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PROFESSOR: Hi.

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00:00:32,240 --> 00:00:35,930
Our main aim today
is to successfully

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00:00:35,930 --> 00:00:39,320
conclude block three,
and to set the stage

11
00:00:39,320 --> 00:00:41,530
for introducing block four.

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00:00:41,530 --> 00:00:45,730
Now by way of brief review,
we may have become so involved

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00:00:45,730 --> 00:00:48,680
with our discussion of
the last few lessons

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where we've done the chain
rule, that we may have forgotten

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00:00:52,000 --> 00:00:56,360
that the basic ingredient
that we had arrived at,

16
00:00:56,360 --> 00:00:58,880
that set up our proof
of the chain rule--

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00:00:58,880 --> 00:01:00,441
and then after that,
we got involved

18
00:01:00,441 --> 00:01:02,440
in the computation of how
to use the chain rule.

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00:01:02,440 --> 00:01:04,849
But the basic
ingredient was the idea

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00:01:04,849 --> 00:01:10,140
that we had obtained a linear
approximation, a delta w tan

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00:01:10,140 --> 00:01:11,190
idea.

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00:01:11,190 --> 00:01:14,680
And this concept leads to the
idea of a differential, which

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00:01:14,680 --> 00:01:17,170
is an extension of the
idea of differentials

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00:01:17,170 --> 00:01:19,510
as we knew them in
part one of our course.

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00:01:19,510 --> 00:01:22,860
And what I would like to do
is to talk somewhat briefly

26
00:01:22,860 --> 00:01:26,570
about this topic, to show
how it introduces what

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00:01:26,570 --> 00:01:29,200
will be the main
body of material

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00:01:29,200 --> 00:01:32,080
in block four and an
important application

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00:01:32,080 --> 00:01:35,780
that we can handle now that
will show why it's worth

30
00:01:35,780 --> 00:01:38,086
understanding certain things
about differentials, even

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00:01:38,086 --> 00:01:40,130
on a very elementary level.

32
00:01:40,130 --> 00:01:42,840
At any rate, the topic
that I've chosen for today

33
00:01:42,840 --> 00:01:45,330
is called "Exact Differentials."

34
00:01:45,330 --> 00:01:47,650
And the idea,
again, is completely

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00:01:47,650 --> 00:01:50,110
analogous to what we did in
calculus of a single variable.

36
00:01:50,110 --> 00:01:52,870
We start with w
equals f of x, y.

37
00:01:52,870 --> 00:01:55,950
And we showed that if w was
a continuously differentiable

38
00:01:55,950 --> 00:01:58,390
function of x and
y, it made sense

39
00:01:58,390 --> 00:02:02,030
to emphasize the expression
called delta w tan, which

40
00:02:02,030 --> 00:02:03,650
was the partial
of f with respect

41
00:02:03,650 --> 00:02:05,930
to x evaluated at a
given point times delta

42
00:02:05,930 --> 00:02:08,080
x, plus the partial
of f with respect

43
00:02:08,080 --> 00:02:11,090
to y at that same
point times delta y.

44
00:02:11,090 --> 00:02:14,040
And what we do is the
same thing that we did,

45
00:02:14,040 --> 00:02:16,260
as I say, in calculus
of a single variable.

46
00:02:16,260 --> 00:02:18,540
We introduce a new language.

47
00:02:18,540 --> 00:02:23,160
We write delta w tan now as dw.

48
00:02:23,160 --> 00:02:29,010
We write delta x as
dx, delta y as dy.

49
00:02:29,010 --> 00:02:34,650
And what we say is that dw is
the total differential of w.

50
00:02:34,650 --> 00:02:37,300
To see how this is analogous
to what happened in calculus

51
00:02:37,300 --> 00:02:40,450
of a single variable, notice,
for the sake of argument,

52
00:02:40,450 --> 00:02:43,890
that if f happens to be
a function of x alone,

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00:02:43,890 --> 00:02:45,440
so that there is
no second variable.

54
00:02:45,440 --> 00:02:47,740
In other words, if f
is independent of y,

55
00:02:47,740 --> 00:02:51,450
notice that the partial of f
with respect to y will be 0.

56
00:02:51,450 --> 00:02:55,170
And we will then have that
w is a function of x alone.

57
00:02:55,170 --> 00:02:56,830
dw, then, will be what?

58
00:02:56,830 --> 00:02:59,590
Well in that case, the
partial of f with respect to x

59
00:02:59,590 --> 00:03:03,470
is the ordinary derivative of
f with respect to x times dx.

60
00:03:03,470 --> 00:03:08,780
And we're back to our old recipe
that dw is dw/dx times dx.

61
00:03:08,780 --> 00:03:12,040
That this is a natural extension
of what happened in calculus

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00:03:12,040 --> 00:03:13,370
of a single variable.

63
00:03:13,370 --> 00:03:14,090
All right?

64
00:03:14,090 --> 00:03:15,700
Now what leads
into the next block

65
00:03:15,700 --> 00:03:19,200
and why we're going to drop this
for the time being but to go up

66
00:03:19,200 --> 00:03:21,290
to a different
aspect of this, is

67
00:03:21,290 --> 00:03:26,130
notice that if we can
assume that dw can replace

68
00:03:26,130 --> 00:03:28,665
delta w-- if we can
assume, as we talked

69
00:03:28,665 --> 00:03:30,040
about in the case
of continuously

70
00:03:30,040 --> 00:03:33,430
differentiable functions, that
the difference between delta w

71
00:03:33,430 --> 00:03:37,010
and delta w tan is negligible
for sufficiently small delta

72
00:03:37,010 --> 00:03:42,780
x and delta y, notice that the
equation dw equals f sub x dx

73
00:03:42,780 --> 00:03:46,265
plus f sub y dy, since f sub
x and f sub y are computed

74
00:03:46,265 --> 00:03:48,500
at a given point,
these become what?

75
00:03:48,500 --> 00:03:52,900
Linear combinations of dx
and dy, a constant times dx

76
00:03:52,900 --> 00:03:54,760
plus a constant times dy.

77
00:03:54,760 --> 00:03:57,480
And we're into the subject
of linear equations, which

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00:03:57,480 --> 00:03:59,640
we will talk about next time.

79
00:03:59,640 --> 00:04:01,490
For the time being,
all I want to point out

80
00:04:01,490 --> 00:04:05,920
is that dw is called the
total differential of w.

81
00:04:05,920 --> 00:04:10,330
And that conversely, starting
with any expression of the form

82
00:04:10,330 --> 00:04:14,430
some function M of x and y
times dx plus some function

83
00:04:14,430 --> 00:04:16,600
N of x and y times
dy-- in other words,

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00:04:16,600 --> 00:04:19,800
M of x, y dx plus N
of x, y dy, no matter

85
00:04:19,800 --> 00:04:22,200
what functions of x
and y M and N are,

86
00:04:22,200 --> 00:04:26,140
any expression of this form
is called a differential.

