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So, so far, we've seen things
about vectors,
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equation of planes,
motions in space,
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and so on.
Basically we've done geometry
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in space.
But, calculus,
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really, is about studying
functions.
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Now, we're going to actually
move on to studying functions of
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several variables.
So, this new unit,
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what we'll do over the next
three weeks or so will be about
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functions of several variables
and their derivatives.
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OK, so first of all,
we should try to figure out how
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we are going to think about
functions.
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So, remember,
if you have a function of one
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variable, that means you have a
quantity that depends on one
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parameter.
Maybe f depends on the variable
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00:01:12,000 --> 00:01:14,000
x.
And, for example,
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00:01:14,000 --> 00:01:19,000
a function that you all know is
f of x equals sin(x).
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00:01:19,000 --> 00:01:22,000
And, the way we would represent
that is maybe just by plotting
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the graph of the function.
So, the graph of a function,
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00:01:28,000 --> 00:01:34,000
we plot y = f(x).
And, the graph of a sine
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00:01:34,000 --> 00:01:45,000
function that looks like this.
OK, so now, let's say that we
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had, actually,
a function of two variables.
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So, that means that the value
of F depends actually on two
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00:01:56,000 --> 00:01:59,000
different parameters,
say, if the variables are x and
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00:01:59,000 --> 00:02:03,000
y,
or they can have any names you
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00:02:03,000 --> 00:02:07,000
want.
So, given values of the two
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00:02:07,000 --> 00:02:13,000
parameters, x and y,
the function will give us a
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00:02:13,000 --> 00:02:17,000
number that we'll call f(x,
y).
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That depends on x and y
according to some formula,
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00:02:21,000 --> 00:02:29,000
OK, not very surprising so far.
So, for example,
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00:02:29,000 --> 00:02:44,000
I can give you the function
f(x, y) = x^2 y^2.
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00:02:44,000 --> 00:02:47,000
And, of course,
as with functions of one
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00:02:47,000 --> 00:02:50,000
variable, we don't need things
to be defined everywhere.
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00:02:50,000 --> 00:02:53,000
Sometimes there is the domain
of definition.
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00:02:53,000 --> 00:02:57,000
So, this one is defined all the
time.
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00:02:57,000 --> 00:03:01,000
But, if I tell you,
say, f of x,
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00:03:01,000 --> 00:03:08,000
y equals square root of y,
well, this is only defined if y
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00:03:08,000 --> 00:03:14,000
is nonnegative.
If I tell you f(x,
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00:03:14,000 --> 00:03:23,000
y) equals one over x y,
that's only defined if x y is
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00:03:23,000 --> 00:03:30,000
not zero, and so on.
Now, so these are mathematical
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00:03:30,000 --> 00:03:33,000
examples given by explicit
formulas.
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00:03:33,000 --> 00:03:35,000
And, of course,
there's physical examples.
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00:03:35,000 --> 00:03:38,000
For example,
so examples coming from real
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life, so for example,
you can look at the temperature
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at the certain point on the
surface of the earth.
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00:03:43,000 --> 00:03:46,000
So, you use maybe longitude and
latitude; that's x and y.
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00:03:46,000 --> 00:03:49,000
And then you have f(x,
y) equals the temperature at
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00:03:49,000 --> 00:03:50,000
that point.
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00:03:50,000 --> 00:04:12,000
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00:04:12,000 --> 00:04:17,000
In fact, because temperature
depends also may be on how high
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up you are.
It depends on elevation.
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00:04:18,000 --> 00:04:21,000
So, it's actually a function of
maybe x, y, z.
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00:04:21,000 --> 00:04:24,000
And, it also depends on time.
So, in fact,
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maybe it's a function of t in x
y z coordinates in space.
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00:04:28,000 --> 00:04:31,000
So, you see that real-world
functions can depends on a lot
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of variables.
So, our goal will be to
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understand how to deal with
that.
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00:04:35,000 --> 00:04:57,000
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00:04:57,000 --> 00:05:01,000
OK, so now you will see very
soon, but actually it's already
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00:05:01,000 --> 00:05:05,000
tricky enough to picture a
function of two variables.
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00:05:05,000 --> 00:05:08,000
So, we are going to focus on
the case of functions of two
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00:05:08,000 --> 00:05:10,000
variables.
And then, we'll see that if we
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00:05:10,000 --> 00:05:12,000
have more than two variables,
then it's harder to plot the
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00:05:12,000 --> 00:05:14,000
function.
We cannot draw with the graph
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00:05:14,000 --> 00:05:17,000
looks like anymore.
But, the tools are the same,
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00:05:17,000 --> 00:05:20,000
the notion of partial
derivatives, grade and vector,
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00:05:20,000 --> 00:05:23,000
and so on,
all the tools that we will
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develop work exactly the same
way no matter how many variables
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00:05:27,000 --> 00:05:30,000
you have.
So, I'll say,
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for simplicity -- -- we'll
focus mostly on two or sometimes
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three variables.
But, it works the same in any
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number of variables.
OK, so the first question is
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00:05:56,000 --> 00:06:05,000
how to visualize a function of
two variables.
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00:06:05,000 --> 00:06:10,000
So, the first thing we can do
is try to draw the graph of f.
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00:06:10,000 --> 00:06:19,000
So, maybe I should say f --
which is a function of two
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variables.
So, the first answer will be,
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00:06:23,000 --> 00:06:26,000
we can try to look at it's
graph.
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00:06:26,000 --> 00:06:29,000
And, the idea is the same as
with one variable,
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namely, we look at all the
possible values of the
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00:06:31,000 --> 00:06:34,000
parameters,
x and y, and for each of them,
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00:06:34,000 --> 00:06:40,000
we plot a point whose height is
the value of a function at these
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parameters.
So, we'll plot,
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00:06:43,000 --> 00:06:47,000
let's say, z equals f(x,
y).
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00:06:47,000 --> 00:06:52,000
And, now that will become,
actually, a surface in space.
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00:06:52,000 --> 00:06:57,000
OK, so for each value of x and
y, yeah, we have x,
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y in the x, y plane,
then we'll plot the point in
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space at position x,
y.
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00:07:05,000 --> 00:07:13,000
And, z equals f of x, y.
OK, and if we take all of these
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points together,
they will give us some surface
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that sits in space.
Yes?
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Oh, a function of two
variables, shorthand.
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Well, let's say how to
visualize a function of two
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variables.
OK, so, how do we do that
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concretely?
