1
00:00:01 --> 00:00:03
The following content is
provided under a Creative
2
00:00:03 --> 00:00:05
Commons license.
Your support will help MIT
3
00:00:05 --> 00:00:08
OpenCourseWare continue to offer
high quality educational
4
00:00:08 --> 00:00:13
resources for free.
To make a donation or to view
5
00:00:13 --> 00:00:18
additional materials from
hundreds of MIT courses,
6
00:00:18 --> 00:00:23
visit MIT OpenCourseWare at
ocw.mit.edu.
7
00:00:23 --> 00:00:25
So, so far, we've seen things
about vectors,
8
00:00:25 --> 00:00:27
equation of planes,
motions in space,
9
00:00:27 --> 00:00:30
and so on.
Basically we've done geometry
10
00:00:30 --> 00:00:31
in space.
But, calculus,
11
00:00:31 --> 00:00:33
really, is about studying
functions.
12
00:00:33 --> 00:00:37
Now, we're going to actually
move on to studying functions of
13
00:00:37 --> 00:00:40
several variables.
So, this new unit,
14
00:00:40 --> 00:00:45
what we'll do over the next
three weeks or so will be about
15
00:00:45 --> 00:00:50
functions of several variables
and their derivatives.
16
00:00:50 --> 00:00:55
OK, so first of all,
we should try to figure out how
17
00:00:55 --> 00:00:58
we are going to think about
functions.
18
00:00:58 --> 00:01:02
So, remember,
if you have a function of one
19
00:01:02 --> 00:01:08
variable, that means you have a
quantity that depends on one
20
00:01:08 --> 00:01:12
parameter.
Maybe f depends on the variable
21
00:01:12 --> 00:01:14
x.
And, for example,
22
00:01:14 --> 00:01:19
a function that you all know is
f of x equals sin(x).
23
00:01:19 --> 00:01:22
And, the way we would represent
that is maybe just by plotting
24
00:01:22 --> 00:01:28
the graph of the function.
So, the graph of a function,
25
00:01:28 --> 00:01:34
we plot y = f(x).
And, the graph of a sine
26
00:01:34 --> 00:01:45
function that looks like this.
OK, so now, let's say that we
27
00:01:45 --> 00:01:53
had, actually,
a function of two variables.
28
00:01:53 --> 00:01:56
So, that means that the value
of F depends actually on two
29
00:01:56 --> 00:01:59
different parameters,
say, if the variables are x and
30
00:01:59 --> 00:02:03
y,
or they can have any names you
31
00:02:03 --> 00:02:07
want.
So, given values of the two
32
00:02:07 --> 00:02:13
parameters, x and y,
the function will give us a
33
00:02:13 --> 00:02:17
number that we'll call f(x,
y).
34
00:02:17 --> 00:02:21
That depends on x and y
according to some formula,
35
00:02:21 --> 00:02:29
OK, not very surprising so far.
So, for example,
36
00:02:29 --> 00:02:44
I can give you the function
f(x, y) = x^2 y^2.
37
00:02:44 --> 00:02:47
And, of course,
as with functions of one
38
00:02:47 --> 00:02:50
variable, we don't need things
to be defined everywhere.
39
00:02:50 --> 00:02:53
Sometimes there is the domain
of definition.
40
00:02:53 --> 00:02:57
So, this one is defined all the
time.
41
00:02:57 --> 00:03:01
But, if I tell you,
say, f of x,
42
00:03:01 --> 00:03:08
y equals square root of y,
well, this is only defined if y
43
00:03:08 --> 00:03:14
is nonnegative.
If I tell you f(x,
44
00:03:14 --> 00:03:23
y) equals one over x y,
that's only defined if x y is
45
00:03:23 --> 00:03:30
not zero, and so on.
Now, so these are mathematical
46
00:03:30 --> 00:03:33
examples given by explicit
formulas.
47
00:03:33 --> 00:03:35
And, of course,
there's physical examples.
48
00:03:35 --> 00:03:38
For example,
so examples coming from real
49
00:03:38 --> 00:03:40
life, so for example,
you can look at the temperature
50
00:03:40 --> 00:03:43
at the certain point on the
surface of the earth.
51
00:03:43 --> 00:03:46
So, you use maybe longitude and
latitude; that's x and y.
52
00:03:46 --> 00:03:49
And then you have f(x,
y) equals the temperature at
53
00:03:49 --> 00:03:50
that point.
54
00:03:50 --> 00:04:12
55
00:04:12 --> 00:04:17
In fact, because temperature
depends also may be on how high
56
00:04:17 --> 00:04:18
up you are.
It depends on elevation.
57
00:04:18 --> 00:04:21
So, it's actually a function of
maybe x, y, z.
58
00:04:21 --> 00:04:24
And, it also depends on time.
So, in fact,
59
00:04:24 --> 00:04:28
maybe it's a function of t in x
y z coordinates in space.
60
00:04:28 --> 00:04:31
So, you see that real-world
functions can depends on a lot
61
00:04:31 --> 00:04:33
of variables.
So, our goal will be to
62
00:04:33 --> 00:04:35
understand how to deal with
that.
63
00:04:35 --> 00:04:57
64
00:04:57 --> 00:05:01
OK, so now you will see very
soon, but actually it's already
65
00:05:01 --> 00:05:05
tricky enough to picture a
function of two variables.
66
00:05:05 --> 00:05:08
So, we are going to focus on
the case of functions of two
67
00:05:08 --> 00:05:10
variables.
And then, we'll see that if we
68
00:05:10 --> 00:05:12
have more than two variables,
then it's harder to plot the
69
00:05:12 --> 00:05:14
function.
We cannot draw with the graph
70
00:05:14 --> 00:05:17
looks like anymore.
But, the tools are the same,
71
00:05:17 --> 00:05:20
the notion of partial
derivatives, grade and vector,
72
00:05:20 --> 00:05:23
and so on,
all the tools that we will
73
00:05:23 --> 00:05:27
develop work exactly the same
way no matter how many variables
74
00:05:27 --> 00:05:30
you have.
So, I'll say,
75
00:05:30 --> 00:05:41
for simplicity -- -- we'll
focus mostly on two or sometimes
76
00:05:41 --> 00:05:48
three variables.