87
00:04:26,140 --> 00:04:27,670
By the way, you
may recall that you

88
00:04:27,670 --> 00:04:31,180
were used to calling dx a
differential back in calculus

89
00:04:31,180 --> 00:04:32,490
of a single variable.

90
00:04:32,490 --> 00:04:35,910
Notice that in particular,
one choice of capital N

91
00:04:35,910 --> 00:04:38,540
is to have N of x,
y be identically 0.

92
00:04:38,540 --> 00:04:41,200
We could have M of x,
y be identically 1.

93
00:04:41,200 --> 00:04:46,470
In which case, M*dx plus N*dy
would simply be 1*dx plus 0*dy,

94
00:04:46,470 --> 00:04:48,130
or dx.

95
00:04:48,130 --> 00:04:51,560
In other words, notice that
this extended definition

96
00:04:51,560 --> 00:04:54,430
of a differential for
two or more variables

97
00:04:54,430 --> 00:04:57,820
includes the definition
for a single variable.

98
00:04:57,820 --> 00:04:59,920
At any rate, any
expression of this type

99
00:04:59,920 --> 00:05:01,740
is called a differential.

100
00:05:01,740 --> 00:05:03,750
And again, a very natural
question that comes up

101
00:05:03,750 --> 00:05:05,250
is who wants differentials?

102
00:05:05,250 --> 00:05:06,880
Why do we need them?

103
00:05:06,880 --> 00:05:10,000
And perhaps the best
application comes

104
00:05:10,000 --> 00:05:12,260
from the clue that
a certain subject is

105
00:05:12,260 --> 00:05:15,370
called "Differential
Equations," rather than

106
00:05:15,370 --> 00:05:17,190
derivative equations.

107
00:05:17,190 --> 00:05:20,000
Let me illustrate this
in terms of what I think

108
00:05:20,000 --> 00:05:21,780
is a rather nice
example, in the sense

109
00:05:21,780 --> 00:05:23,610
that it will be hard
to guess the answer,

110
00:05:23,610 --> 00:05:26,060
but rather easy to visualize
what the problem says,

111
00:05:26,060 --> 00:05:27,077
at least.

112
00:05:27,077 --> 00:05:28,785
Let's imagine that I
have a certain curve

113
00:05:28,785 --> 00:05:31,220
S in the xy-plane.

114
00:05:31,220 --> 00:05:34,030
I don't tell you anything
else about that curve

115
00:05:34,030 --> 00:05:38,160
except that its slope
at any point x comma y

116
00:05:38,160 --> 00:05:41,660
is given by the very
interesting relationship

117
00:05:41,660 --> 00:05:44,680
that it's the quotient of
the square of its distance

118
00:05:44,680 --> 00:05:49,280
from the origin over twice the
product of its coordinates.

119
00:05:49,280 --> 00:05:51,525
In other words, to find
the slope of the curve S

120
00:05:51,525 --> 00:05:56,720
at the point x comma y, I
square the distance of the point

121
00:05:56,720 --> 00:06:00,340
from the origin, take
the negative of that,

122
00:06:00,340 --> 00:06:02,720
divide by twice the
product of the coordinates,

123
00:06:02,720 --> 00:06:04,956
and that's going
to be the slope.

124
00:06:04,956 --> 00:06:06,830
Now certainly that makes
sense geometrically.

125
00:06:06,830 --> 00:06:10,190
But what type of a curve
would have that property?

126
00:06:10,190 --> 00:06:13,090
And even assuming that we knew
such a curve, how in the world

127
00:06:13,090 --> 00:06:15,540
can you express it in
a convenient manner?

128
00:06:15,540 --> 00:06:16,780
Well here's the idea.

129
00:06:16,780 --> 00:06:18,250
Again, back to
the same technique

130
00:06:18,250 --> 00:06:20,240
that we used in calculus
of a single variable.

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00:06:20,240 --> 00:06:23,030
Algebraically, this
is our equation.

132
00:06:23,030 --> 00:06:26,260
If we cross-multiply and
collect all our terms

133
00:06:26,260 --> 00:06:28,300
onto one side of
the equation, we

134
00:06:28,300 --> 00:06:31,650
obtain what we call the
differential equation, see?

135
00:06:31,650 --> 00:06:34,900
It's an equation
involving a differential.

136
00:06:34,900 --> 00:06:38,550
In fact, in this case, using
our general notation, M of x, y

137
00:06:38,550 --> 00:06:40,390
is x squared plus y squared.

138
00:06:40,390 --> 00:06:42,790
And N of x, y is 2x*y.

139
00:06:42,790 --> 00:06:45,510
We have a differential
equal to 0.

140
00:06:45,510 --> 00:06:47,340
Now without going
into this right now,

141
00:06:47,340 --> 00:06:50,576
suppose I just happen to
have a very quick eye.

142
00:06:50,576 --> 00:06:53,280
Well in fact, before I even say
that, let me point this out.

143
00:06:53,280 --> 00:06:56,020
We've had problems like this
in part one of our course.

144
00:06:56,020 --> 00:06:58,660
The main difference was
that we could separate out

145
00:06:58,660 --> 00:06:59,310
the variables.

146
00:06:59,310 --> 00:07:01,690
We could get the x's
and the y's separated.

147
00:07:01,690 --> 00:07:04,790
Remember now, y is implicitly
a function of x here,

148
00:07:04,790 --> 00:07:07,240
in this expression,
because of the equation.

149
00:07:07,240 --> 00:07:09,580
See? x and y are no
longer independent

150
00:07:09,580 --> 00:07:11,570
when we equate this to 0.

151
00:07:11,570 --> 00:07:15,450
The question is though,
we cannot separate

152
00:07:15,450 --> 00:07:17,540
the x's and the y's here.

153
00:07:17,540 --> 00:07:20,500
But if we were lucky-- that's
where luck comes in-- we would

154
00:07:20,500 --> 00:07:23,790
recognize that treating
y as a function of x,

155
00:07:23,790 --> 00:07:25,910
the differential on
the left-hand side

156
00:07:25,910 --> 00:07:30,650
is precisely the differential
of 1/3 x cubed plus x squared y.

157
00:07:30,650 --> 00:07:33,160
In other words, the partial
of this with respect to x is

158
00:07:33,160 --> 00:07:37,890
simply x squared-- do I have...

159
00:07:37,890 --> 00:07:40,522
I think I may have
this thing in reverse.

160
00:07:40,522 --> 00:07:41,230
Let me make sure.

161
00:07:41,230 --> 00:07:44,060
If I take the partial of
this with respect to x,

162
00:07:44,060 --> 00:07:48,170
I have 1/3 x cubed
plus 2x*y, right?

163
00:07:48,170 --> 00:07:51,290
And if I take the partial
with respect to y,

164
00:07:51,290 --> 00:07:53,455
I simply have x squared.

