

587
00:32:50,420 --> 00:32:53,490
That's what it means for a
point to verify the plane.

588
00:32:53,490 --> 00:32:59,035
When you take the x0, y0, z0 and
you plug it into the equation,

589
00:32:59,035 --> 00:33:02,770
you have to have an
identity 0 equals 0.

590
00:33:02,770 --> 00:33:06,800
So this can be rewritten
zx plus by plus cz

591
00:33:06,800 --> 00:33:10,890
just like we did there plus a d.

592
00:33:10,890 --> 00:33:12,535
And who in the world is the d?

593
00:33:12,535 --> 00:33:18,730
The d will be exactly minus
ax0 minus by0 minus cz0.

594
00:33:18,730 --> 00:33:22,940
If that makes you uncomfortable,
this is in chapter nine.

595
00:33:22,940 --> 00:33:28,890
Look at the equation of a
plane and the normal to it.

596
00:33:28,890 --> 00:33:32,760
Now I know that I can do
better than that if I'm smart.

597
00:33:32,760 --> 00:33:35,850
So again, I collect the ABC.

598
00:33:35,850 --> 00:33:37,176
Now I know my ABC.

599
00:33:37,176 --> 00:33:40,325


600
00:33:40,325 --> 00:33:42,225
I put them in here.

601
00:33:42,225 --> 00:33:46,170
So I have f sub x at
the point in time.

602
00:33:46,170 --> 00:33:50,600
Oh, OK, x minus x0
plus, who is my b?

603
00:33:50,600 --> 00:33:57,120
F sub y computed at the
point p times y minus y0.

604
00:33:57,120 --> 00:33:58,760
And, what?

605
00:33:58,760 --> 00:33:59,930
Minus, right?

606
00:33:59,930 --> 00:34:03,500
Minus-- minus 1.

607
00:34:03,500 --> 00:34:05,040
I'm not going to write minus 1.

608
00:34:05,040 --> 00:34:07,330
You're going to make fun of me.

609
00:34:07,330 --> 00:34:10,237
Minus z minus cz.

610
00:34:10,237 --> 00:34:12,060
And my proof is done.

611
00:34:12,060 --> 00:34:17,020
QED-- what does it mean, QED?

612
00:34:17,020 --> 00:34:19,620
In Latin.

613
00:34:19,620 --> 00:34:22,530
QED means I proved
what I wanted to prove.

614
00:34:22,530 --> 00:34:23,960
Do you know what it stands for?

615
00:34:23,960 --> 00:34:26,717
Did you take Latin, any of you?

616
00:34:26,717 --> 00:34:29,090
You took Latin?

617
00:34:29,090 --> 00:34:33,190
Quod erat demonstrandum.

618
00:34:33,190 --> 00:34:36,699


619
00:34:36,699 --> 00:34:39,431
So this was to be proved.

620
00:34:39,431 --> 00:34:42,050
That's exactly what
it was to be proved.

621
00:34:42,050 --> 00:34:44,647
That, what, that
c minus z0, which

622
00:34:44,647 --> 00:34:50,179
was my fellow over here pretty
in pink, is going to be f sub x

623
00:34:50,179 --> 00:34:54,620
times x minus x0 plus yf
sub y times y minus y0.

624
00:34:54,620 --> 00:35:01,140
So now you know why the equation
of the tangent plane is that.

625
00:35:01,140 --> 00:35:04,525
I proved it more or less,
making some assumptions,

626
00:35:04,525 --> 00:35:06,830
some axioms as assumption.

627
00:35:06,830 --> 00:35:09,540
But you don't know
how to use it.

628
00:35:09,540 --> 00:35:10,860
So let's use it.

629
00:35:10,860 --> 00:35:14,255
So for the same valley--
not valley, hill--

630
00:35:14,255 --> 00:35:16,170
it was full of snow.

631
00:35:16,170 --> 00:35:19,375
Z equals 1 minus x
squared-- what was you

632
00:35:19,375 --> 00:35:20,810
guys have forgotten?