But, it works the same in any
77
00:05:48 --> 00:05:56
number of variables.
OK, so the first question is
78
00:05:56 --> 00:06:05
how to visualize a function of
two variables.
79
00:06:05 --> 00:06:10
So, the first thing we can do
is try to draw the graph of f.
80
00:06:10 --> 00:06:19
So, maybe I should say f --
which is a function of two
81
00:06:19 --> 00:06:23
variables.
So, the first answer will be,
82
00:06:23 --> 00:06:26
we can try to look at it's
graph.
83
00:06:26 --> 00:06:29
And, the idea is the same as
with one variable,
84
00:06:29 --> 00:06:31
namely, we look at all the
possible values of the
85
00:06:31 --> 00:06:34
parameters,
x and y, and for each of them,
86
00:06:34 --> 00:06:40
we plot a point whose height is
the value of a function at these
87
00:06:40 --> 00:06:43
parameters.
So, we'll plot,
88
00:06:43 --> 00:06:47
let's say, z equals f(x,
y).
89
00:06:47 --> 00:06:52
And, now that will become,
actually, a surface in space.
90
00:06:52 --> 00:06:57
OK, so for each value of x and
y, yeah, we have x,
91
00:06:57 --> 00:07:02
y in the x, y plane,
then we'll plot the point in
92
00:07:02 --> 00:07:05
space at position x,
y.
93
00:07:05 --> 00:07:13
And, z equals f of x, y.
OK, and if we take all of these
94
00:07:13 --> 00:07:17
points together,
they will give us some surface
95
00:07:17 --> 00:07:24
that sits in space.
Yes?
96
00:07:24 --> 00:07:29
Oh, a function of two
variables, shorthand.
97
00:07:29 --> 00:07:36
Well, let's say how to
visualize a function of two
98
00:07:36 --> 00:07:39
variables.
OK, so, how do we do that
99
00:07:39 --> 00:07:41
concretely?
Say that I give you a formula
100
00:07:41 --> 00:07:46
for f.
How do we try to represent it?
101
00:07:46 --> 00:07:57
So, let's do our first example.
Let's say I give you a function
102
00:07:57 --> 00:08:02
f(x, y) = -y.
OK, so it looks a little bit
103
00:08:02 --> 00:08:04
silly because it doesn't depend
on x.
104
00:08:04 --> 00:08:10
But, that's not the problem.
It's still a valid function of
105
00:08:10 --> 00:08:13
x and y.
It just happens to be constant
106
00:08:13 --> 00:08:17
with respect to x.
So, to draw the graph we look
107
00:08:17 --> 00:08:22
at the surface in space defined
by z equals y.
108
00:08:22 --> 00:08:26
What kind of surface is that?
It's a plane, OK?
109
00:08:26 --> 00:08:32
And, if we want to draw it,
z equals minus y will look,
110
00:08:32 --> 00:08:36
well, let's put y axis.
Let's put x axis.
111
00:08:36 --> 00:08:39
Let's put z axis.
If I look at what happens in
112
00:08:39 --> 00:08:43
the y, z plane in the plane of a
blackboard, it will just look
113
00:08:43 --> 00:08:45
like a line that goes downward
with slope one.
114
00:08:45 --> 00:08:51
OK, so it will be this.
And, what happens if I change x?
115
00:08:51 --> 00:08:54
Well, if I change x,
nothing happens because x
116
00:08:54 --> 00:08:57
doesn't appear in this equation.
So, in fact,
117
00:08:57 --> 00:09:01
if instead of setting x equal
to zero I set x equal to one,
118
00:09:01 --> 00:09:04
I'm in front of the blackboard,
or minus one at the back.
119
00:09:04 --> 00:09:07
Well, it still looks exactly
the same.
120
00:09:07 --> 00:09:15
So, I have this plane that
actually contains the x axis and
121
00:09:15 --> 00:09:22
slopes downwards with slope one.
It's kind of hard to draw.
122
00:09:22 --> 00:09:25
Now, you see immediately what
the big problem with graphs will
123
00:09:25 --> 00:09:29
be.
But, these pictures are hard to
124
00:09:29 --> 00:09:34
read.
But that's our first graph.
125
00:09:34 --> 00:09:41
OK, a question so far?
OK, so let's say that we have a
126
00:09:41 --> 00:09:43
slightly more complicated
function.
127
00:09:43 --> 00:09:50
How do we see it?
So, let's draw another example.
128
00:09:50 --> 00:09:58
Let's say I give you f(x,
y) = 1 - x^2-y^2.
129
00:09:58 --> 00:10:06
So, we should try to picture
what the surface z=1- x^2-y^2
130
00:10:06 --> 00:10:10
looks like.
So, how do we do that?
131
00:10:10 --> 00:10:15
Well, maybe you are very fast
and figured out what it looks
132
00:10:15 --> 00:10:17
like.
But, if not,
133
00:10:17 --> 00:10:21
then we need to work piece by
piece.
134
00:10:21 --> 00:10:27
So, maybe it will help if we
understand first what it does in
135
00:10:27 --> 00:10:35
the plane of the blackboard.
So, if we look at it in the y,
136
00:10:35 --> 00:10:42
z plane, that means we set x
equal to zero.
137
00:10:42 --> 00:10:48
And then, z becomes 1 - y^2.
What is that?
138
00:10:48 --> 00:10:54
It's a parabola pointing
downwards, and starting at one.
139
00:10:54 --> 00:11:00
So, we should draw maybe this
downward parabola.
140
00:11:00 --> 00:11:08
It starts at one and it cuts
the y axis at one.
141
00:11:08 --> 00:11:11
When y is one,
that gives us zero.
142
00:11:11 --> 00:11:15
So, we might have an idea of
what it might look like,
143
00:11:15 --> 00:11:19
or maybe not.
Let's get more slices.
144
00:11:19 --> 00:11:27
Let's see what it does in the
x, z plane, this other vertical
145
00:11:27 --> 00:11:33
plane that's coming towards us.
So, in the x,
146
00:11:33 --> 00:11:38
z plane, if we set y equal to
zero, we get z equals one minus
147
00:11:38 --> 00:11:40
x^2.
It's, again,
148
00:11:40 --> 00:11:46
a parabola going downwards.