Say that I give you a formula
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00:07:41,000 --> 00:07:46,000
for f.
How do we try to represent it?
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So, let's do our first example.
Let's say I give you a function
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f(x, y) = -y.
OK, so it looks a little bit
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silly because it doesn't depend
on x.
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00:08:04,000 --> 00:08:10,000
But, that's not the problem.
It's still a valid function of
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x and y.
It just happens to be constant
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with respect to x.
So, to draw the graph we look
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at the surface in space defined
by z equals y.
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What kind of surface is that?
It's a plane, OK?
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00:08:26,000 --> 00:08:32,000
And, if we want to draw it,
z equals minus y will look,
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00:08:32,000 --> 00:08:36,000
well, let's put y axis.
Let's put x axis.
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00:08:36,000 --> 00:08:39,000
Let's put z axis.
If I look at what happens in
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the y, z plane in the plane of a
blackboard, it will just look
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00:08:43,000 --> 00:08:45,000
like a line that goes downward
with slope one.
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00:08:45,000 --> 00:08:51,000
OK, so it will be this.
And, what happens if I change x?
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00:08:51,000 --> 00:08:54,000
Well, if I change x,
nothing happens because x
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doesn't appear in this equation.
So, in fact,
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00:08:57,000 --> 00:09:01,000
if instead of setting x equal
to zero I set x equal to one,
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00:09:01,000 --> 00:09:04,000
I'm in front of the blackboard,
or minus one at the back.
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00:09:04,000 --> 00:09:07,000
Well, it still looks exactly
the same.
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00:09:07,000 --> 00:09:15,000
So, I have this plane that
actually contains the x axis and
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00:09:15,000 --> 00:09:22,000
slopes downwards with slope one.
It's kind of hard to draw.
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00:09:22,000 --> 00:09:25,000
Now, you see immediately what
the big problem with graphs will
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be.
But, these pictures are hard to
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00:09:29,000 --> 00:09:34,000
read.
But that's our first graph.
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OK, a question so far?
OK, so let's say that we have a
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slightly more complicated
function.
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00:09:43,000 --> 00:09:50,000
How do we see it?
So, let's draw another example.
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00:09:50,000 --> 00:09:58,000
Let's say I give you f(x,
y) = 1 - x^2-y^2.
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00:09:58,000 --> 00:10:06,000
So, we should try to picture
what the surface z=1- x^2-y^2
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00:10:06,000 --> 00:10:10,000
looks like.
So, how do we do that?
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00:10:10,000 --> 00:10:15,000
Well, maybe you are very fast
and figured out what it looks
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00:10:15,000 --> 00:10:17,000
like.
But, if not,
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00:10:17,000 --> 00:10:21,000
then we need to work piece by
piece.
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00:10:21,000 --> 00:10:27,000
So, maybe it will help if we
understand first what it does in
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00:10:27,000 --> 00:10:35,000
the plane of the blackboard.
So, if we look at it in the y,
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00:10:35,000 --> 00:10:42,000
z plane, that means we set x
equal to zero.
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00:10:42,000 --> 00:10:48,000
And then, z becomes 1 - y^2.
What is that?
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00:10:48,000 --> 00:10:54,000
It's a parabola pointing
downwards, and starting at one.
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00:10:54,000 --> 00:11:00,000
So, we should draw maybe this
downward parabola.
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00:11:00,000 --> 00:11:08,000
It starts at one and it cuts
the y axis at one.
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00:11:08,000 --> 00:11:11,000
When y is one,
that gives us zero.
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00:11:11,000 --> 00:11:15,000
So, we might have an idea of
what it might look like,
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00:11:15,000 --> 00:11:19,000
or maybe not.
Let's get more slices.
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00:11:19,000 --> 00:11:27,000
Let's see what it does in the
x, z plane, this other vertical
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00:11:27,000 --> 00:11:33,000
plane that's coming towards us.
So, in the x,
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00:11:33,000 --> 00:11:38,000
z plane, if we set y equal to
zero, we get z equals one minus
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00:11:38,000 --> 00:11:40,000
x^2.
It's, again,
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00:11:40,000 --> 00:11:46,000
a parabola going downwards.
OK, so I'm going to try to draw
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00:11:46,000 --> 00:11:51,000
a parabola that goes downward,
but now to the front and to the
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00:11:51,000 --> 00:11:54,000
back.
So, we are starting to have a
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00:11:54,000 --> 00:11:57,000
slightly better idea but we
still don't know whether the
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00:11:57,000 --> 00:11:59,000
cross section of this thing
might be round,
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00:11:59,000 --> 00:12:04,000
square, something else.
So, it wants more confirmation.
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00:12:04,000 --> 00:12:16,000
We might want to also figure
out where the surface intersects
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the x, y plane.
So, we hit the x,
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00:12:22,000 --> 00:12:29,000
y plane when z equals zero.
That means 1-x^2-y^2 should be
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00:12:29,000 --> 00:12:38,000
0, that becomes x^2 y^2 = 1.
That is a circle of radius 1.
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00:12:38,000 --> 00:12:46,000
That's the unit size.
So, that means that here,
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00:12:46,000 --> 00:12:50,000
we actually have the unit
circle.
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00:12:50,000 --> 00:12:54,000
And now, you should imagine
that you have this thing that
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00:12:54,000 --> 00:12:57,000
when you slice it by a vertical
plane, looks like a downwards
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00:12:57,000 --> 00:13:00,000
parabola.
And, it's actually a surface of
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00:13:00,000 --> 00:13:03,000
revolution.
You can rotate it around the z
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00:13:03,000 --> 00:13:06,000
axis, OK?
Now, if you stare long enough
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00:13:06,000 --> 00:13:09,000
at that equation,
you'll actually see that,
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yes, we know that it had to be
like that.
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00:13:12,000 --> 00:13:17,000
But, see, so these are useful
ways of trying to guess what the
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00:13:17,000 --> 00:13:19,000
graph looks like.
Of course, the other way is to
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00:13:19,000 --> 00:13:24,000
just ask your computer to do it.
And then, you know,
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00:13:24,000 --> 00:13:30,000
you will get that kind of
formula.
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00:13:30,000 --> 00:13:36,000
OK, well, I can leave it on if
you want.
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No, because I plotted a
different function that I will
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show you later.
So, it goes this way.
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00:13:43,000 --> 00:13:46,000
I mean, if you want,
it's really going downward.
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00:13:46,000 --> 00:13:49,000
Yes, I agree that the sheet is
upside down.
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That's because I plotted
something else.