633
00:35:20,810 --> 00:35:24,970
OK, 1 minus x squared
minus y squared.

634
00:35:24,970 --> 00:35:32,570
Find the tangent plane
at the following points.

635
00:35:32,570 --> 00:35:37,045
Ah, x0, y0 to be origin.

636
00:35:37,045 --> 00:35:39,470
And you say, did you
say that that's trivial?

637
00:35:39,470 --> 00:35:40,480
Yes, it is trivial.

638
00:35:40,480 --> 00:35:42,950
But I'm going to do
it one more time.

639
00:35:42,950 --> 00:35:47,190
And what was my
[INAUDIBLE] point before?

640
00:35:47,190 --> 00:35:48,630
STUDENT: [INAUDIBLE]

641
00:35:48,630 --> 00:35:53,430
PROFESSOR: 1 over
2 and 1 over 2.

642
00:35:53,430 --> 00:35:57,790
OK, and what will be the
corresponding point in 3D?

643
00:35:57,790 --> 00:36:01,500
1 over 2, 1 over 2, I plug in.

644
00:36:01,500 --> 00:36:03,540
Ah, yes.

645
00:36:03,540 --> 00:36:07,066
And with this, I hope
to finish the day so we

646
00:36:07,066 --> 00:36:10,236
can go to our other businesses.

647
00:36:10,236 --> 00:36:11,620
Is this hard?

648
00:36:11,620 --> 00:36:15,960
Now, I was not able-- I
have to be honest with you.

649
00:36:15,960 --> 00:36:20,670
I was not able to memorize the
equation of a tangent plane

650
00:36:20,670 --> 00:36:27,010
when I was-- when I was young,
like a freshman and sophomore.

651
00:36:27,010 --> 00:36:29,990
I wasn't ready to
understand that this

652
00:36:29,990 --> 00:36:33,220
is a linear approximation
of a curved something.

653
00:36:33,220 --> 00:36:35,530
This practically like
the Taylor equation

654
00:36:35,530 --> 00:36:39,400
for functions of
two variables when

655
00:36:39,400 --> 00:36:42,710
you neglect the quadratic
third term and so on.

656
00:36:42,710 --> 00:36:46,115
You just take the--
I'll teach you

657
00:36:46,115 --> 00:36:52,150
next time when this is, a first
order linear approximation.

658
00:36:52,150 --> 00:36:54,010
All right, can we do
this really quickly?

659
00:36:54,010 --> 00:36:55,800
It's going to be
a piece of cake.

660
00:36:55,800 --> 00:36:56,630
Let's see.

661
00:36:56,630 --> 00:36:58,360
Again, how do we do that?

662
00:36:58,360 --> 00:36:59,820
This is f of x and y.

663
00:36:59,820 --> 00:37:01,700
We computed that again.

664
00:37:01,700 --> 00:37:04,440
F of 0, 0 was this 0.

665
00:37:04,440 --> 00:37:08,490
Guys, if I say something
silly, will you stop me?

666
00:37:08,490 --> 00:37:12,760
F of f sub x-- f
of y at 0, 0 is 0.

667
00:37:12,760 --> 00:37:14,330
So I have two slopes.

668
00:37:14,330 --> 00:37:15,520
Those are my hands.

669
00:37:15,520 --> 00:37:19,430
The slopes of my hands are 0.

670
00:37:19,430 --> 00:37:27,270
So the tangent plane will
be z minus z0 equals 0.

671
00:37:27,270 --> 00:37:29,050
What is the 0?

672
00:37:29,050 --> 00:37:29,550
STUDENT: 1

673
00:37:29,550 --> 00:37:30,550
PROFESSOR: 1, excellent.

674
00:37:30,550 --> 00:37:31,754
STUDENT: [INAUDIBLE]

675
00:37:31,754 --> 00:37:32,795
PROFESSOR: Why is that 1?

676
00:37:32,795 --> 00:37:36,060
0 and 0 give me 1.