165
00:07:53,455 --> 00:07:55,670
At any rate-- what
I'm saying is,

166
00:07:55,670 --> 00:07:58,837
if we knew a function whose--
see, I have this in reverse.

167
00:07:58,837 --> 00:08:00,170
That's what's bothering me here.

168
00:08:00,170 --> 00:08:03,051
I think the square
should go this way.

169
00:08:03,051 --> 00:08:03,925
That's not important.

170
00:08:03,925 --> 00:08:05,780
All I'm saying is if I've
done this thing correctly,

171
00:08:05,780 --> 00:08:07,710
if you take the partial
of this with respect

172
00:08:07,710 --> 00:08:11,220
to y-- with respect x,
you should get this.

173
00:08:11,220 --> 00:08:13,887
If you take the partial
of this with respect to y,

174
00:08:13,887 --> 00:08:14,720
you should get this.

175
00:08:14,720 --> 00:08:16,350
And that comes
out correctly now.

176
00:08:16,350 --> 00:08:19,220
In other words, the partial
of this with respect to x

177
00:08:19,220 --> 00:08:21,350
is x squared plus y squared.

178
00:08:21,350 --> 00:08:25,250
The partial of this with respect
to y-- this term drops out,

179
00:08:25,250 --> 00:08:26,670
and there's just 2x*y.

180
00:08:26,670 --> 00:08:28,330
I'm sorry for that
little mistake.

181
00:08:28,330 --> 00:08:31,000
But I won't really apologize
for it in the sense

182
00:08:31,000 --> 00:08:34,559
that once we've found it--
or even if we didn't find it,

183
00:08:34,559 --> 00:08:35,820
the key theory is the same.

184
00:08:35,820 --> 00:08:39,850
The point is that if we knew a
function whose differential was

185
00:08:39,850 --> 00:08:43,110
this, then the fact that
this differential is 0

186
00:08:43,110 --> 00:08:46,760
means the expression
itself must be a constant.

187
00:08:46,760 --> 00:08:54,570
In other words, 1/3 x cubed plus
x y squared must be a constant.

188
00:08:54,570 --> 00:08:58,520
And therefore, what
we're saying is if we now

189
00:08:58,520 --> 00:09:01,525
solve the equation-- I
really am sorry to have

190
00:09:01,525 --> 00:09:02,400
botched this for you.

191
00:09:02,400 --> 00:09:05,060
But again, as I say,
the idea basically

192
00:09:05,060 --> 00:09:06,440
comes through unimpeded.

193
00:09:06,440 --> 00:09:10,080
I now solve for y in terms
of x, and I get what?

194
00:09:10,080 --> 00:09:13,460
That y is equal to--
transposing here--

195
00:09:13,460 --> 00:09:16,720
plus or minus the square
root of c minus 1/3 x

196
00:09:16,720 --> 00:09:18,960
cubed, all over x.

197
00:09:18,960 --> 00:09:21,330
In other words, the
family of curves

198
00:09:21,330 --> 00:09:24,900
that satisfies this
differential equation

199
00:09:24,900 --> 00:09:27,714
is precisely this family here,
whatever that may look like.

200
00:09:27,714 --> 00:09:29,130
In other words,
this is the family

201
00:09:29,130 --> 00:09:32,110
of curves that has the
interesting slope property

202
00:09:32,110 --> 00:09:33,495
that we just discussed.

203
00:09:33,495 --> 00:09:35,350
Now you see, again,
the reason I say

204
00:09:35,350 --> 00:09:37,340
that I wasn't too
concerned about what

205
00:09:37,340 --> 00:09:39,720
this function actually
was is, notice

206
00:09:39,720 --> 00:09:42,580
that I was just pulling it out
of the hat for you here anyway.

207
00:09:42,580 --> 00:09:45,620
I said suppose we could
find a function w such

208
00:09:45,620 --> 00:09:47,800
that dw was this.

209
00:09:47,800 --> 00:09:51,950
The major question that comes
up in how one uses differentials

210
00:09:51,950 --> 00:09:54,990
and the like is as follows.

211
00:09:54,990 --> 00:09:58,440
Suppose all I told you was
that we had the differential x

212
00:09:58,440 --> 00:10:02,620
squared plus y squared dx plus
2x*y*dy, and I said to you,

213
00:10:02,620 --> 00:10:07,390
find a function w such that
dw will be x squared plus y

214
00:10:07,390 --> 00:10:10,590
squared dx plus 2x*y*dy.

215
00:10:10,590 --> 00:10:14,740
The key computational aid in
doing this involves the fact

216
00:10:14,740 --> 00:10:16,670
that if u and v are
independent variables--

217
00:10:16,670 --> 00:10:18,550
and it's crucial that
they be independent--

218
00:10:18,550 --> 00:10:22,580
that if a*u plus b*v
equals c*u plus d*v,

219
00:10:22,580 --> 00:10:27,780
it must be that a equals
c, and b equals d.

220
00:10:27,780 --> 00:10:31,360
In other words, the only way
that two linear combinations

221
00:10:31,360 --> 00:10:33,860
of independent
variables can be equal

222
00:10:33,860 --> 00:10:37,310
is if they're equal
coefficient by coefficient.

223
00:10:37,310 --> 00:10:39,150
You see, the proof
is very simple.

224
00:10:39,150 --> 00:10:42,540
Namely, after all, if u and
v are independent variables,

225
00:10:42,540 --> 00:10:47,660
why can't I pick v to be 0,
let u be any non-zero number.

226
00:10:47,660 --> 00:10:50,500
Then if v is 0, when
I equate these two,

227
00:10:50,500 --> 00:10:52,740
it says a*u equals c*u.

228
00:10:52,740 --> 00:10:56,550
Since u is not 0, I can cancel,
and obtain that a equals c.

229
00:10:56,550 --> 00:11:00,690
And in a similar way letting u
equal 0, I have b*v equals d*v.

230
00:11:00,690 --> 00:11:03,620
I can cancel the v's
and get that b equals d.

231
00:11:03,620 --> 00:11:06,410
But it's crucial that
u and v be independent.

232
00:11:06,410 --> 00:11:08,440
Because if u and v
are not independent,

233
00:11:08,440 --> 00:11:13,320
how do I know that I can have
u equal to 0 when v is not 0?

234
00:11:13,320 --> 00:11:16,470
For example, let's suppose
that v and u were dependent.

235
00:11:16,470 --> 00:11:19,500
Suppose, for example,
that v equals twice u.

236
00:11:19,500 --> 00:11:25,410
Observe, in this case, that
9u plus 4v-- since v is 2u--

237
00:11:25,410 --> 00:11:27,540
turns out to be 17u.

238
00:11:27,540 --> 00:11:34,570
On the other hand, 7u plus 5v is
7u plus 10u, which is also 17u.