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00:13:52,000 --> 00:13:58,000
OK, so, in fact,
so I plotted in my computer was
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00:13:58,000 --> 00:14:03,000
actually x^2 y^2 that looks like
that.
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00:14:03,000 --> 00:14:08,000
See, it's the same with a
parabola going upwards.
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00:14:08,000 --> 00:14:14,000
If you want to see more
examples, I have various
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00:14:14,000 --> 00:14:19,000
examples to show,
well, here's the graph,
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00:14:19,000 --> 00:14:21,000
y^2-x^2.
See, so that one is kind of
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interesting.
It looks like a saddle.
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00:14:23,000 --> 00:14:29,000
If you look at it in the y,
z plane, then it's a parabola
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00:14:29,000 --> 00:14:34,000
going up, z = y^2.
And, that's what we see to the
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00:14:34,000 --> 00:14:38,000
left and to the right.
But, if you put it in the x,
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00:14:38,000 --> 00:14:42,000
z plane, then that's a parabola
going downwards,
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00:14:42,000 --> 00:14:45,000
z = - x^2.
So, we have a parabola going
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00:14:45,000 --> 00:14:48,000
downwards in one direction,
upwards in the other one.
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00:14:48,000 --> 00:14:53,000
And together,
they form this surface.
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00:14:53,000 --> 00:14:55,000
And of course,
you can plot much more
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00:14:55,000 --> 00:14:58,000
complicated functions.
So, this one,
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well, if you can read very
small things,
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00:15:00,000 --> 00:15:03,000
you can see the formula.
It doesn't matter,
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00:15:03,000 --> 00:15:09,000
just to show you that you can
put a formula into a computer:
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it will show you a picture.
OK, so that's pretty good.
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00:15:18,000 --> 00:15:20,000
I mean, you can see that it can
get a bit cluttered because
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00:15:20,000 --> 00:15:22,000
maybe those features that are
hidden behind,
200
00:15:22,000 --> 00:15:25,000
or maybe we have trouble seeing
if we don't have a computer,
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00:15:25,000 --> 00:15:29,000
that looks very readable.
But, this is kind of hard to
202
00:15:29,000 --> 00:15:33,000
visualize sometimes.
So, there is another way to
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00:15:33,000 --> 00:15:36,000
plot the functions of two
variables.
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00:15:36,000 --> 00:15:47,000
And, let's call it the contour
plot.
205
00:15:47,000 --> 00:15:51,000
So, the contour plot is a very
elegant solution to the problem
206
00:15:51,000 --> 00:15:55,000
that it's difficult to draw to
space pictures on a sheet of
207
00:15:55,000 --> 00:15:58,000
paper or on a blackboard.
So, instead,
208
00:15:58,000 --> 00:16:02,000
let's try to represent the
function of two variables by
209
00:16:02,000 --> 00:16:04,000
just the map,
you know, the same way that
210
00:16:04,000 --> 00:16:07,000
when you walk around,
you have actually geographical
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00:16:07,000 --> 00:16:11,000
maps that fit on a piece of
paper that tell you about what
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00:16:11,000 --> 00:16:17,000
the real world looks like.
So, what contour plot looks
213
00:16:17,000 --> 00:16:22,000
something like this?
So, it's an x, y plot.
214
00:16:22,000 --> 00:16:25,000
And, that, you have a bunch of
curves.
215
00:16:25,000 --> 00:16:32,000
And, what the curves represent
are the elevations on the graph.
216
00:16:32,000 --> 00:16:35,000
So, for example,
this curve might correspond to
217
00:16:35,000 --> 00:16:37,000
all the points where f(x,
y) = 1.
218
00:16:37,000 --> 00:16:46,000
And, that curve might be all
the points where f=2 and f=3 and
219
00:16:46,000 --> 00:16:49,000
so on, OK?
So, when you see you this kind
220
00:16:49,000 --> 00:16:53,000
of plot, you're supposed to
think that the graph of the
221
00:16:53,000 --> 00:16:56,000
function sits somewhere in space
above that.
222
00:16:56,000 --> 00:17:00,000
It's like a map telling you how
high things are.
223
00:17:00,000 --> 00:17:03,000
And, what you would want to do
with the function,
224
00:17:03,000 --> 00:17:06,000
really, is be able to tell
quickly what's the value at a
225
00:17:06,000 --> 00:17:08,000
given point?
Well, let's say I want to look
226
00:17:08,000 --> 00:17:11,000
at that point.
I know that f is somewhere
227
00:17:11,000 --> 00:17:14,000
between 1 and 2.
Actually, it's much faster to
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00:17:14,000 --> 00:17:17,000
read than the graph.
On the graph I might have to
229
00:17:17,000 --> 00:17:18,000
look carefully,
and then measure things,
230
00:17:18,000 --> 00:17:22,000
and so on.
Here, I can just raise the
231
00:17:22,000 --> 00:17:27,000
value of f by comparing with the
nearby lines.
232
00:17:27,000 --> 00:17:31,000
OK, so let me try to make that
more precise.
233
00:17:31,000 --> 00:17:41,000
So, it shows all the points --
-- where f(x,
234
00:17:41,000 --> 00:17:53,000
y) equals some fixed values,
some fixed constants.
235
00:17:53,000 --> 00:18:05,000
And, these constants typically
are chosen at regular intervals.
236
00:18:05,000 --> 00:18:07,000
For example,
here I chose one,
237
00:18:07,000 --> 00:18:11,000
two, three, and they could
continue with zero minus one,
238
00:18:11,000 --> 00:18:16,000
and so on.
So, one way to think about it,
239
00:18:16,000 --> 00:18:21,000
how does this relate to the
graph?
240
00:18:21,000 --> 00:18:31,000
Well, that's the same thing as
cutting, I mean,
241
00:18:31,000 --> 00:18:40,000
we slice the graph by
horizontal planes.
242
00:18:40,000 --> 00:18:44,000
So, horizontal planes have
equations of a form z equals
243
00:18:44,000 --> 00:18:46,000
some constant,
z equals zero,
244
00:18:46,000 --> 00:18:48,000
z equals one,
z equals two,
245
00:18:48,000 --> 00:18:51,000
and so on.
So, maybe the graph of my
246
00:18:51,000 --> 00:18:55,000
function will be some sort of
plot out there.
247
00:18:55,000 --> 00:19:00,000
And, if I slice it by the plane
z equals one,
248
00:19:00,000 --> 00:19:04,000
then I will get the level
curve,
249
00:19:04,000 --> 00:19:14,000
which is the point where f(x,
y) = 1,
250
00:19:14,000 --> 00:19:24,000
and so, that's called a level
curve of f.