OK, so I'm going to try to draw
149
00:11:46 --> 00:11:51
a parabola that goes downward,
but now to the front and to the
150
00:11:51 --> 00:11:54
back.
So, we are starting to have a
151
00:11:54 --> 00:11:57
slightly better idea but we
still don't know whether the
152
00:11:57 --> 00:11:59
cross section of this thing
might be round,
153
00:11:59 --> 00:12:04
square, something else.
So, it wants more confirmation.
154
00:12:04 --> 00:12:16
We might want to also figure
out where the surface intersects
155
00:12:16 --> 00:12:22
the x, y plane.
So, we hit the x,
156
00:12:22 --> 00:12:29
y plane when z equals zero.
That means 1-x^2-y^2 should be
157
00:12:29 --> 00:12:38
0, that becomes x^2 y^2 = 1.
That is a circle of radius 1.
158
00:12:38 --> 00:12:46
That's the unit size.
So, that means that here,
159
00:12:46 --> 00:12:50
we actually have the unit
circle.
160
00:12:50 --> 00:12:54
And now, you should imagine
that you have this thing that
161
00:12:54 --> 00:12:57
when you slice it by a vertical
plane, looks like a downwards
162
00:12:57 --> 00:13:00
parabola.
And, it's actually a surface of
163
00:13:00 --> 00:13:03
revolution.
You can rotate it around the z
164
00:13:03 --> 00:13:06
axis, OK?
Now, if you stare long enough
165
00:13:06 --> 00:13:09
at that equation,
you'll actually see that,
166
00:13:09 --> 00:13:12
yes, we know that it had to be
like that.
167
00:13:12 --> 00:13:17
But, see, so these are useful
ways of trying to guess what the
168
00:13:17 --> 00:13:19
graph looks like.
Of course, the other way is to
169
00:13:19 --> 00:13:24
just ask your computer to do it.
And then, you know,
170
00:13:24 --> 00:13:30
you will get that kind of
formula.
171
00:13:30 --> 00:13:36
OK, well, I can leave it on if
you want.
172
00:13:36 --> 00:13:40
No, because I plotted a
different function that I will
173
00:13:40 --> 00:13:43
show you later.
So, it goes this way.
174
00:13:43 --> 00:13:46
I mean, if you want,
it's really going downward.
175
00:13:46 --> 00:13:49
Yes, I agree that the sheet is
upside down.
176
00:13:49 --> 00:13:52
That's because I plotted
something else.
177
00:13:52 --> 00:13:58
OK, so, in fact,
so I plotted in my computer was
178
00:13:58 --> 00:14:03
actually x^2 y^2 that looks like
that.
179
00:14:03 --> 00:14:08
See, it's the same with a
parabola going upwards.
180
00:14:08 --> 00:14:14
If you want to see more
examples, I have various
181
00:14:14 --> 00:14:19
examples to show,
well, here's the graph,
182
00:14:19 --> 00:14:21
y^2-x^2.
See, so that one is kind of
183
00:14:21 --> 00:14:23
interesting.
It looks like a saddle.
184
00:14:23 --> 00:14:29
If you look at it in the y,
z plane, then it's a parabola
185
00:14:29 --> 00:14:34
going up, z = y^2.
And, that's what we see to the
186
00:14:34 --> 00:14:38
left and to the right.
But, if you put it in the x,
187
00:14:38 --> 00:14:42
z plane, then that's a parabola
going downwards,
188
00:14:42 --> 00:14:45
z = - x^2.
So, we have a parabola going
189
00:14:45 --> 00:14:48
downwards in one direction,
upwards in the other one.
190
00:14:48 --> 00:14:53
And together,
they form this surface.
191
00:14:53 --> 00:14:55
And of course,
you can plot much more
192
00:14:55 --> 00:14:58
complicated functions.
So, this one,
193
00:14:58 --> 00:15:00
well, if you can read very
small things,
194
00:15:00 --> 00:15:03
you can see the formula.
It doesn't matter,
195
00:15:03 --> 00:15:09
just to show you that you can
put a formula into a computer:
196
00:15:09 --> 00:15:18
it will show you a picture.
OK, so that's pretty good.
197
00:15:18 --> 00:15:20
I mean, you can see that it can
get a bit cluttered because
198
00:15:20 --> 00:15:22
maybe those features that are
hidden behind,
199
00:15:22 --> 00:15:25
or maybe we have trouble seeing
if we don't have a computer,
200
00:15:25 --> 00:15:29
that looks very readable.
But, this is kind of hard to
201
00:15:29 --> 00:15:33
visualize sometimes.
So, there is another way to
202
00:15:33 --> 00:15:36
plot the functions of two
variables.
203
00:15:36 --> 00:15:47
And, let's call it the contour
plot.
204
00:15:47 --> 00:15:51
So, the contour plot is a very
elegant solution to the problem
205
00:15:51 --> 00:15:55
that it's difficult to draw to
space pictures on a sheet of
206
00:15:55 --> 00:15:58
paper or on a blackboard.
So, instead,
207
00:15:58 --> 00:16:02
let's try to represent the
function of two variables by
208
00:16:02 --> 00:16:04
just the map,
you know, the same way that
209
00:16:04 --> 00:16:07
when you walk around,
you have actually geographical
210
00:16:07 --> 00:16:11
maps that fit on a piece of
paper that tell you about what
211
00:16:11 --> 00:16:17
the real world looks like.
So, what contour plot looks
212
00:16:17 --> 00:16:22
something like this?
So, it's an x, y plot.
213
00:16:22 --> 00:16:25
And, that, you have a bunch of
curves.
214
00:16:25 --> 00:16:32
And, what the curves represent
are the elevations on the graph.
215
00:16:32 --> 00:16:35
So, for example,
this curve might correspond to
216
00:16:35 --> 00:16:37
all the points where f(x,
y) = 1.
217
00:16:37 --> 00:16:46
And, that curve might be all
the points where f=2 and f=3 and
218
00:16:46 --> 00:16:49
so on, OK?
So, when you see you this kind
219
00:16:49 --> 00:16:53
of plot, you're supposed to
think that the graph of the
220
00:16:53 --> 00:16:56
function sits somewhere in space
above that.
221
00:16:56 --> 00:17:00
It's like a map telling you how
high things are.