239
00:11:34,570 --> 00:11:39,410
Notice that 9u plus
4v equals 7u plus 5v,

240
00:11:39,410 --> 00:11:42,780
even though the coefficients
don't line up as far

241
00:11:42,780 --> 00:11:45,540
as being equal is concerned.

242
00:11:45,540 --> 00:11:46,200
You see?

243
00:11:46,200 --> 00:11:49,130
In other words, notice that
being able to equate things

244
00:11:49,130 --> 00:11:51,860
coefficient by coefficient
hinges on the fact

245
00:11:51,860 --> 00:11:54,740
that the variables that we're
dealing with are independent.

246
00:11:54,740 --> 00:11:56,810
And the key thing going
back to this problem

247
00:11:56,810 --> 00:11:58,780
is, what were dx and dy?

248
00:11:58,780 --> 00:12:00,800
They were delta x and delta y.

249
00:12:00,800 --> 00:12:03,620
And as long as x and y
are independent variables,

250
00:12:03,620 --> 00:12:07,185
certainly the change in x
can be done independently

251
00:12:07,185 --> 00:12:08,670
of the change in y.

252
00:12:08,670 --> 00:12:12,540
In other words, dx and dy
are independent variables.

253
00:12:12,540 --> 00:12:17,550
So if we now say, let's find
that function w such that dw is

254
00:12:17,550 --> 00:12:23,550
x squared plus y
squared dx plus 2x*y*dy,

255
00:12:23,550 --> 00:12:26,650
the one thing we know
about dw, if it exists,

256
00:12:26,650 --> 00:12:30,070
by our basic definition of
this section of this lecture,

257
00:12:30,070 --> 00:12:34,530
is that dw is the partial of f
with respect to x times dx plus

258
00:12:34,530 --> 00:12:37,540
the partial of f with
respect to y times dy.

259
00:12:37,540 --> 00:12:40,420
Since dx and dy are
independent variables,

260
00:12:40,420 --> 00:12:43,170
the only way these two
expressions can be identical

261
00:12:43,170 --> 00:12:45,170
is coefficient by coefficient.

262
00:12:45,170 --> 00:12:47,230
That means in
particular, therefore,

263
00:12:47,230 --> 00:12:48,740
that the partial
of f with respect

264
00:12:48,740 --> 00:12:51,750
to x, which is the
coefficient of dx here,

265
00:12:51,750 --> 00:12:53,670
must be x squared
plus y squared, which

266
00:12:53,670 --> 00:12:55,350
is the coefficient of dx here.

267
00:12:55,350 --> 00:12:58,990
And similarly, the partial of
f with respect to y must equal

268
00:12:58,990 --> 00:13:00,800
2x*y.

269
00:13:00,800 --> 00:13:03,030
Now, armed with
this information,

270
00:13:03,030 --> 00:13:05,750
I can actually go
out and construct f.

271
00:13:05,750 --> 00:13:07,420
And the way I do
this is remember--

272
00:13:07,420 --> 00:13:10,170
as soon as I see the partial
of f with respect to x,

273
00:13:10,170 --> 00:13:12,460
it means I'm treating
y as a constant.

274
00:13:12,460 --> 00:13:16,170
If I'm treating y as a constant,
I just integrate this thing

275
00:13:16,170 --> 00:13:18,080
as if x were the only variable.

276
00:13:18,080 --> 00:13:20,820
Treating x as a variable
and y as a constant,

277
00:13:20,820 --> 00:13:23,680
the integral of x
squared is 1/3 x cubed.

278
00:13:23,680 --> 00:13:25,830
The integral of y
squared is simply

279
00:13:25,830 --> 00:13:29,160
y squared x, because y is
being treated as a constant.

280
00:13:29,160 --> 00:13:33,180
And finally, there must be
a constant of integration,

281
00:13:33,180 --> 00:13:36,460
but my constant now
is any function of y.

282
00:13:36,460 --> 00:13:38,700
In other words, if
I have any function

283
00:13:38,700 --> 00:13:42,360
of y, its partial with respect
to x, by definition of x and y

284
00:13:42,360 --> 00:13:44,890
being independent-- the
partial of any function

285
00:13:44,890 --> 00:13:46,890
of y with respect to x is 0.

286
00:13:46,890 --> 00:13:49,640
Therefore my constant of
integration, in this case,

287
00:13:49,640 --> 00:13:51,349
is a function of y alone.

288
00:13:51,349 --> 00:13:53,640
Again, to look at this in a
different perspective, what

289
00:13:53,640 --> 00:13:55,920
I'm saying is the
most general function

290
00:13:55,920 --> 00:13:57,910
of two variables
in the whole world

291
00:13:57,910 --> 00:13:59,760
whose derivative
with respect to x

292
00:13:59,760 --> 00:14:04,020
is x squared plus y squared
is 1/3 x cubed plus y squared

293
00:14:04,020 --> 00:14:07,410
x plus a function of y alone.

294
00:14:07,410 --> 00:14:11,920
Now all I don't know so far
is what specific function of y

295
00:14:11,920 --> 00:14:13,400
is g?

296
00:14:13,400 --> 00:14:15,620
In other words, I've
determined f now up

297
00:14:15,620 --> 00:14:20,940
to this particular arbitrary
constant of integration.

298
00:14:20,940 --> 00:14:22,850
How do I get a
hold of this thing?

299
00:14:22,850 --> 00:14:26,340
Notice that as yet, I have not
used a piece of information

300
00:14:26,340 --> 00:14:30,020
that tells me the partial of
f with respect to y is 2x*y.

301
00:14:30,020 --> 00:14:32,320
And to utilize that piece
of information, what I do

302
00:14:32,320 --> 00:14:34,570
is I come to this equation,
and I say, lookit.

303
00:14:34,570 --> 00:14:37,720
Let me, from here, just
differentiate this with respect

304
00:14:37,720 --> 00:14:39,880
to y, treating x as a constant.

305
00:14:39,880 --> 00:14:42,070
If I do that, this
term drops out.

306
00:14:42,070 --> 00:14:44,250
This term becomes 2x*y.

307
00:14:44,250 --> 00:14:46,910
And this, being a
function of y alone,

308
00:14:46,910 --> 00:14:49,940
its derivative with respect
to y is just g prime of y.

309
00:14:49,940 --> 00:14:55,810
So whatever f is, the derivative
of f with respect to y is 2x*y

310
00:14:55,810 --> 00:14:57,640
plus g prime of y.

311
00:14:57,640 --> 00:15:00,310
On the other hand, we also
know that the partial of f with

312
00:15:00,310 --> 00:15:02,860
respect to y is 2x*y.

313
00:15:02,860 --> 00:15:05,640
Consequently, since these
are two different expressions

314
00:15:05,640 --> 00:15:09,040
for the same quantity,
they must be equal.

315
00:15:09,040 --> 00:15:12,210
Equating these two,
the 2x*y's cancel.