251
00:19:24,000 --> 00:19:32,000
OK, and so we repeat the
process with maybe z equals two,
252
00:19:32,000 --> 00:19:38,000
and we get another level curve,
and so on.
253
00:19:38,000 --> 00:19:44,000
And, then we squish all of them
up, and that's how we get the
254
00:19:44,000 --> 00:19:47,000
contour plot.
OK, so each of these lines,
255
00:19:47,000 --> 00:19:50,000
imagine that this is like some
mountain or something that you
256
00:19:50,000 --> 00:19:52,000
are hiking on.
Each of these lines tells you
257
00:19:52,000 --> 00:19:55,000
how you could move to stay at a
constant height if you want to
258
00:19:55,000 --> 00:19:58,000
get to the other side of the
mountain but without ever going
259
00:19:58,000 --> 00:20:03,000
up or down.
You would just walk along that
260
00:20:03,000 --> 00:20:08,000
path, and it will get you there
without effort.
261
00:20:08,000 --> 00:20:11,000
So, in fact,
if you have been talking about
262
00:20:11,000 --> 00:20:15,000
hiking on mountains,
well, that's exactly what a
263
00:20:15,000 --> 00:20:21,000
topographical map is about.
So, I need to zoom a bit.
264
00:20:21,000 --> 00:20:27,000
So, a topographic map,
this one from the US geological
265
00:20:27,000 --> 00:20:32,000
survey shows you,
basically, all the level curves
266
00:20:32,000 --> 00:20:37,000
of an altitude function on a
piece of land.
267
00:20:37,000 --> 00:20:40,000
So, you know that if you walk
right along these curves,
268
00:20:40,000 --> 00:20:42,000
you will stay along the same
height.
269
00:20:42,000 --> 00:20:46,000
And you know that if you walk
towards, these don't have
270
00:20:46,000 --> 00:20:48,000
numbers.
Let me find a place with
271
00:20:48,000 --> 00:20:53,000
numbers.
Here, there is a 500 in the
272
00:20:53,000 --> 00:20:56,000
middle.
So, you know that if you walk
273
00:20:56,000 --> 00:20:59,000
on the line that says 500,
you stay always at 500 meters
274
00:20:59,000 --> 00:21:02,000
in elevation.
If you walk towards the
275
00:21:02,000 --> 00:21:05,000
mountain that I think is below
it, then you will go up,
276
00:21:05,000 --> 00:21:07,000
and so on.
So, you can see,
277
00:21:07,000 --> 00:21:10,000
for example,
here there's a peak,
278
00:21:10,000 --> 00:21:13,000
and here there is a valley with
the river in it,
279
00:21:13,000 --> 00:21:17,000
and the altitudes go down,
and then back up again on the
280
00:21:17,000 --> 00:21:19,000
other side.
OK, so that's an example of a
281
00:21:19,000 --> 00:21:22,000
contour plot of a function.
Of course, we don't have a
282
00:21:22,000 --> 00:21:25,000
formula for that function,
but we have a contour plot,
283
00:21:25,000 --> 00:21:29,000
and that's what we need
actually to understand what's
284
00:21:29,000 --> 00:21:36,000
going on there.
OK, any questions?
285
00:21:36,000 --> 00:21:39,000
No?
OK, so another example of
286
00:21:39,000 --> 00:21:42,000
contour plots,
well, you've probably seen
287
00:21:42,000 --> 00:21:46,000
various versions of these
temperature maps.
288
00:21:46,000 --> 00:21:51,000
So, that's supposed to be how
warm it is right now.
289
00:21:51,000 --> 00:21:55,000
So, this one is color-coded.
Instead of having curves,
290
00:21:55,000 --> 00:21:58,000
it has all these colors.
But, the effect is the same.
291
00:21:58,000 --> 00:22:01,000
If you look at the separations
between consecutive colors,
292
00:22:01,000 --> 00:22:05,000
these are the level curves of a
function that tells you the
293
00:22:05,000 --> 00:22:12,000
temperature at a given point.
OK, so these are examples of
294
00:22:12,000 --> 00:22:24,000
contour plots in real life.
OK, no questions?
295
00:22:24,000 --> 00:22:26,000
No?
OK, so basically,
296
00:22:26,000 --> 00:22:31,000
one of the goals that one
should try to achieve at this
297
00:22:31,000 --> 00:22:35,000
point is becoming familiar with
the contour plot,
298
00:22:35,000 --> 00:22:38,000
the graph,
and how to view and deal with
299
00:22:38,000 --> 00:22:39,000
functions.
300
00:22:39,000 --> 00:22:54,000
301
00:22:54,000 --> 00:23:02,000
[APPLAUSE]
OK, so -- Let's do an example.
302
00:23:02,000 --> 00:23:04,000
Well, let's do a couple of
examples.
303
00:23:04,000 --> 00:23:08,000
So, let's start with f(x,y) = -
y.
304
00:23:08,000 --> 00:23:12,000
And, I'm going to take the same
two examples as there to start
305
00:23:12,000 --> 00:23:16,000
with so that we see the relation
between the graph and the
306
00:23:16,000 --> 00:23:23,000
contour plots.
So, let's try to plot it.
307
00:23:23,000 --> 00:23:30,000
So, we are asked for the level
curve, f equals 0 for this one?
308
00:23:30,000 --> 00:23:38,000
Well, f is zero when y is zero.
So, that's the x axis.
309
00:23:38,000 --> 00:23:44,000
OK, so that's the level, zero.
Where's the level one?
310
00:23:44,000 --> 00:23:48,000
Well, f is one when negative y
is one.
311
00:23:48,000 --> 00:23:51,000
That means when y is negative
one.
312
00:23:51,000 --> 00:23:57,000
So, you go to minus one,
and that will be where my level
313
00:23:57,000 --> 00:24:02,000
one is, and so on.
f is two when y is negative
314
00:24:02,000 --> 00:24:06,000
two.
F is negative one when y is
315
00:24:06,000 --> 00:24:10,000
one, and so on.
Is that convincing?
316
00:24:10,000 --> 00:24:15,000
Do you see how we got that?
OK, let me do it again.
317
00:24:15,000 --> 00:24:18,000
I don't see anybody nodding,
so that's kind of bad news.