222
00:17:00 --> 00:17:03
And, what you would want to do
with the function,
223
00:17:03 --> 00:17:06
really, is be able to tell
quickly what's the value at a
224
00:17:06 --> 00:17:08
given point?
Well, let's say I want to look
225
00:17:08 --> 00:17:11
at that point.
I know that f is somewhere
226
00:17:11 --> 00:17:14
between 1 and 2.
Actually, it's much faster to
227
00:17:14 --> 00:17:17
read than the graph.
On the graph I might have to
228
00:17:17 --> 00:17:18
look carefully,
and then measure things,
229
00:17:18 --> 00:17:22
and so on.
Here, I can just raise the
230
00:17:22 --> 00:17:27
value of f by comparing with the
nearby lines.
231
00:17:27 --> 00:17:31
OK, so let me try to make that
more precise.
232
00:17:31 --> 00:17:41
So, it shows all the points --
-- where f(x,
233
00:17:41 --> 00:17:53
y) equals some fixed values,
some fixed constants.
234
00:17:53 --> 00:18:05
And, these constants typically
are chosen at regular intervals.
235
00:18:05 --> 00:18:07
For example,
here I chose one,
236
00:18:07 --> 00:18:11
two, three, and they could
continue with zero minus one,
237
00:18:11 --> 00:18:16
and so on.
So, one way to think about it,
238
00:18:16 --> 00:18:21
how does this relate to the
graph?
239
00:18:21 --> 00:18:31
Well, that's the same thing as
cutting, I mean,
240
00:18:31 --> 00:18:40
we slice the graph by
horizontal planes.
241
00:18:40 --> 00:18:44
So, horizontal planes have
equations of a form z equals
242
00:18:44 --> 00:18:46
some constant,
z equals zero,
243
00:18:46 --> 00:18:48
z equals one,
z equals two,
244
00:18:48 --> 00:18:51
and so on.
So, maybe the graph of my
245
00:18:51 --> 00:18:55
function will be some sort of
plot out there.
246
00:18:55 --> 00:19:00
And, if I slice it by the plane
z equals one,
247
00:19:00 --> 00:19:04
then I will get the level
curve,
248
00:19:04 --> 00:19:14
which is the point where f(x,
y) = 1,
249
00:19:14 --> 00:19:24
and so, that's called a level
curve of f.
250
00:19:24 --> 00:19:32
OK, and so we repeat the
process with maybe z equals two,
251
00:19:32 --> 00:19:38
and we get another level curve,
and so on.
252
00:19:38 --> 00:19:44
And, then we squish all of them
up, and that's how we get the
253
00:19:44 --> 00:19:47
contour plot.
OK, so each of these lines,
254
00:19:47 --> 00:19:50
imagine that this is like some
mountain or something that you
255
00:19:50 --> 00:19:52
are hiking on.
Each of these lines tells you
256
00:19:52 --> 00:19:55
how you could move to stay at a
constant height if you want to
257
00:19:55 --> 00:19:58
get to the other side of the
mountain but without ever going
258
00:19:58 --> 00:20:03
up or down.
You would just walk along that
259
00:20:03 --> 00:20:08
path, and it will get you there
without effort.
260
00:20:08 --> 00:20:11
So, in fact,
if you have been talking about
261
00:20:11 --> 00:20:15
hiking on mountains,
well, that's exactly what a
262
00:20:15 --> 00:20:21
topographical map is about.
So, I need to zoom a bit.
263
00:20:21 --> 00:20:27
So, a topographic map,
this one from the US geological
264
00:20:27 --> 00:20:32
survey shows you,
basically, all the level curves
265
00:20:32 --> 00:20:37
of an altitude function on a
piece of land.
266
00:20:37 --> 00:20:40
So, you know that if you walk
right along these curves,
267
00:20:40 --> 00:20:42
you will stay along the same
height.
268
00:20:42 --> 00:20:46
And you know that if you walk
towards, these don't have
269
00:20:46 --> 00:20:48
numbers.
Let me find a place with
270
00:20:48 --> 00:20:53
numbers.
Here, there is a 500 in the
271
00:20:53 --> 00:20:56
middle.
So, you know that if you walk
272
00:20:56 --> 00:20:59
on the line that says 500,
you stay always at 500 meters
273
00:20:59 --> 00:21:02
in elevation.
If you walk towards the
274
00:21:02 --> 00:21:05
mountain that I think is below
it, then you will go up,
275
00:21:05 --> 00:21:07
and so on.
So, you can see,
276
00:21:07 --> 00:21:10
for example,
here there's a peak,
277
00:21:10 --> 00:21:13
and here there is a valley with
the river in it,
278
00:21:13 --> 00:21:17
and the altitudes go down,
and then back up again on the
279
00:21:17 --> 00:21:19
other side.
OK, so that's an example of a
280
00:21:19 --> 00:21:22
contour plot of a function.
Of course, we don't have a
281
00:21:22 --> 00:21:25
formula for that function,
but we have a contour plot,
282
00:21:25 --> 00:21:29
and that's what we need
actually to understand what's
283
00:21:29 --> 00:21:36
going on there.
OK, any questions?
284
00:21:36 --> 00:21:39
No?
OK, so another example of
285
00:21:39 --> 00:21:42
contour plots,
well, you've probably seen
286
00:21:42 --> 00:21:46
various versions of these
temperature maps.
287
00:21:46 --> 00:21:51
So, that's supposed to be how
warm it is right now.
288
00:21:51 --> 00:21:55
So, this one is color-coded.
Instead of having curves,
289
00:21:55 --> 00:21:58
it has all these colors.
But, the effect is the same.
290
00:21:58 --> 00:22:01
If you look at the separations
between consecutive colors,
291
00:22:01 --> 00:22:05
these are the level curves of a
function that tells you the
292
00:22:05 --> 00:22:12
temperature at a given point.
OK, so these are examples of
293
00:22:12 --> 00:22:24
contour plots in real life.
OK, no questions?
294
00:22:24 --> 00:22:26
No?
OK, so basically,
295
00:22:26 --> 00:22:31
one of the goals that one
should try to achieve at this
296
00:22:31 --> 00:22:35
point is becoming familiar with
the contour plot,
297
00:22:35 --> 00:22:38
the graph,
and how to view and deal with
298
00:22:38 --> 00:22:39
functions.