316
00:15:12,210 --> 00:15:16,060
We obtain that g prime of y is
0, which means, in this case,

317
00:15:16,060 --> 00:15:18,960
that g of y was a bona
fide constant, that not

318
00:15:18,960 --> 00:15:22,702
only is g independent of x,
it's also independent of y.

319
00:15:22,702 --> 00:15:24,410
And we see that the
most general function

320
00:15:24,410 --> 00:15:27,410
f which has the desired
property is what?

321
00:15:27,410 --> 00:15:30,560
All we have to do is now go
back to this expression here,

322
00:15:30,560 --> 00:15:34,140
which gave us the
answer up to g of y,

323
00:15:34,140 --> 00:15:37,780
put this value of g of y in,
and we see that the most general

324
00:15:37,780 --> 00:15:42,530
function whose total
differential is x squared plus

325
00:15:42,530 --> 00:15:48,070
y squared dx plus 2x*y*dy is 1/3
x cubed plus y squared x plus

326
00:15:48,070 --> 00:15:48,660
c.

327
00:15:48,660 --> 00:15:51,700
And I guess there's a moral
to this story, after all.

328
00:15:51,700 --> 00:15:52,200
You know?

329
00:15:52,200 --> 00:15:53,690
When I did it the
quick way, I made

330
00:15:53,690 --> 00:15:55,910
a careless computational
mistake earlier

331
00:15:55,910 --> 00:15:58,220
in the lecture that sort
of threw me off a bit

332
00:15:58,220 --> 00:15:59,640
and perplexed me.

333
00:15:59,640 --> 00:16:03,520
And it seems it
was poetic justice.

334
00:16:03,520 --> 00:16:05,360
Because, you know,
when I took this method

335
00:16:05,360 --> 00:16:07,900
where we pretended we didn't
know the answer in advance

336
00:16:07,900 --> 00:16:10,270
and just worked the
thing out systematically,

337
00:16:10,270 --> 00:16:12,150
we came up with the right term.

338
00:16:12,150 --> 00:16:15,810
In other words, it should be
y squared x, not y x squared.

339
00:16:15,810 --> 00:16:19,417
And that was what we saw
should be the correct answer.

340
00:16:19,417 --> 00:16:21,750
So maybe there's something
to be said about doing things

341
00:16:21,750 --> 00:16:23,870
systematically, after all.

342
00:16:23,870 --> 00:16:24,370
You see?

343
00:16:24,370 --> 00:16:25,810
If I can't win
them all, at least

344
00:16:25,810 --> 00:16:29,780
I have the gift of rationalizing
to think that I win them all.

345
00:16:29,780 --> 00:16:34,320
Time after time, I snatch
defeat from the jaws of victory.

346
00:16:34,320 --> 00:16:34,820
No?

347
00:16:34,820 --> 00:16:35,330
All right.

348
00:16:35,330 --> 00:16:36,240
You know what I mean.

349
00:16:36,240 --> 00:16:36,890
Lookit.

350
00:16:36,890 --> 00:16:40,200
Anyway, the question that
we've now solved is what?

351
00:16:40,200 --> 00:16:42,180
We have worked out this routine.

352
00:16:42,180 --> 00:16:44,560
Notice that what
we are really doing

353
00:16:44,560 --> 00:16:48,220
is the counterpart of calculus
of a single variable, that

354
00:16:48,220 --> 00:16:51,680
knowing what a partial is,
we can integrate with respect

355
00:16:51,680 --> 00:16:54,440
to that variable, treating
all the other variables

356
00:16:54,440 --> 00:16:55,650
as a constant.

357
00:16:55,650 --> 00:16:57,690
And this is the
general technique.

358
00:16:57,690 --> 00:17:00,250
Now let's summarize
this more generally.

359
00:17:00,250 --> 00:17:01,760
The general definition is this.

360
00:17:01,760 --> 00:17:05,579
And by the way, for the sake
of conserving space, I will now

361
00:17:05,579 --> 00:17:09,619
abbreviate M of x, y and
N of x, y just by M and N.

362
00:17:09,619 --> 00:17:11,270
But whenever I
write M and N here,

363
00:17:11,270 --> 00:17:14,339
it's assumed that M and N
are functions of x and y.

364
00:17:14,339 --> 00:17:15,420
Well what we say is this.

365
00:17:15,420 --> 00:17:18,170
M*dx plus N*dy,
which by the way,

366
00:17:18,170 --> 00:17:21,510
by a previous definition
is always a differential.

367
00:17:21,510 --> 00:17:25,380
But this differential is
called an exact differential

368
00:17:25,380 --> 00:17:30,010
if and only if there exists
a function w-- f of x, y--

369
00:17:30,010 --> 00:17:33,720
such that dw is M*dx plus N*dy.

370
00:17:33,720 --> 00:17:37,340
Or in terms of our
previous discussion,

371
00:17:37,340 --> 00:17:38,790
what this means is what?

372
00:17:38,790 --> 00:17:45,330
dw is also equal to f
sub x dx plus f sub y dy.

373
00:17:45,330 --> 00:17:47,410
Since x and y are
independent variables,

374
00:17:47,410 --> 00:17:49,520
dx and dy are
independent variables.

375
00:17:49,520 --> 00:17:51,870
So an alternative
definition is what?

376
00:17:51,870 --> 00:17:55,940
That M*dx plus N*dy is an exact
differential if there exists

377
00:17:55,940 --> 00:17:59,950
a function f such that the
partial of f with respect to x

378
00:17:59,950 --> 00:18:04,310
is M, and the partial of
f with respect to y is N.

379
00:18:04,310 --> 00:18:07,190
Now, by the way, let's just
pause here for a moment.

380
00:18:07,190 --> 00:18:09,700
You might suspect that
with all the functions

381
00:18:09,700 --> 00:18:12,920
to choose from, that no
matter how M and N were given,

382
00:18:12,920 --> 00:18:17,235
we're bound to find at least
one function which satisfies--

383
00:18:17,235 --> 00:18:18,610
I should put this
in, because you

384
00:18:18,610 --> 00:18:21,340
want both of these conditions
fulfilled at the same time.

385
00:18:21,340 --> 00:18:24,260
The amazing thing is
that it is not always

386
00:18:24,260 --> 00:18:27,240
possible to find
a function which

387
00:18:27,240 --> 00:18:31,040
has a given differential
as its total differential.

388
00:18:31,040 --> 00:18:33,570
In fact, let me show you why,
if we can just do something

389
00:18:33,570 --> 00:18:34,990
a little bit tricky here.

390
00:18:34,990 --> 00:18:37,630
Let's suppose that the
function f exists such

391
00:18:37,630 --> 00:18:39,100
that its partial
with respect to x

392
00:18:39,100 --> 00:18:41,190
is M and its partial
with respect to y

393
00:18:41,190 --> 00:18:45,640
is N. Suppose that both of these
happen to be differentiable.

394
00:18:45,640 --> 00:18:48,520
Take the partial of f
sub x with respect to y.