318
00:24:18,000 --> 00:24:22,000
So, if I want to know,
where is the level curve,
319
00:24:22,000 --> 00:24:26,000
say, one, I try to set f equals
to one.
320
00:24:26,000 --> 00:24:31,000
Let's do this one.
f equals one means that
321
00:24:31,000 --> 00:24:36,000
negative y is one means that y
is minus one,
322
00:24:36,000 --> 00:24:43,000
and y equals minus one is this
horizontal line on this chart.
323
00:24:43,000 --> 00:24:47,000
OK, and same with the others.
So, you can see on the map that
324
00:24:47,000 --> 00:24:49,000
the value of a function doesn't
depend on x.
325
00:24:49,000 --> 00:24:52,000
If you move parallel to the x
axis, nothing happens.
326
00:24:52,000 --> 00:24:56,000
If you move in the y direction,
it decreases at a constant
327
00:24:56,000 --> 00:24:59,000
rate.
That's why the contours are
328
00:24:59,000 --> 00:25:03,000
evenly spaced.
How spaced out they are tells
329
00:25:03,000 --> 00:25:06,000
you, actually,
how steep things are.
330
00:25:06,000 --> 00:25:08,000
So, that corresponds exactly to
that picture,
331
00:25:08,000 --> 00:25:11,000
except that here we draw x
coming to the front,
332
00:25:11,000 --> 00:25:14,000
and y to the right.
So, you have to rotate the map
333
00:25:14,000 --> 00:25:19,000
by 90� to get to that.
It's an unfortunate consequence
334
00:25:19,000 --> 00:25:24,000
of the usual way of plotting
things in space.
335
00:25:24,000 --> 00:25:32,000
OK, so these horizontal lines
here correspond actually to
336
00:25:32,000 --> 00:25:35,000
horizontal lines here.
337
00:25:35,000 --> 00:25:43,000
338
00:25:43,000 --> 00:25:54,000
OK, second example.
Let's do 1-x^2-y^2.
339
00:25:54,000 --> 00:26:00,000
OK, or maybe I will write it as
1 - (x^2 y^2).
340
00:26:00,000 --> 00:26:06,000
It's really the same thing.
So, x, y, let's see,
341
00:26:06,000 --> 00:26:12,000
where is this function equal to
zero?
342
00:26:12,000 --> 00:26:21,000
Well, we said f is zero in the
unit circle.
343
00:26:21,000 --> 00:26:32,000
OK, so, the zero level,
well, let's say that this is my
344
00:26:32,000 --> 00:26:36,000
unit.
That's where it's at zero.
345
00:26:36,000 --> 00:26:48,000
What about f equals one?
Well, that's when x^2 y^2 = 0.
346
00:26:48,000 --> 00:26:49,000
Well, that's only going to be
here.
347
00:26:49,000 --> 00:27:00,000
So, that's just a single point.
What about f equals minus one?
348
00:27:00,000 --> 00:27:07,000
That's when x^2 y^2 =2.
That's a circle of radius
349
00:27:07,000 --> 00:27:10,000
square root of two,
which is about 1.4.
350
00:27:10,000 --> 00:27:17,000
So, it's somewhere here.
Then, minus two,
351
00:27:17,000 --> 00:27:24,000
similarly, will be x^2 y^2 = 3.
Square root of three is about
352
00:27:24,000 --> 00:27:27,000
1.7.
And then, minus three will be
353
00:27:27,000 --> 00:27:30,000
of radius two,
and so on.
354
00:27:30,000 --> 00:27:38,000
So, what I want to show here is
that they are getting closer and
355
00:27:38,000 --> 00:27:41,000
closer apart,
OK?
356
00:27:41,000 --> 00:27:44,000
So, first it's concentric
circles that tells us that we
357
00:27:44,000 --> 00:27:47,000
have a shape that's a solid of
the graph is going to be a
358
00:27:47,000 --> 00:27:52,000
surface of revolution.
Things don't change if I rotate.
359
00:27:52,000 --> 00:27:56,000
And second, the level curves
are getting closer and closer to
360
00:27:56,000 --> 00:27:59,000
each other.
That means it's getting steeper
361
00:27:59,000 --> 00:28:03,000
and steeper because I have to
travel a shorter distance if I
362
00:28:03,000 --> 00:28:06,000
want my altitude to change by
one.
363
00:28:06,000 --> 00:28:09,000
OK, so, this top here is kind
of flat.
364
00:28:09,000 --> 00:28:11,000
And then it gets steeper and
steeper.
365
00:28:11,000 --> 00:28:16,000
And, that's what we observe on
that picture there.
366
00:28:16,000 --> 00:28:24,000
OK, so just to show you a few
more, where did I put my,
367
00:28:24,000 --> 00:28:30,000
so, for x^2 y^2,
the contour plot looks like
368
00:28:30,000 --> 00:28:37,000
this.
Maybe actually I'll make it.
369
00:28:37,000 --> 00:28:41,000
OK, so it looks exactly the
same as this one.
370
00:28:41,000 --> 00:28:44,000
But, the difference is if you
can see the numbers which are
371
00:28:44,000 --> 00:28:45,000
not there,
so you can see them,
372
00:28:45,000 --> 00:28:49,000
then you would know that
instead of decreasing as we move
373
00:28:49,000 --> 00:28:52,000
out,
this one is increasing as we go
374
00:28:52,000 --> 00:28:54,000
out.
OK, so that's where we use,
375
00:28:54,000 --> 00:28:57,000
actually, the labels on the
level curves that tell us
376
00:28:57,000 --> 00:29:00,000
whether things are going up or
down.
377
00:29:00,000 --> 00:29:04,000
But, the contour plots look
exactly the same.
378
00:29:04,000 --> 00:29:14,000
For the next one I had,
I think, was y^2-x^2.
379
00:29:14,000 --> 00:29:18,000
So, the contour plot,
well, let me actually zoom out.
380
00:29:18,000 --> 00:29:20,000
So, the contour plot looks like
that.
381
00:29:20,000 --> 00:29:23,000
So, the level curve
corresponding to zero is
382
00:29:23,000 --> 00:29:27,000
actually two diagonal lines.
And, if you look on the plot,
383
00:29:27,000 --> 00:29:30,000
say that you started at the
saddle point in the middle and
384
00:29:30,000 --> 00:29:33,000
you try to stay at the same
level.
385
00:29:33,000 --> 00:29:35,000
So, it looks like a mountain
pass, right?
386
00:29:35,000 --> 00:29:38,000
Well, there's actually four
directions from that point that
387
00:29:38,000 --> 00:29:41,000
you can go staying at the same
height.