299
00:22:39 --> 00:22:54
300
00:22:54 --> 00:23:02
[APPLAUSE]
OK, so -- Let's do an example.
301
00:23:02 --> 00:23:04
Well, let's do a couple of
examples.
302
00:23:04 --> 00:23:08
So, let's start with f(x,y) = -
y.
303
00:23:08 --> 00:23:12
And, I'm going to take the same
two examples as there to start
304
00:23:12 --> 00:23:16
with so that we see the relation
between the graph and the
305
00:23:16 --> 00:23:23
contour plots.
So, let's try to plot it.
306
00:23:23 --> 00:23:30
So, we are asked for the level
curve, f equals 0 for this one?
307
00:23:30 --> 00:23:38
Well, f is zero when y is zero.
So, that's the x axis.
308
00:23:38 --> 00:23:44
OK, so that's the level, zero.
Where's the level one?
309
00:23:44 --> 00:23:48
Well, f is one when negative y
is one.
310
00:23:48 --> 00:23:51
That means when y is negative
one.
311
00:23:51 --> 00:23:57
So, you go to minus one,
and that will be where my level
312
00:23:57 --> 00:24:02
one is, and so on.
f is two when y is negative
313
00:24:02 --> 00:24:06
two.
F is negative one when y is
314
00:24:06 --> 00:24:10
one, and so on.
Is that convincing?
315
00:24:10 --> 00:24:15
Do you see how we got that?
OK, let me do it again.
316
00:24:15 --> 00:24:18
I don't see anybody nodding,
so that's kind of bad news.
317
00:24:18 --> 00:24:22
So, if I want to know,
where is the level curve,
318
00:24:22 --> 00:24:26
say, one, I try to set f equals
to one.
319
00:24:26 --> 00:24:31
Let's do this one.
f equals one means that
320
00:24:31 --> 00:24:36
negative y is one means that y
is minus one,
321
00:24:36 --> 00:24:43
and y equals minus one is this
horizontal line on this chart.
322
00:24:43 --> 00:24:47
OK, and same with the others.
So, you can see on the map that
323
00:24:47 --> 00:24:49
the value of a function doesn't
depend on x.
324
00:24:49 --> 00:24:52
If you move parallel to the x
axis, nothing happens.
325
00:24:52 --> 00:24:56
If you move in the y direction,
it decreases at a constant
326
00:24:56 --> 00:24:59
rate.
That's why the contours are
327
00:24:59 --> 00:25:03
evenly spaced.
How spaced out they are tells
328
00:25:03 --> 00:25:06
you, actually,
how steep things are.
329
00:25:06 --> 00:25:08
So, that corresponds exactly to
that picture,
330
00:25:08 --> 00:25:11
except that here we draw x
coming to the front,
331
00:25:11 --> 00:25:14
and y to the right.
So, you have to rotate the map
332
00:25:14 --> 00:25:19
by 90� to get to that.
It's an unfortunate consequence
333
00:25:19 --> 00:25:24
of the usual way of plotting
things in space.
334
00:25:24 --> 00:25:32
OK, so these horizontal lines
here correspond actually to
335
00:25:32 --> 00:25:35
horizontal lines here.
336
00:25:35 --> 00:25:43
337
00:25:43 --> 00:25:54
OK, second example.
Let's do 1-x^2-y^2.
338
00:25:54 --> 00:26:00
OK, or maybe I will write it as
1 - (x^2 y^2).
339
00:26:00 --> 00:26:06
It's really the same thing.
So, x, y, let's see,
340
00:26:06 --> 00:26:12
where is this function equal to
zero?
341
00:26:12 --> 00:26:21
Well, we said f is zero in the
unit circle.
342
00:26:21 --> 00:26:32
OK, so, the zero level,
well, let's say that this is my
343
00:26:32 --> 00:26:36
unit.
That's where it's at zero.
344
00:26:36 --> 00:26:48
What about f equals one?
Well, that's when x^2 y^2 = 0.
345
00:26:48 --> 00:26:49
Well, that's only going to be
here.
346
00:26:49 --> 00:27:00
So, that's just a single point.
What about f equals minus one?
347
00:27:00 --> 00:27:07
That's when x^2 y^2 =2.
That's a circle of radius
348
00:27:07 --> 00:27:10
square root of two,
which is about 1.4.
349
00:27:10 --> 00:27:17
So, it's somewhere here.
Then, minus two,
350
00:27:17 --> 00:27:24
similarly, will be x^2 y^2 = 3.
Square root of three is about
351
00:27:24 --> 00:27:27
1.7.
And then, minus three will be
352
00:27:27 --> 00:27:30
of radius two,
and so on.
353
00:27:30 --> 00:27:38
So, what I want to show here is
that they are getting closer and
354
00:27:38 --> 00:27:41
closer apart,
OK?
355
00:27:41 --> 00:27:44
So, first it's concentric
circles that tells us that we
356
00:27:44 --> 00:27:47
have a shape that's a solid of
the graph is going to be a
357
00:27:47 --> 00:27:52
surface of revolution.
Things don't change if I rotate.
358
00:27:52 --> 00:27:56
And second, the level curves
are getting closer and closer to
359
00:27:56 --> 00:27:59
each other.
That means it's getting steeper
360
00:27:59 --> 00:28:03
and steeper because I have to
travel a shorter distance if I
361
00:28:03 --> 00:28:06
want my altitude to change by
one.
362
00:28:06 --> 00:28:09
OK, so, this top here is kind
of flat.
363
00:28:09 --> 00:28:11
And then it gets steeper and
steeper.
364
00:28:11 --> 00:28:16
And, that's what we observe on
that picture there.
365
00:28:16 --> 00:28:24
OK, so just to show you a few
more, where did I put my,
366
00:28:24 --> 00:28:30
so, for x^2 y^2,
the contour plot looks like
367
00:28:30 --> 00:28:37
this.
Maybe actually I'll make it.
368
00:28:37 --> 00:28:41
OK, so it looks exactly the
same as this one.