395
00:18:48,520 --> 00:18:51,980
That tells me that f sub
x, y is the partial of M

396
00:18:51,980 --> 00:18:54,010
with respect to y.

397
00:18:54,010 --> 00:18:57,860
Let's take the partial of
f sub y with respect to x.

398
00:18:57,860 --> 00:19:00,160
That tells me that the partial
of f sub y with respect

399
00:19:00,160 --> 00:19:02,070
to x is N sub x.

400
00:19:02,070 --> 00:19:04,820
In other words, f sub
y, x equals N sub x.

401
00:19:04,820 --> 00:19:08,100
Now if f is
continuous-- see, if we

402
00:19:08,100 --> 00:19:09,605
have the right
amount of continuity

403
00:19:09,605 --> 00:19:11,230
and differentiability
that we've talked

404
00:19:11,230 --> 00:19:13,670
about in previous
lectures, notice

405
00:19:13,670 --> 00:19:16,690
that for most well-defined
functions-- in particular,

406
00:19:16,690 --> 00:19:19,430
if these two partials-- see,
we don't know what f is.

407
00:19:19,430 --> 00:19:20,790
We're trying to find f.

408
00:19:20,790 --> 00:19:22,034
And we're assuming it exists.

409
00:19:22,034 --> 00:19:23,700
The thing that's given
in our definition

410
00:19:23,700 --> 00:19:26,900
are M and N. So to word this
more succinctly, if it turns

411
00:19:26,900 --> 00:19:29,570
out that the partial
of M with respect to y

412
00:19:29,570 --> 00:19:31,750
and the partial of
N with respect to x

413
00:19:31,750 --> 00:19:34,550
happen to be
continuous functions,

414
00:19:34,550 --> 00:19:37,820
notice that these
two things are equal.

415
00:19:37,820 --> 00:19:40,690
And in particular, that
means that the partial of M

416
00:19:40,690 --> 00:19:43,440
with respect to y has to
equal the partial of N

417
00:19:43,440 --> 00:19:44,940
with respect to x.

418
00:19:44,940 --> 00:19:47,490
Coming back to the
original definition here,

419
00:19:47,490 --> 00:19:49,290
this is a rather amazing thing.

420
00:19:49,290 --> 00:19:52,060
It says that if this
differential is exact,

421
00:19:52,060 --> 00:19:55,450
if you take the
coefficient of dx--

422
00:19:55,450 --> 00:19:58,265
see, think of M as being
the coefficient of dx and N

423
00:19:58,265 --> 00:20:00,450
as being the coefficient of dy.

424
00:20:00,450 --> 00:20:02,870
If you take the
coefficient of dx

425
00:20:02,870 --> 00:20:05,390
and differentiate that
with respect to y,

426
00:20:05,390 --> 00:20:08,360
you must get the
same answer as if you

427
00:20:08,360 --> 00:20:11,790
took the coefficient of
dy and differentiated that

428
00:20:11,790 --> 00:20:13,160
with respect to x.

429
00:20:13,160 --> 00:20:15,440
That's an amazing
coincidence if that happens.

430
00:20:15,440 --> 00:20:19,020
Given two arbitrary functions,
this indeed won't happen.

431
00:20:19,020 --> 00:20:21,560
Now before I show you
that, let's summarize this

432
00:20:21,560 --> 00:20:23,900
in words written down
so that I make sure

433
00:20:23,900 --> 00:20:26,260
that you understand
what I'm saying.

434
00:20:26,260 --> 00:20:28,880
What I'm saying is, if the
partial of M with respect to y

435
00:20:28,880 --> 00:20:32,220
and the partial of N with
respect to x are continuous,

436
00:20:32,220 --> 00:20:37,470
then if M*dx plus N*dy is
exact, the implication is that

437
00:20:37,470 --> 00:20:40,820
the partial of M with respect
to y must equal the partial of N

438
00:20:40,820 --> 00:20:42,100
with respect to x.

439
00:20:42,100 --> 00:20:47,130
And to invert the emphasis here,
another way of saying this is

440
00:20:47,130 --> 00:20:50,290
that if the partial of M with
respect to y is not equal

441
00:20:50,290 --> 00:20:52,310
to the partial of N
with respect to x,

442
00:20:52,310 --> 00:20:55,860
then M*dx plus
N*dy is not exact.

443
00:20:55,860 --> 00:20:58,690
And let me show you that
by means of an example.

444
00:20:58,690 --> 00:21:01,890
Let's look at the differential--
a very simple-looking one.

445
00:21:01,890 --> 00:21:05,600
In fact, it looks far less
complicated than the expression

446
00:21:05,600 --> 00:21:07,610
that we worked with previously.

447
00:21:07,610 --> 00:21:10,720
Let's look simply
at y*dx minus x*dy.

448
00:21:10,720 --> 00:21:14,070
In this problem, the
coefficient of dx is y.

449
00:21:14,070 --> 00:21:19,050
That plays the role of M. The
coefficient of dy is minus x.

450
00:21:19,050 --> 00:21:21,450
That's what we're
calling N, you see.

451
00:21:21,450 --> 00:21:24,060
And the partial of
M with respect to y

452
00:21:24,060 --> 00:21:27,040
is just the partial of y with
respect to y, which is 1.

453
00:21:27,040 --> 00:21:29,400
The partial of N
with respect to x

454
00:21:29,400 --> 00:21:33,120
is the partial of minus x with
respect to x, which is minus 1.

455
00:21:33,120 --> 00:21:36,542
Since 1 is not equal to
minus 1, this is not exact.

456
00:21:36,542 --> 00:21:38,375
Because if it were
exact, these two partials

457
00:21:38,375 --> 00:21:39,600
would have to be equal.

458
00:21:39,600 --> 00:21:41,800
What does it mean to say
that this is not exact?

459
00:21:41,800 --> 00:21:44,130
What it means is
that there is not

460
00:21:44,130 --> 00:21:46,550
a single function in
the whole world that

461
00:21:46,550 --> 00:21:50,250
has the property that its
partial with respect to x is y,

462
00:21:50,250 --> 00:21:53,280
and it's partial with
respect to y is minus x.

463
00:21:53,280 --> 00:21:56,040
You can look forever.

464
00:21:56,040 --> 00:21:58,800
And not only won't you
find one, there isn't one.

465
00:21:58,800 --> 00:22:01,470
And by the way, let me make
one very quick aside here.

466
00:22:01,470 --> 00:22:04,080
Notice the difference between
saying you can't find one,

467
00:22:04,080 --> 00:22:05,250
and there isn't one.

468
00:22:05,250 --> 00:22:06,940
There may be one,
and you're just not

469
00:22:06,940 --> 00:22:08,444
lucky enough to find it.

470
00:22:08,444 --> 00:22:10,860
I'm saying that the reason you
don't find an answer here--

471
00:22:10,860 --> 00:22:13,750
unless you make a mistake and
think you've found an answer--

472
00:22:13,750 --> 00:22:16,320
the reason you don't find
an answer is there is none.