388
00:29:41,000 --> 00:29:44,000
And actually,
on the map, they look exactly
389
00:29:44,000 --> 00:29:46,000
like this, too,
these crossing lines.
390
00:29:46,000 --> 00:29:49,000
OK, so, these are things that
go on the side of the two
391
00:29:49,000 --> 00:29:53,000
mountains that are to the left
and right, and stay at the same
392
00:29:53,000 --> 00:29:57,000
height as the mountain pass.
On the other hand,
393
00:29:57,000 --> 00:30:01,000
if you go along the y
direction, to the left or to the
394
00:30:01,000 --> 00:30:05,000
right, then you go towards
positive values.
395
00:30:05,000 --> 00:30:11,000
And, if you go along the x
axis, then you get towards
396
00:30:11,000 --> 00:30:18,000
negative values.
OK, the equation for,
397
00:30:18,000 --> 00:30:25,000
the function was y^2-x^2.
So, you can try to plot them by
398
00:30:25,000 --> 00:30:27,000
hand and confirmed that it does
look like that.
399
00:30:27,000 --> 00:30:33,000
But, I trust my computer.
And, finally,
400
00:30:33,000 --> 00:30:39,000
this one, well,
so the contour plot looks a bit
401
00:30:39,000 --> 00:30:43,000
complicated.
But, you can see two things.
402
00:30:43,000 --> 00:30:45,000
In the middle,
you can see these two origins
403
00:30:45,000 --> 00:30:47,000
with these concentric circles
which are not really circles,
404
00:30:47,000 --> 00:30:50,000
but, you know,
these closed curves that are
405
00:30:50,000 --> 00:30:53,000
concentric.
And, they correspond to the two
406
00:30:53,000 --> 00:30:56,000
mountains.
And then, at some point in the
407
00:30:56,000 --> 00:31:00,000
middle, we have a mountain pass.
And there, we see the two
408
00:31:00,000 --> 00:31:05,000
crossing lines again,
like, on the plot of y^2-x^2.
409
00:31:05,000 --> 00:31:11,000
And so, at this saddle point
here, if we go north or south,
410
00:31:11,000 --> 00:31:15,000
then we go down on either side
to the Valley.
411
00:31:15,000 --> 00:31:17,000
And, if we go east or west,
then we go towards the
412
00:31:17,000 --> 00:31:21,000
mountains.
We'll go up.
413
00:31:21,000 --> 00:31:26,000
OK, does that make sense a
little bit?
414
00:31:26,000 --> 00:31:31,000
OK, so, reading plots is not
easy, but hopefully we'll get
415
00:31:31,000 --> 00:31:32,000
used to it very soon.
416
00:31:32,000 --> 00:31:44,000
417
00:31:44,000 --> 00:31:49,000
OK, so actually let's say,
well, OK, so,
418
00:31:49,000 --> 00:31:55,000
I want to point out one thing.
The contour plot tells us,
419
00:31:55,000 --> 00:32:00,000
actually, what happens when we
move, when we change x and y.
420
00:32:00,000 --> 00:32:05,000
So, if I change the value of x
and y, that means I'm moving
421
00:32:05,000 --> 00:32:08,000
east-west or north-south on the
map.
422
00:32:08,000 --> 00:32:12,000
And then, I can ask myself,
is the value of the function
423
00:32:12,000 --> 00:32:15,000
increase or decrease in each of
these situations?
424
00:32:15,000 --> 00:32:18,000
Well, that's the kind of thing
that the contour plot can tell
425
00:32:18,000 --> 00:32:19,000
me very quickly.
426
00:32:19,000 --> 00:32:54,000
427
00:32:54,000 --> 00:32:56,000
So -- OK, so say,
for example,
428
00:32:56,000 --> 00:32:59,000
that I have a piece of contour
plot.
429
00:32:59,000 --> 00:33:01,000
That looks, you know,
like that.
430
00:33:01,000 --> 00:33:06,000
Maybe this is f equals one,
and this is f equals two.
431
00:33:06,000 --> 00:33:13,000
And here, this is f equals zero.
And, let's say that I start at
432
00:33:13,000 --> 00:33:17,000
the point, say,
at this point.
433
00:33:17,000 --> 00:33:23,000
OK, so here I have (x0, y0).
And, the question I might ask
434
00:33:23,000 --> 00:33:26,000
myself is if I change x or y,
how does f change?
435
00:33:26,000 --> 00:33:34,000
Well, the contour plot tells me
that if x increases,
436
00:33:34,000 --> 00:33:41,000
and I keep y constant,
then what happens to f(x,
437
00:33:41,000 --> 00:33:44,000
y)?
Well, it will increase because
438
00:33:44,000 --> 00:33:47,000
if I move to the right,
then I go from one to a value
439
00:33:47,000 --> 00:33:50,000
bigger than one.
I don't know exactly how much,
440
00:33:50,000 --> 00:33:53,000
but I know that somewhere
between one and two,
441
00:33:53,000 --> 00:33:57,000
it's more than one.
If x decreases,
442
00:33:57,000 --> 00:34:02,000
then f decreases.
And, similarly,
443
00:34:02,000 --> 00:34:07,000
I can tell that if y increases,
then f, well,
444
00:34:07,000 --> 00:34:14,000
it looks like if I increase y,
then f will also increase.
445
00:34:14,000 --> 00:34:20,000
And, if y decreases,
then f decreases.
446
00:34:20,000 --> 00:34:23,000
And, that's the kind of
qualitative analysis that we can
447
00:34:23,000 --> 00:34:27,000
do easily from the contour plot.
But, maybe we'd like to
448
00:34:27,000 --> 00:34:30,000
actually be more precise in
that, and tell how fast f
449
00:34:30,000 --> 00:34:34,000
changes if I change x or y.
OK, so to find the rate of
450
00:34:34,000 --> 00:34:39,000
change, that's exactly where we
use derivatives.
451
00:34:39,000 --> 00:34:47,000
So -- So, we are going to have
to deal with partial
452
00:34:47,000 --> 00:34:58,000
derivatives.
So, I will explain to you soon
453
00:34:58,000 --> 00:35:05,000
why partial.
So, let me just remind you
454
00:35:05,000 --> 00:35:12,000
first, if you have a function of
one variable,
455
00:35:12,000 --> 00:35:18,000
then so let's say f of x,
then you have a derivative,
456
00:35:18,000 --> 00:35:22,000
f prime of x is also called
df/dx.