677
00:37:36,060 --> 00:37:39,875
So that was the picture
that I had z equals 1

678
00:37:39,875 --> 00:37:42,845
as the tangent plane at
the point corresponding

679
00:37:42,845 --> 00:37:44,825
to the origin.

680
00:37:44,825 --> 00:37:48,340
That look like the
north pole, 0, 0, 1.

681
00:37:48,340 --> 00:37:50,052
OK, no.

682
00:37:50,052 --> 00:37:52,550
It's the top of a hill.

683
00:37:52,550 --> 00:37:56,330
And finally, one last
thing [INAUDIBLE].

684
00:37:56,330 --> 00:37:58,200
Maybe you can do
this by yourselves,

685
00:37:58,200 --> 00:38:01,060
but I will shut up if I can.

686
00:38:01,060 --> 00:38:03,390
I can't in general,
but I'll shut up.

687
00:38:03,390 --> 00:38:09,080
Let's see-- f sub x at 1
over root 2, 1 over root 2.

688
00:38:09,080 --> 00:38:10,195
Why was that?

689
00:38:10,195 --> 00:38:11,895
What is f sub x?

690
00:38:11,895 --> 00:38:14,320
STUDENT: The square root of--
negative square root of 2.

691
00:38:14,320 --> 00:38:17,370
PROFESSOR: Right,
we've done that before.

692
00:38:17,370 --> 00:38:20,240
And you got exactly what
you said-- [INAUDIBLE]

693
00:38:20,240 --> 00:38:24,760
2 f sub y at the same point.

694
00:38:24,760 --> 00:38:29,520
I am too lazy to write it
down again-- minus root 2.

695
00:38:29,520 --> 00:38:32,940
And how do we actually
express the final answer

696
00:38:32,940 --> 00:38:37,260
so we can go home and
whatever-- to the next class?

697
00:38:37,260 --> 00:38:39,261
Is it hard?

698
00:38:39,261 --> 00:38:39,760
No.

699
00:38:39,760 --> 00:38:40,970
What's the answer?

700
00:38:40,970 --> 00:38:44,140
Z minus-- now, attention.

701
00:38:44,140 --> 00:38:45,804
What is z0?

702
00:38:45,804 --> 00:38:46,730
STUDENT: 0.

703
00:38:46,730 --> 00:38:48,820
PROFESSOR: 0, right.

704
00:38:48,820 --> 00:38:49,470
Why is that?

705
00:38:49,470 --> 00:38:54,280
Because when I plug 1 over
a 2, 1 over a 2, I got 0.

706
00:38:54,280 --> 00:38:56,780
0-- do I have to write it down?

707
00:38:56,780 --> 00:38:59,300
No, not unless I
want to be silly.

708
00:38:59,300 --> 00:39:02,140
But if you do write
down everything

709
00:39:02,140 --> 00:39:04,980
and you don't simplify
the equation of the plane,

710
00:39:04,980 --> 00:39:08,650
we don't penalize you in
any way in the final, OK?

711
00:39:08,650 --> 00:39:14,140
So if you show your work like
that, you're going to be fine.

712
00:39:14,140 --> 00:39:16,960
What is that 1 over 2?

713
00:39:16,960 --> 00:39:25,160
Plus minus root 2 times
y minus 1 over root 2.

714
00:39:25,160 --> 00:39:27,780
Is it elegant?

715
00:39:27,780 --> 00:39:30,820
No, it's not elegant at all.

716
00:39:30,820 --> 00:39:35,960
So as the last row for
today, one final line.

717
00:39:35,960 --> 00:39:40,060
Can we make it
look more elegant?

718
00:39:40,060 --> 00:39:43,540
Do we care to make
it more elegant?

719
00:39:43,540 --> 00:39:47,610
Definitely some of you care.

720
00:39:47,610 --> 00:39:52,180
Z will be minus root 2x.

721
00:39:52,180 --> 00:39:56,660
I want to be consistent and
keep the same style in y.