473
00:22:16,320 --> 00:22:18,310
And how can I show you
that there is none?

474
00:22:18,310 --> 00:22:20,320
Well the best way is
to do the same thing

475
00:22:20,320 --> 00:22:22,220
that we did in the
previous example,

476
00:22:22,220 --> 00:22:23,970
and see something go wrong.

477
00:22:23,970 --> 00:22:25,810
Let's suppose there
were an answer to this.

478
00:22:25,810 --> 00:22:29,350
Well if the partial of f
with respect to x equals y,

479
00:22:29,350 --> 00:22:31,330
let me integrate
this with respect

480
00:22:31,330 --> 00:22:33,470
to x, treating y as a constant.

481
00:22:33,470 --> 00:22:35,760
It means that the function
that I'm looking for

482
00:22:35,760 --> 00:22:37,140
must have what form?

483
00:22:37,140 --> 00:22:41,500
It's x times y plus some
function of y alone.

484
00:22:41,500 --> 00:22:43,570
And now we say, gee, we're
in pretty good shape.

485
00:22:43,570 --> 00:22:46,730
All we've got to do is find what
g of y is, and we're home free.

486
00:22:46,730 --> 00:22:49,210
Let's continue to mimic
what we did before.

487
00:22:49,210 --> 00:22:52,090
Knowing this, I can take the
partial of this with respect

488
00:22:52,090 --> 00:22:53,860
to y, OK?

489
00:22:53,860 --> 00:22:54,860
Which is what?

490
00:22:54,860 --> 00:22:58,230
If I take the partial of
this with respect to y,

491
00:22:58,230 --> 00:23:04,830
this is f sub y is
x plus g prime of y.

492
00:23:04,830 --> 00:23:06,050
All right?

493
00:23:06,050 --> 00:23:07,850
Now this is the partial
of f with respect

494
00:23:07,850 --> 00:23:10,010
to y, computed from here.

495
00:23:10,010 --> 00:23:12,750
I know by hypothesis that
the function I'm looking for

496
00:23:12,750 --> 00:23:15,580
must have its partial with
respect to y equal to this.

497
00:23:15,580 --> 00:23:18,780
Consequently, these two
expressions must be equal.

498
00:23:18,780 --> 00:23:20,840
And equating these
two expressions says

499
00:23:20,840 --> 00:23:24,080
that g prime of y--
see, x plus g prime of y

500
00:23:24,080 --> 00:23:25,550
must equal minus x.

501
00:23:25,550 --> 00:23:29,660
Transposing says that g
prime of y is minus 2x.

502
00:23:29,660 --> 00:23:30,880
And this is a contradiction.

503
00:23:30,880 --> 00:23:33,930
Because lookit, g prime of
y is a function of y alone.

504
00:23:33,930 --> 00:23:36,780
And here we have it equal
to some function of x.

505
00:23:36,780 --> 00:23:38,730
Or another way of
looking at it, this

506
00:23:38,730 --> 00:23:42,590
says that x depends on y. x
is some function of y, which

507
00:23:42,590 --> 00:23:46,330
is a contradiction since
we assumed that x and y are

508
00:23:46,330 --> 00:23:48,400
independent variables.

509
00:23:48,400 --> 00:23:51,470
By the way, just
as a quick aside,

510
00:23:51,470 --> 00:23:54,910
notice that in the case where we
were able to succeed at this--

511
00:23:54,910 --> 00:23:59,570
namely the previous example, x
squared plus y dx plus 2x*y*dy,

512
00:23:59,570 --> 00:24:03,010
M was x squared plus
y squared, N was 2x*y.

513
00:24:03,010 --> 00:24:07,270
Notice that the partial of
M with respect to y is 2y.

514
00:24:07,270 --> 00:24:11,800
And the partial of N with
respect to x is also 2y.

515
00:24:11,800 --> 00:24:14,240
So it seems that the
deciding factor really

516
00:24:14,240 --> 00:24:17,530
seems to be that the partial
of M with respect to y

517
00:24:17,530 --> 00:24:20,100
has to equal the partial
of N with respect to x.

518
00:24:20,100 --> 00:24:22,552
And the major result is--
and I didn't write this in,

519
00:24:22,552 --> 00:24:24,510
because I didn't want to
use up too much space.

520
00:24:24,510 --> 00:24:27,700
But this is emphasized in the
text and in the exercises.

521
00:24:27,700 --> 00:24:34,870
If M, N, M sub y, and N sub x
all exist and are continuous,

522
00:24:34,870 --> 00:24:40,240
then not only can we say that
M*dx plus N*dy is exact implies

523
00:24:40,240 --> 00:24:43,640
this part, but we can
do the converse too.

524
00:24:43,640 --> 00:24:46,860
That in particular, if
M sub y equals N sub x,

525
00:24:46,860 --> 00:24:48,914
we can conclude
that this is exact.

526
00:24:48,914 --> 00:24:50,330
And the reason for
this-- and I'll

527
00:24:50,330 --> 00:24:52,371
go through this very
quickly because the proof is

528
00:24:52,371 --> 00:24:53,464
given in the text.

529
00:24:53,464 --> 00:24:55,880
I just want to show you the
highlights of what happens is,

530
00:24:55,880 --> 00:24:58,090
what really happened
when we tried

531
00:24:58,090 --> 00:25:01,270
to construct f that made
things work in one case

532
00:25:01,270 --> 00:25:02,480
but not in the other?

533
00:25:02,480 --> 00:25:05,200
I think the best way to do
this is to work abstractly

534
00:25:05,200 --> 00:25:08,070
here without specifying
what M and N are,

535
00:25:08,070 --> 00:25:09,420
and see what happens.

536
00:25:09,420 --> 00:25:12,490
What we were trying to solve
was a pair of equations--

537
00:25:12,490 --> 00:25:15,140
we were trying to find f
such that the partial of f

538
00:25:15,140 --> 00:25:17,970
with respect to x is M,
partial of f with respect to y

539
00:25:17,970 --> 00:25:20,080
is N. The first
thing that we did

540
00:25:20,080 --> 00:25:23,140
was we treated y as a
constant and integrated this

541
00:25:23,140 --> 00:25:24,650
with respect to x.

542
00:25:24,650 --> 00:25:26,680
Well, I can't do
this specifically,

543
00:25:26,680 --> 00:25:28,050
because I don't know what M is.

544
00:25:28,050 --> 00:25:31,095
So let me just put the integral
sign in here to say what?

545
00:25:31,095 --> 00:25:32,470
I'm integrating
this with respect

546
00:25:32,470 --> 00:25:34,480
to x, treating y as a constant.

547
00:25:34,480 --> 00:25:37,150
That's some function of x
and y, you see over here.

548
00:25:37,150 --> 00:25:38,800
See M is a function of x and y.