457
00:35:22,000 --> 00:35:31,000
And, it's defined as a limit
when delta x goes to zero of the
458
00:35:31,000 --> 00:35:35,000
change in f.
Sorry, it's not going to fit.
459
00:35:35,000 --> 00:35:42,000
I have to go to the next line.
It's going to be the limit as
460
00:35:42,000 --> 00:35:47,000
delta x goes to zero of the rate
of change.
461
00:35:47,000 --> 00:35:52,000
So, the change in f between x
and x plus delta x divided by
462
00:35:52,000 --> 00:35:56,000
delta x.
Sometimes you write delta f for
463
00:35:56,000 --> 00:35:59,000
the change in f divided by delta
x.
464
00:35:59,000 --> 00:36:04,000
And then, you take the limit of
this rate of increase as delta x
465
00:36:04,000 --> 00:36:05,000
goes to zero.
Now, of course,
466
00:36:05,000 --> 00:36:08,000
if you have a formula for f,
then you know,
467
00:36:08,000 --> 00:36:12,000
at least you should know,
I suspect most of you know how
468
00:36:12,000 --> 00:36:19,000
to actually take the derivative
of a function from its formula.
469
00:36:19,000 --> 00:36:30,000
So -- Now, how do we do that?
Sorry, and I should also tell
470
00:36:30,000 --> 00:36:32,000
you what this means on the
graph.
471
00:36:32,000 --> 00:36:36,000
So, if I plot the graph of a
function, and to have my point,
472
00:36:36,000 --> 00:36:41,000
x, and here I have f of x,
how do I see the derivative?
473
00:36:41,000 --> 00:36:48,000
Well, I look at the tangent
line to the graph,
474
00:36:48,000 --> 00:36:55,000
and the slope of the tangent
line is f prime of x,
475
00:36:55,000 --> 00:36:59,000
OK?
And, not every function has a
476
00:36:59,000 --> 00:37:03,000
derivative.
You have functions that are not
477
00:37:03,000 --> 00:37:05,000
regular enough to actually have
a derivative.
478
00:37:05,000 --> 00:37:08,000
So, in this class,
we are not going to actually
479
00:37:08,000 --> 00:37:11,000
worry too much about
differentiability.
480
00:37:11,000 --> 00:37:16,000
But, it's good,
at least, to be aware that you
481
00:37:16,000 --> 00:37:19,000
can't always take the
derivative.
482
00:37:19,000 --> 00:37:24,000
So, yes, and what else do I
want to remind you of?
483
00:37:24,000 --> 00:37:32,000
Well, they also have an
approximation formula -- --
484
00:37:32,000 --> 00:37:39,000
which says that,
you know, if we have the value
485
00:37:39,000 --> 00:37:41,000
of f at some point,
x0,
486
00:37:41,000 --> 00:37:47,000
and that we want to find the
value at a nearby point,
487
00:37:47,000 --> 00:37:51,000
x close to x0,
then our best guess is that
488
00:37:51,000 --> 00:37:58,000
it's f of x0 plus the derivative
f prime at x0 times delta x,
489
00:37:58,000 --> 00:38:02,000
or if you want, x minus x0,
OK?
490
00:38:02,000 --> 00:38:06,000
Is this kind of familiar to you?
Yeah, I mean,
491
00:38:06,000 --> 00:38:09,000
maybe with different notations.
Maybe you called that delta x
492
00:38:09,000 --> 00:38:12,000
or something.
Maybe you called that x0 plus h
493
00:38:12,000 --> 00:38:14,000
or something.
But, it's the usual
494
00:38:14,000 --> 00:38:18,000
approximation formula using the
derivative.
495
00:38:18,000 --> 00:38:21,000
If you put more terms,
then you get the dreaded Taylor
496
00:38:21,000 --> 00:38:24,000
approximation that I know you
guys don't like.
497
00:38:24,000 --> 00:38:36,000
So, the question is how do we
do the same for a function of
498
00:38:36,000 --> 00:38:41,000
two variables,
f(x, y)?
499
00:38:41,000 --> 00:38:45,000
So, the difficulty there is we
can change x,
500
00:38:45,000 --> 00:38:49,000
or we can change y,
or we can change both.
501
00:38:49,000 --> 00:38:52,000
And, presumably,
the manner in which f changes
502
00:38:52,000 --> 00:38:56,000
will be different depending on
whether we change x or y.
503
00:38:56,000 --> 00:39:00,000
So, that's why we have several
different notions of derivative.
504
00:39:00,000 --> 00:39:24,000
505
00:39:24,000 --> 00:39:37,000
So, OK, we have a notation.
OK, so this is a curly d,
506
00:39:37,000 --> 00:39:41,000
and it is not a straight d,
and it is not a delta.
507
00:39:41,000 --> 00:39:44,000
It's a d that kind of curves
backwards like that.
508
00:39:44,000 --> 00:39:50,000
And, this symbol is partial.
OK, so it's a special notation
509
00:39:50,000 --> 00:39:54,000
for partial derivatives.
And, essentially what it means
510
00:39:54,000 --> 00:39:56,000
is we are going to do a
derivative where we care about
511
00:39:56,000 --> 00:39:59,000
only one variable at a time.
That's why it's partial.
512
00:39:59,000 --> 00:40:02,000
It's missing the other
variables.
513
00:40:02,000 --> 00:40:06,000
So, a function of several
variables doesn't have the usual
514
00:40:06,000 --> 00:40:10,000
derivative.
It has only partial derivatives
515
00:40:10,000 --> 00:40:15,000
for each variable.
So, the partial derivative,
516
00:40:15,000 --> 00:40:23,000
the partial f partial x at (x0,
y0) is defined to be the limit
517
00:40:23,000 --> 00:40:29,000
when I take a small change in x,
delta x,
518
00:40:29,000 --> 00:40:43,000
of the change in f -- --
divided by delta x.
519
00:40:43,000 --> 00:40:47,000
OK, so here I'm actually not
changing y at all.
520
00:40:47,000 --> 00:40:51,000
I'm just changing x and looking
at the rate of change with
521
00:40:51,000 --> 00:40:54,000
respect to x.
And, I have the same with
522
00:40:54,000 --> 00:40:58,000
respect to y.
Partial f partial y is the
523
00:40:58,000 --> 00:41:04,000
limit, so I should say,
at a point x0 y0 is the limit
524
00:41:04,000 --> 00:41:13,000
as delta y turns to zero.