369
00:28:41 --> 00:28:44
But, the difference is if you
can see the numbers which are
370
00:28:44 --> 00:28:45
not there,
so you can see them,
371
00:28:45 --> 00:28:49
then you would know that
instead of decreasing as we move
372
00:28:49 --> 00:28:52
out,
this one is increasing as we go
373
00:28:52 --> 00:28:54
out.
OK, so that's where we use,
374
00:28:54 --> 00:28:57
actually, the labels on the
level curves that tell us
375
00:28:57 --> 00:29:00
whether things are going up or
down.
376
00:29:00 --> 00:29:04
But, the contour plots look
exactly the same.
377
00:29:04 --> 00:29:14
For the next one I had,
I think, was y^2-x^2.
378
00:29:14 --> 00:29:18
So, the contour plot,
well, let me actually zoom out.
379
00:29:18 --> 00:29:20
So, the contour plot looks like
that.
380
00:29:20 --> 00:29:23
So, the level curve
corresponding to zero is
381
00:29:23 --> 00:29:27
actually two diagonal lines.
And, if you look on the plot,
382
00:29:27 --> 00:29:30
say that you started at the
saddle point in the middle and
383
00:29:30 --> 00:29:33
you try to stay at the same
level.
384
00:29:33 --> 00:29:35
So, it looks like a mountain
pass, right?
385
00:29:35 --> 00:29:38
Well, there's actually four
directions from that point that
386
00:29:38 --> 00:29:41
you can go staying at the same
height.
387
00:29:41 --> 00:29:44
And actually,
on the map, they look exactly
388
00:29:44 --> 00:29:46
like this, too,
these crossing lines.
389
00:29:46 --> 00:29:49
OK, so, these are things that
go on the side of the two
390
00:29:49 --> 00:29:53
mountains that are to the left
and right, and stay at the same
391
00:29:53 --> 00:29:57
height as the mountain pass.
On the other hand,
392
00:29:57 --> 00:30:01
if you go along the y
direction, to the left or to the
393
00:30:01 --> 00:30:05
right, then you go towards
positive values.
394
00:30:05 --> 00:30:11
And, if you go along the x
axis, then you get towards
395
00:30:11 --> 00:30:18
negative values.
OK, the equation for,
396
00:30:18 --> 00:30:25
the function was y^2-x^2.
So, you can try to plot them by
397
00:30:25 --> 00:30:27
hand and confirmed that it does
look like that.
398
00:30:27 --> 00:30:33
But, I trust my computer.
And, finally,
399
00:30:33 --> 00:30:39
this one, well,
so the contour plot looks a bit
400
00:30:39 --> 00:30:43
complicated.
But, you can see two things.
401
00:30:43 --> 00:30:45
In the middle,
you can see these two origins
402
00:30:45 --> 00:30:47
with these concentric circles
which are not really circles,
403
00:30:47 --> 00:30:50
but, you know,
these closed curves that are
404
00:30:50 --> 00:30:53
concentric.
And, they correspond to the two
405
00:30:53 --> 00:30:56
mountains.
And then, at some point in the
406
00:30:56 --> 00:31:00
middle, we have a mountain pass.
And there, we see the two
407
00:31:00 --> 00:31:05
crossing lines again,
like, on the plot of y^2-x^2.
408
00:31:05 --> 00:31:11
And so, at this saddle point
here, if we go north or south,
409
00:31:11 --> 00:31:15
then we go down on either side
to the Valley.
410
00:31:15 --> 00:31:17
And, if we go east or west,
then we go towards the
411
00:31:17 --> 00:31:21
mountains.
We'll go up.
412
00:31:21 --> 00:31:26
OK, does that make sense a
little bit?
413
00:31:26 --> 00:31:31
OK, so, reading plots is not
easy, but hopefully we'll get
414
00:31:31 --> 00:31:32
used to it very soon.
415
00:31:32 --> 00:31:44
416
00:31:44 --> 00:31:49
OK, so actually let's say,
well, OK, so,
417
00:31:49 --> 00:31:55
I want to point out one thing.
The contour plot tells us,
418
00:31:55 --> 00:32:00
actually, what happens when we
move, when we change x and y.
419
00:32:00 --> 00:32:05
So, if I change the value of x
and y, that means I'm moving
420
00:32:05 --> 00:32:08
east-west or north-south on the
map.
421
00:32:08 --> 00:32:12
And then, I can ask myself,
is the value of the function
422
00:32:12 --> 00:32:15
increase or decrease in each of
these situations?
423
00:32:15 --> 00:32:18
Well, that's the kind of thing
that the contour plot can tell
424
00:32:18 --> 00:32:19
me very quickly.
425
00:32:19 --> 00:32:54
426
00:32:54 --> 00:32:56
So -- OK, so say,
for example,
427
00:32:56 --> 00:32:59
that I have a piece of contour
plot.
428
00:32:59 --> 00:33:01
That looks, you know,
like that.
429
00:33:01 --> 00:33:06
Maybe this is f equals one,
and this is f equals two.
430
00:33:06 --> 00:33:13
And here, this is f equals zero.
And, let's say that I start at
431
00:33:13 --> 00:33:17
the point, say,
at this point.
432
00:33:17 --> 00:33:23
OK, so here I have (x0, y0).
And, the question I might ask
433
00:33:23 --> 00:33:26
myself is if I change x or y,
how does f change?
434
00:33:26 --> 00:33:34
Well, the contour plot tells me
that if x increases,
435
00:33:34 --> 00:33:41
and I keep y constant,
then what happens to f(x,
436
00:33:41 --> 00:33:44
y)?
Well, it will increase because
437
00:33:44 --> 00:33:47
if I move to the right,
then I go from one to a value
438
00:33:47 --> 00:33:50
bigger than one.
I don't know exactly how much,
439
00:33:50 --> 00:33:53
but I know that somewhere
between one and two,
440
00:33:53 --> 00:33:57
it's more than one.
If x decreases,
441
00:33:57 --> 00:34:02
then f decreases.
And, similarly,
442
00:34:02 --> 00:34:07
I can tell that if y increases,
then f, well,
443
00:34:07 --> 00:34:14
it looks like if I increase y,
then f will also increase.
444
00:34:14 --> 00:34:20
And, if y decreases,
then f decreases.
445
00:34:20 --> 00:34:23
And, that's the kind of
qualitative analysis that we can
446
00:34:23 --> 00:34:27
do easily from the contour plot.