549
00:25:38,800 --> 00:25:39,840
I'm holding y constant.

550
00:25:39,840 --> 00:25:40,890
I'm integrating this.

551
00:25:40,890 --> 00:25:45,000
Plus a constant of integration
which depends only on y.

552
00:25:45,000 --> 00:25:45,500
All right?

553
00:25:45,500 --> 00:25:46,760
Same as before.

554
00:25:46,760 --> 00:25:48,080
What was my next step?

555
00:25:48,080 --> 00:25:52,470
I took the partial of
this with respect to y,

556
00:25:52,470 --> 00:25:54,500
and I was going to
compare that with what

557
00:25:54,500 --> 00:25:56,420
I knew the partial of
f with respect to y

558
00:25:56,420 --> 00:25:59,440
had to be-- namely, N.
So I then took what?

559
00:25:59,440 --> 00:26:01,530
The partial of this
with respect to y.

560
00:26:01,530 --> 00:26:04,040
That gave me the partial of
this integral with respect

561
00:26:04,040 --> 00:26:06,510
to y plus g prime of y.

562
00:26:06,510 --> 00:26:10,070
And equating that to N,
I get that g prime of y

563
00:26:10,070 --> 00:26:14,390
was N minus the partial with
respect to y integral M dx.

564
00:26:14,390 --> 00:26:17,200
And again, don't be
alarmed by this expression.

565
00:26:17,200 --> 00:26:19,550
If M had been
given explicitly, I

566
00:26:19,550 --> 00:26:22,530
would have solved explicitly
for what this function was.

567
00:26:22,530 --> 00:26:26,570
The key point is to
notice that this side here

568
00:26:26,570 --> 00:26:29,120
is independent of x.

569
00:26:29,120 --> 00:26:32,250
Consequently, this equation
will be a contradiction

570
00:26:32,250 --> 00:26:37,500
unless the right-hand
side is independent of x.

571
00:26:37,500 --> 00:26:39,820
In other words, we
have determined f up

572
00:26:39,820 --> 00:26:42,450
to this function g of y.

573
00:26:42,450 --> 00:26:44,570
And what's going
to happen is we are

574
00:26:44,570 --> 00:26:46,670
going to be able
to compute g of y

575
00:26:46,670 --> 00:26:49,140
if this expression is
independent of x, namely just

576
00:26:49,140 --> 00:26:50,270
by integrating this.

577
00:26:50,270 --> 00:26:56,070
But if g of y is not independent
of x, we are in trouble.

578
00:26:56,070 --> 00:26:57,880
It means that it's
not going to exist.

579
00:26:57,880 --> 00:27:01,880
The key step is, is
this independent of x?

580
00:27:01,880 --> 00:27:03,710
And the answer is, the
best way to find out

581
00:27:03,710 --> 00:27:06,820
in terms of calculus is to take
its derivative with respect

582
00:27:06,820 --> 00:27:07,570
to x.

583
00:27:07,570 --> 00:27:11,300
If this depends on y alone,
its partial with respect to x

584
00:27:11,300 --> 00:27:12,600
must be 0.

585
00:27:12,600 --> 00:27:15,570
In other words, if this partial
with respect to x is not 0,

586
00:27:15,570 --> 00:27:18,080
it means that this
expression varies with x.

587
00:27:18,080 --> 00:27:20,720
At any rate, to see
what this derivative is,

588
00:27:20,720 --> 00:27:23,770
we just differentiate
term by term.

589
00:27:23,770 --> 00:27:24,990
And then we say, you know?

590
00:27:24,990 --> 00:27:26,540
It's rather interesting.

591
00:27:26,540 --> 00:27:31,530
If this thing had been first--
notice that if you integrate

592
00:27:31,530 --> 00:27:33,445
with respect to x and
then differentiate

593
00:27:33,445 --> 00:27:37,150
with respect to x, you wind
up with just the integrand.

594
00:27:37,150 --> 00:27:40,360
Well again, if we have
enough continuity,

595
00:27:40,360 --> 00:27:43,020
the order of differentiation
is irrelevant.

596
00:27:43,020 --> 00:27:45,690
We can reverse the order
without changing anything

597
00:27:45,690 --> 00:27:47,820
if we now interchange
this order,

598
00:27:47,820 --> 00:27:50,230
take the partial first
with respect to x.

599
00:27:50,230 --> 00:27:52,640
That will give us just
M. And the partial

600
00:27:52,640 --> 00:27:54,800
of M with respect to y
is just the partial of M

601
00:27:54,800 --> 00:27:56,180
with respect for y, of course.

602
00:27:56,180 --> 00:27:58,930
What we're saying is that from
this step, we get to this step.

603
00:27:58,930 --> 00:28:00,920
And this in turn implies this.

604
00:28:00,920 --> 00:28:04,940
Notice that the only
way that this can be 0

605
00:28:04,940 --> 00:28:07,510
is if the partial of
N with respect to x

606
00:28:07,510 --> 00:28:10,630
equals the partial of
M with respect to y.

607
00:28:10,630 --> 00:28:14,850
So in summary then, this is
a very interesting device.

608
00:28:14,850 --> 00:28:19,000
It tells us how to tell whether
a differential is exact.

609
00:28:19,000 --> 00:28:20,950
And the proof is very
nice in this case

610
00:28:20,950 --> 00:28:24,290
because the proof
exactly imitates

611
00:28:24,290 --> 00:28:28,020
how we construct the function
f if it turns out to be exact.

612
00:28:28,020 --> 00:28:31,880
But the point is, if
given M*dx plus N*dy,

613
00:28:31,880 --> 00:28:34,240
if the partial of M with
respect to y is not equal

614
00:28:34,240 --> 00:28:35,920
to the partial of N
with respect to x,

615
00:28:35,920 --> 00:28:39,330
there's no hope that the
function will be exact--

616
00:28:39,330 --> 00:28:41,160
the differential be exact.

617
00:28:41,160 --> 00:28:45,220
At any rate, we will
have sufficient exercises

618
00:28:45,220 --> 00:28:48,670
for drill in how one
uses exact differentials.

619
00:28:48,670 --> 00:28:52,180
This concludes block
three of our material.

620
00:28:52,180 --> 00:28:54,670
Our next block of
material which will

621
00:28:54,670 --> 00:28:57,690
concern linear
systems of equations

622
00:28:57,690 --> 00:28:59,710
will begin with
our next lecture.

623
00:28:59,710 --> 00:29:03,500
And until next time, goodbye.

624
00:29:03,500 --> 00:29:05,880
Funding for the
publication of this video

625
00:29:05,880 --> 00:29:10,750
was provided by the Gabriella
and Paul Rosenbaum Foundation.

626
00:29:10,750 --> 00:29:14,920
Help OCW continue to provide
free and open access to MIT

627
00:29:14,920 --> 00:29:19,338
courses by making a donation
at ocw.mit.edu/donate.