So, this time I keep x the
525
00:41:13,000 --> 00:41:21,000
same, but I change y.
OK, so that's the definition of
526
00:41:21,000 --> 00:41:26,000
a partial derivative.
And, we say that a function is
527
00:41:26,000 --> 00:41:29,000
differentiable if these things
exist.
528
00:41:29,000 --> 00:41:31,000
OK, so most of the functions
we'll see are differentiable.
529
00:41:31,000 --> 00:41:34,000
And, we'll actually learn how
to compute their partial
530
00:41:34,000 --> 00:41:38,000
derivatives without having to do
this because we'll just have the
531
00:41:38,000 --> 00:41:41,000
usual methods for computing
derivatives.
532
00:41:41,000 --> 00:41:46,000
So, in fact,
I claim you already know how to
533
00:41:46,000 --> 00:41:49,000
take partial derivatives.
So, let's see what it means
534
00:41:49,000 --> 00:41:50,000
geometrically.
535
00:41:50,000 --> 00:42:00,000
536
00:42:00,000 --> 00:42:07,000
So, geometrically,
my function can be represented
537
00:42:07,000 --> 00:42:12,000
by this graph,
and I fix some point,
538
00:42:12,000 --> 00:42:18,000
(x0, y0).
And then, I'm going to ask
539
00:42:18,000 --> 00:42:24,000
myself what happens if I change
the value of,
540
00:42:24,000 --> 00:42:30,000
well, x, keeping y constant.
So, if I keep y constant and
541
00:42:30,000 --> 00:42:33,000
change x, it means that I'm
moving forwards or backwards
542
00:42:33,000 --> 00:42:38,000
parallel to the x axis.
So, that determines for me the
543
00:42:38,000 --> 00:42:46,000
vertical plane parallel to the
x, z plane when I fix y equals
544
00:42:46,000 --> 00:42:51,000
constant.
And now, if I slice the graph
545
00:42:51,000 --> 00:42:59,000
by that, I will get some curve
that goes, it's a slice of the
546
00:42:59,000 --> 00:43:03,000
graph of f.
And now, what I want to find is
547
00:43:03,000 --> 00:43:06,000
how f depends on x if I keep y
constant.
548
00:43:06,000 --> 00:43:09,000
That means it's the rate of
change if I move along this
549
00:43:09,000 --> 00:43:11,000
curve.
So, in fact,
550
00:43:11,000 --> 00:43:17,000
if I look at the slope of this
thing.
551
00:43:17,000 --> 00:43:22,000
So, if I draw the tangent line
to this slice,
552
00:43:22,000 --> 00:43:28,000
then the slope will be partial
f of partial x.
553
00:43:28,000 --> 00:43:32,000
I think I have a better picture
of that somewhere.
554
00:43:32,000 --> 00:43:40,000
Yes, here it is.
OK, that's the same picture,
555
00:43:40,000 --> 00:43:43,000
just with different colors.
So, I take the graph.
556
00:43:43,000 --> 00:43:46,000
I slice it by a vertical plane.
I get the curve,
557
00:43:46,000 --> 00:43:50,000
and now I take the slope of
that curve, and that gives me
558
00:43:50,000 --> 00:43:54,000
the partial derivative.
And, to finish,
559
00:43:54,000 --> 00:43:59,000
let me just tell you how,
and I should say,
560
00:43:59,000 --> 00:44:02,000
partial f partial y is the same
thing but slicing now by your
561
00:44:02,000 --> 00:44:05,000
plane that goes in the y,
z directions.
562
00:44:05,000 --> 00:44:11,000
OK, so I fix x equals constant.
That means that I slice by a
563
00:44:11,000 --> 00:44:13,000
plane that's parallel to the
blackboard.
564
00:44:13,000 --> 00:44:17,000
I get a curve,
and I looked at the slope of
565
00:44:17,000 --> 00:44:20,000
that curve.
OK, so it's just a matter of
566
00:44:20,000 --> 00:44:23,000
formatting one variable,
setting it constant,
567
00:44:23,000 --> 00:44:27,000
and looking at the other one.
So, how to compute these
568
00:44:27,000 --> 00:44:29,000
things, well,
we actually,
569
00:44:29,000 --> 00:44:33,000
to find, well,
there's a piece of notation I
570
00:44:33,000 --> 00:44:38,000
haven't told you yet.
So, another notation you will
571
00:44:38,000 --> 00:44:42,000
see, I think this is what one
uses a lot in physics.
572
00:44:42,000 --> 00:44:45,000
And, this is what one uses a
lot in applied math,
573
00:44:45,000 --> 00:44:47,000
which is the same thing as
physics but with different
574
00:44:47,000 --> 00:44:50,000
notations.
OK, so it just too different
575
00:44:50,000 --> 00:44:54,000
notations: partial f partial x,
or f subscript x.
576
00:44:54,000 --> 00:45:01,000
And, they are the same thing.
Well, we just treat y as a
577
00:45:01,000 --> 00:45:10,000
constant, and x as a variable.
And, vice versa if we want to
578
00:45:10,000 --> 00:45:16,000
find partial with aspect to y.
So, let me just finish with one
579
00:45:16,000 --> 00:45:22,000
quick example.
Let's say that they give you f
580
00:45:22,000 --> 00:45:28,000
of x, y equals x^3y y^2,
then partial f partial x.
581
00:45:28,000 --> 00:45:32,000
Well, let's take the derivative.
So, here it's x^3 times a
582
00:45:32,000 --> 00:45:37,000
constant.
Derivative of x^3 is 3x^2 times
583
00:45:37,000 --> 00:45:42,000
the constant plus what's the
derivative of y^2?
584
00:45:42,000 --> 00:45:45,000
Zero, because it's a constant.
If you do, instead,
585
00:45:45,000 --> 00:45:48,000
partial f partial y,
then this is actually a
586
00:45:48,000 --> 00:45:51,000
constant times y.
The derivative of y is one.
587
00:45:51,000 --> 00:45:57,000
So, that's just x^3.
And, the derivative of y^2 is
588
00:45:57,000 --> 00:45:59,000
2y.
OK, so it's fairly easy.
589
00:45:59,000 --> 00:46:02,000
You just have to keep
remembering which one is a
590
00:46:02,000 --> 00:46:06,000
variable, and which one isn't.
OK, so more about this next
591
00:46:06,000 --> 00:46:10,000
time, and we will also learn
about maxima and minima in
592
00:46:10,000 --> 00:46:13,000
several variables.
593
00:46:13,000 --> 00:46:18,000