But, maybe we'd like to
447
00:34:27 --> 00:34:30
actually be more precise in
that, and tell how fast f
448
00:34:30 --> 00:34:34
changes if I change x or y.
OK, so to find the rate of
449
00:34:34 --> 00:34:39
change, that's exactly where we
use derivatives.
450
00:34:39 --> 00:34:47
So -- So, we are going to have
to deal with partial
451
00:34:47 --> 00:34:58
derivatives.
So, I will explain to you soon
452
00:34:58 --> 00:35:05
why partial.
So, let me just remind you
453
00:35:05 --> 00:35:12
first, if you have a function of
one variable,
454
00:35:12 --> 00:35:18
then so let's say f of x,
then you have a derivative,
455
00:35:18 --> 00:35:22
f prime of x is also called
df/dx.
456
00:35:22 --> 00:35:31
And, it's defined as a limit
when delta x goes to zero of the
457
00:35:31 --> 00:35:35
change in f.
Sorry, it's not going to fit.
458
00:35:35 --> 00:35:42
I have to go to the next line.
It's going to be the limit as
459
00:35:42 --> 00:35:47
delta x goes to zero of the rate
of change.
460
00:35:47 --> 00:35:52
So, the change in f between x
and x plus delta x divided by
461
00:35:52 --> 00:35:56
delta x.
Sometimes you write delta f for
462
00:35:56 --> 00:35:59
the change in f divided by delta
x.
463
00:35:59 --> 00:36:04
And then, you take the limit of
this rate of increase as delta x
464
00:36:04 --> 00:36:05
goes to zero.
Now, of course,
465
00:36:05 --> 00:36:08
if you have a formula for f,
then you know,
466
00:36:08 --> 00:36:12
at least you should know,
I suspect most of you know how
467
00:36:12 --> 00:36:19
to actually take the derivative
of a function from its formula.
468
00:36:19 --> 00:36:30
So -- Now, how do we do that?
Sorry, and I should also tell
469
00:36:30 --> 00:36:32
you what this means on the
graph.
470
00:36:32 --> 00:36:36
So, if I plot the graph of a
function, and to have my point,
471
00:36:36 --> 00:36:41
x, and here I have f of x,
how do I see the derivative?
472
00:36:41 --> 00:36:48
Well, I look at the tangent
line to the graph,
473
00:36:48 --> 00:36:55
and the slope of the tangent
line is f prime of x,
474
00:36:55 --> 00:36:59
OK?
And, not every function has a
475
00:36:59 --> 00:37:03
derivative.
You have functions that are not
476
00:37:03 --> 00:37:05
regular enough to actually have
a derivative.
477
00:37:05 --> 00:37:08
So, in this class,
we are not going to actually
478
00:37:08 --> 00:37:11
worry too much about
differentiability.
479
00:37:11 --> 00:37:16
But, it's good,
at least, to be aware that you
480
00:37:16 --> 00:37:19
can't always take the
derivative.
481
00:37:19 --> 00:37:24
So, yes, and what else do I
want to remind you of?
482
00:37:24 --> 00:37:32
Well, they also have an
approximation formula -- --
483
00:37:32 --> 00:37:39
which says that,
you know, if we have the value
484
00:37:39 --> 00:37:41
of f at some point,
x0,
485
00:37:41 --> 00:37:47
and that we want to find the
value at a nearby point,
486
00:37:47 --> 00:37:51
x close to x0,
then our best guess is that
487
00:37:51 --> 00:37:58
it's f of x0 plus the derivative
f prime at x0 times delta x,
488
00:37:58 --> 00:38:02
or if you want, x minus x0,
OK?
489
00:38:02 --> 00:38:06
Is this kind of familiar to you?
Yeah, I mean,
490
00:38:06 --> 00:38:09
maybe with different notations.
Maybe you called that delta x
491
00:38:09 --> 00:38:12
or something.
Maybe you called that x0 plus h
492
00:38:12 --> 00:38:14
or something.
But, it's the usual
493
00:38:14 --> 00:38:18
approximation formula using the
derivative.
494
00:38:18 --> 00:38:21
If you put more terms,
then you get the dreaded Taylor
495
00:38:21 --> 00:38:24
approximation that I know you
guys don't like.
496
00:38:24 --> 00:38:36
So, the question is how do we
do the same for a function of
497
00:38:36 --> 00:38:41
two variables,
f(x, y)?
498
00:38:41 --> 00:38:45
So, the difficulty there is we
can change x,
499
00:38:45 --> 00:38:49
or we can change y,
or we can change both.
500
00:38:49 --> 00:38:52
And, presumably,
the manner in which f changes
501
00:38:52 --> 00:38:56
will be different depending on
whether we change x or y.
502
00:38:56 --> 00:39:00
So, that's why we have several
different notions of derivative.
503
00:39:00 --> 00:39:24
504
00:39:24 --> 00:39:37
So, OK, we have a notation.
OK, so this is a curly d,
505
00:39:37 --> 00:39:41
and it is not a straight d,
and it is not a delta.
506
00:39:41 --> 00:39:44
It's a d that kind of curves
backwards like that.
507
00:39:44 --> 00:39:50
And, this symbol is partial.
OK, so it's a special notation
508
00:39:50 --> 00:39:54
for partial derivatives.
And, essentially what it means
509
00:39:54 --> 00:39:56
is we are going to do a
derivative where we care about
510
00:39:56 --> 00:39:59
only one variable at a time.
That's why it's partial.
511
00:39:59 --> 00:40:02
It's missing the other
variables.
512
00:40:02 --> 00:40:06
So, a function of several
variables doesn't have the usual
513
00:40:06 --> 00:40:10
derivative.
It has only partial derivatives
514
00:40:10 --> 00:40:15
for each variable.
So, the partial derivative,
515
00:40:15 --> 00:40:23
the partial f partial x at (x0,
y0) is defined to be the limit
516
00:40:23 --> 00:40:29
when I take a small change in x,
delta x,
517
00:40:29 --> 00:40:43
of the change in f -- --
divided by delta x.
518
00:40:43 --> 00:40:47
OK, so here I'm actually not
changing y at all.