722
00:39:56,660 --> 00:39:58,930
And yet the constant
goes wherever

723
00:39:58,930 --> 00:40:00,365
it wants to go at the end.

724
00:40:00,365 --> 00:40:01,994
What's that constant?

725
00:40:01,994 --> 00:40:03,380
STUDENT: 2 [INAUDIBLE].

726
00:40:03,380 --> 00:40:05,111
PROFESSOR: So you
see what you have.

727
00:40:05,111 --> 00:40:06,152
You have this times that.

728
00:40:06,152 --> 00:40:07,910
It's a 1, this then that is a 1.

729
00:40:07,910 --> 00:40:10,100
1 plus 1 is 2.

730
00:40:10,100 --> 00:40:12,490
All right, are you
happy with this?

731
00:40:12,490 --> 00:40:13,910
I'm not.

732
00:40:13,910 --> 00:40:15,930
I'm happy.

733
00:40:15,930 --> 00:40:18,260
You-- if this were
a multiple choice,

734
00:40:18,260 --> 00:40:21,500
you would be able to
recognize it right away.

735
00:40:21,500 --> 00:40:25,440
What's the standardized general
equation of a plane, though?

736
00:40:25,440 --> 00:40:29,140
Something x plus something y
plus something z plus something

737
00:40:29,140 --> 00:40:31,050
equals 0.

738
00:40:31,050 --> 00:40:34,260
So if you wanted to
make me very happy,

739
00:40:34,260 --> 00:40:38,555
you would still move everybody
to the left hand side.

740
00:40:38,555 --> 00:40:41,080


741
00:40:41,080 --> 00:40:43,220
Do you want equal to or minus 3?

742
00:40:43,220 --> 00:40:45,100
Yes, it does.

743
00:40:45,100 --> 00:40:46,040
STUDENT: [INAUDIBLE]

744
00:40:46,040 --> 00:40:46,980
PROFESSOR: Huh?

745
00:40:46,980 --> 00:40:49,790
Negative 2-- is that OK?

746
00:40:49,790 --> 00:40:50,680
Is that fine?

747
00:40:50,680 --> 00:40:51,610
Are you guys done?

748
00:40:51,610 --> 00:40:52,400
Is this hard?

749
00:40:52,400 --> 00:40:53,560
Mm-mm.

750
00:40:53,560 --> 00:40:55,200
It's hard?

751
00:40:55,200 --> 00:40:56,280
No.

752
00:40:56,280 --> 00:40:58,770
Who said it's hard?

753
00:40:58,770 --> 00:41:05,180
So-- so I would work more
tangent planes next time.

754
00:41:05,180 --> 00:41:08,320
But I think it's something
that we can practice on.

755
00:41:08,320 --> 00:41:12,600
And do expect one exercise
like that from one

756
00:41:12,600 --> 00:41:16,410
of those, God knows,
15, 16 on the final.

757
00:41:16,410 --> 00:41:18,010
I'm not sure about the midterm.

758
00:41:18,010 --> 00:41:19,860
I like this type of problem.

759
00:41:19,860 --> 00:41:23,230
So you might even see
something with tangent planes

760
00:41:23,230 --> 00:41:26,956
on the midterm-- normal to
a surface tangent plane.

761
00:41:26,956 --> 00:41:28,070
It's a good topic.

762
00:41:28,070 --> 00:41:29,470
It's really pretty.

763
00:41:29,470 --> 00:41:33,200
For people who like to draw,
it's also nice to draw them.

764
00:41:33,200 --> 00:41:34,590
But do you have to?

765
00:41:34,590 --> 00:41:35,730
No.

766
00:41:35,730 --> 00:41:39,360
Some of you don't like to.

767
00:41:39,360 --> 00:41:43,160
OK, so now I say thank
you for the attendance

768
00:41:43,160 --> 00:41:48,130
and I'll see you next time
on Thursday-- on Tuesday.

769
00:41:48,130 --> 00:41:50,580
Happy Valentine's Day.

770
00:41:50,580 --> 00:41:52,189