519
00:40:47 --> 00:40:51
I'm just changing x and looking
at the rate of change with
520
00:40:51 --> 00:40:54
respect to x.
And, I have the same with
521
00:40:54 --> 00:40:58
respect to y.
Partial f partial y is the
522
00:40:58 --> 00:41:04
limit, so I should say,
at a point x0 y0 is the limit
523
00:41:04 --> 00:41:13
as delta y turns to zero.
So, this time I keep x the
524
00:41:13 --> 00:41:21
same, but I change y.
OK, so that's the definition of
525
00:41:21 --> 00:41:26
a partial derivative.
And, we say that a function is
526
00:41:26 --> 00:41:29
differentiable if these things
exist.
527
00:41:29 --> 00:41:31
OK, so most of the functions
we'll see are differentiable.
528
00:41:31 --> 00:41:34
And, we'll actually learn how
to compute their partial
529
00:41:34 --> 00:41:38
derivatives without having to do
this because we'll just have the
530
00:41:38 --> 00:41:41
usual methods for computing
derivatives.
531
00:41:41 --> 00:41:46
So, in fact,
I claim you already know how to
532
00:41:46 --> 00:41:49
take partial derivatives.
So, let's see what it means
533
00:41:49 --> 00:41:50
geometrically.
534
00:41:50 --> 00:42:00
535
00:42:00 --> 00:42:07
So, geometrically,
my function can be represented
536
00:42:07 --> 00:42:12
by this graph,
and I fix some point,
537
00:42:12 --> 00:42:18
(x0, y0).
And then, I'm going to ask
538
00:42:18 --> 00:42:24
myself what happens if I change
the value of,
539
00:42:24 --> 00:42:30
well, x, keeping y constant.
So, if I keep y constant and
540
00:42:30 --> 00:42:33
change x, it means that I'm
moving forwards or backwards
541
00:42:33 --> 00:42:38
parallel to the x axis.
So, that determines for me the
542
00:42:38 --> 00:42:46
vertical plane parallel to the
x, z plane when I fix y equals
543
00:42:46 --> 00:42:51
constant.
And now, if I slice the graph
544
00:42:51 --> 00:42:59
by that, I will get some curve
that goes, it's a slice of the
545
00:42:59 --> 00:43:03
graph of f.
And now, what I want to find is
546
00:43:03 --> 00:43:06
how f depends on x if I keep y
constant.
547
00:43:06 --> 00:43:09
That means it's the rate of
change if I move along this
548
00:43:09 --> 00:43:11
curve.
So, in fact,
549
00:43:11 --> 00:43:17
if I look at the slope of this
thing.
550
00:43:17 --> 00:43:22
So, if I draw the tangent line
to this slice,
551
00:43:22 --> 00:43:28
then the slope will be partial
f of partial x.
552
00:43:28 --> 00:43:32
I think I have a better picture
of that somewhere.
553
00:43:32 --> 00:43:40
Yes, here it is.
OK, that's the same picture,
554
00:43:40 --> 00:43:43
just with different colors.
So, I take the graph.
555
00:43:43 --> 00:43:46
I slice it by a vertical plane.
I get the curve,
556
00:43:46 --> 00:43:50
and now I take the slope of
that curve, and that gives me
557
00:43:50 --> 00:43:54
the partial derivative.
And, to finish,
558
00:43:54 --> 00:43:59
let me just tell you how,
and I should say,
559
00:43:59 --> 00:44:02
partial f partial y is the same
thing but slicing now by your
560
00:44:02 --> 00:44:05
plane that goes in the y,
z directions.
561
00:44:05 --> 00:44:11
OK, so I fix x equals constant.
That means that I slice by a
562
00:44:11 --> 00:44:13
plane that's parallel to the
blackboard.
563
00:44:13 --> 00:44:17
I get a curve,
and I looked at the slope of
564
00:44:17 --> 00:44:20
that curve.
OK, so it's just a matter of
565
00:44:20 --> 00:44:23
formatting one variable,
setting it constant,
566
00:44:23 --> 00:44:27
and looking at the other one.
So, how to compute these
567
00:44:27 --> 00:44:29
things, well,
we actually,
568
00:44:29 --> 00:44:33
to find, well,
there's a piece of notation I
569
00:44:33 --> 00:44:38
haven't told you yet.
So, another notation you will
570
00:44:38 --> 00:44:42
see, I think this is what one
uses a lot in physics.
571
00:44:42 --> 00:44:45
And, this is what one uses a
lot in applied math,
572
00:44:45 --> 00:44:47
which is the same thing as
physics but with different
573
00:44:47 --> 00:44:50
notations.
OK, so it just too different
574
00:44:50 --> 00:44:54
notations: partial f partial x,
or f subscript x.
575
00:44:54 --> 00:45:01
And, they are the same thing.
Well, we just treat y as a
576
00:45:01 --> 00:45:10
constant, and x as a variable.
And, vice versa if we want to
577
00:45:10 --> 00:45:16
find partial with aspect to y.
So, let me just finish with one
578
00:45:16 --> 00:45:22
quick example.
Let's say that they give you f
579
00:45:22 --> 00:45:28
of x, y equals x^3y y^2,
then partial f partial x.
580
00:45:28 --> 00:45:32
Well, let's take the derivative.
So, here it's x^3 times a
581
00:45:32 --> 00:45:37
constant.
Derivative of x^3 is 3x^2 times
582
00:45:37 --> 00:45:42
the constant plus what's the
derivative of y^2?
583
00:45:42 --> 00:45:45
Zero, because it's a constant.
If you do, instead,
584
00:45:45 --> 00:45:48
partial f partial y,
then this is actually a
585
00:45:48 --> 00:45:51
constant times y.
The derivative of y is one.
586
00:45:51 --> 00:45:57
So, that's just x^3.
And, the derivative of y^2 is
587
00:45:57 --> 00:45:59
2y.
OK, so it's fairly easy.
588
00:45:59 --> 00:46:02
You just have to keep
remembering which one is a
589
00:46:02 --> 00:46:06
variable, and which one isn't.
OK, so more about this next
590
00:46:06 --> 00:46:10
time, and we will also learn
about maxima and minima in
591
00:46:10 --> 00:46:13
several variables.
592
00:46:13 --> 00:46:18