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PROFESSOR: Hi.
Today, we would like to begin our study of calculus of
several real variables.
And as I mentioned in our last lecture, I would like to begin
it from the traditional point of view.
And I guess I'd like to make a little bit of a short speech
even before we begin.
There may be a little bit of danger in the way I've been
emphasizing the modern approach to mathematics: that
one may actually believe that the traditional approach to
mathematics left very, very much to be desired.
The answer is, it didn't really.
That the role of logic in good, traditional mathematics
was about the same as it is in good, modern mathematics.
That in fact, if we define new mathematics as meaningful
mathematics, there really was no difference between the two
approaches.
Except for the fact that after you have a couple of hundred
years of hindsight and can take a subject apart from a
different point of view, then obviously it's no great
surprise that you can find cleaner ways of really
presenting the topic.
It's the old cliche of hindsight being better than
foresight by a darn sight, or some such thing.
But at any rate, without further ado, let's actually
look at the introduction to calculus of several real
variables from a traditional point of view.
And I've abbreviated that title to simply say, "Calculus
of Several Variables." Whereas I say that,
think of two things.
First of all, it's an introduction.
And second of all, we're going to be doing this from the
traditional point of view.
Now, what we're assuming is remember, last time we talked
about functions of several real variables.
We talked about the limit concept, gave you exercises
that drilled you on what limits meant, and the like.
And as a result, it now makes sense to talk about the
calculus, the instantaneous rate of change.
Except that you have an awful lot of variables in here.
In other words, the general case that we'd like to tackle
is the case where we have w being some real number, being
a function of the n independent
variables x1 up to xn.
And of course, you see, the calculus applies to the f.
The w used here is to indicate the fact that the output is a
real number-- right?
It's like talking about f(x) of x versus y = f(x).
That in a manner of speaking, it's
the f(x) that's important.
We won't get into that right now.
What was important, though, about y = f(x)?
It indicated a graph--
graph.
And because it was a graph, we could visualize things
pictorially that might have been harder to understand
analytically.
For this reason, even though n may be any positive integer
whatsoever--
preferably, of course, greater than 1, otherwise we wouldn't
have a function of several variables.
In fact, that's the definition of a function of several
variables--
it's an expression of this type where n is
greater than 1.
But in general, and you'll notice this in the text as
well, we begin with the special case that n equals 2.
Because, you see, if n is 2 we need-- what?
Two degrees of freedom to plot the independent variables.
That would be the xy plane, for example.
A third degree of freedom--
say, the z-axis, often called w-axis
because of the symbolism.
In other words, in the same way that I could graph a real
valued function of a single real variable as a curve in
the plane, I can graph a real valued function of two real
variables as a surface in three dimensional space.
And as a result, when one is adjusting to new language,
it's nice to take the case n equals 2 simply because you
can view things pictorially.
The danger, of course, with n equals 2, is you get so used
to the picture that after a while you forget that n could
have been more than 2.
So I will try to pay homage to both points of view.
I will simply mention here that we will often let n equal
2 to take advantage of geometry.
When we do this, the usual notation is to let w equal
f(x,y) where x and y are independent.
And I should mention why it's so important to talk about
independent variables.
It'll become clear in a moment.
Let me just tell you what independent means, and then
I'll show you why it becomes clear in a moment.
To say that x and y are independent simply means that
you can choose one of the two variables without
determining the other.
For example, if I were to say let y = 2x, then certainly y
and x are not independent, because the choice of x
determines the choice of y.
But to say that x and y are independent--
and that's what we mean by two degrees of freedom--
is I can take any value of x that I want and not be impeded
as to the choice of y that I want.
And that's why we say the domain is the xy plane.
I have two degrees of freedom.
I can roam anywhere through the plane this way.
But the reason that we want this is as follows.
And this is where the traditional mathematics was
every bit as logical as the modern mathematics.
Let's keep in mind that logic is the art of taking known
situations that we know how to handle, you see, and reducing
unfamiliar situations to these more familiar situations.
The point is that even in traditional mathematics, one
knew how to handle the calculus of
a single real variable.
So what one said was this, given that w is a function of
both x and y, let's hold y fixed, for
example, and let x vary.
You see, if I fix y at some particular value and let x
vary, once I've made that fixed choice of y, I have w as
a function of x alone.
And as long as w is a function of x alone, I already know how
to take the derivative of w with respect to x.
And by the way, there's no favoritism here
between x and y.
In a similar way, if I hold x constant and let y vary, then
w is a function of y alone.
And I can then find the derivative of w
with respect to y.
And these things were called partial derivatives, you see,
because it involved holding-- what?
All but one of the variables constant.
We'll go into that in more detail as we go along.
But the whole idea is-- what?
By choosing the variables to be independent, we can hold
all of them but one at a time constant.
And as long as we're doing that, the resulting function
is a function of a single real variable.
And this reduces us to part one of our course.
So at any rate, what we're saying is, fix y at some
constant value-- say, y equals y sub 0.
And we then let x vary as in ordinary one dimensional
calculus between x0 minus the magnitude of delta x and x0
plus the magnitude of delta x.
In other words, we mark off an interval of magnitude, delta
x, on either side of x0 and let x be any place in here.
Then what we do, is remembering that y0 is a
constant, and that this function, no matter how
complicated it looks like, is a function of
just delta x alone.
We mimic the definition of an ordinary derivative.
In other words, we take the limit as delta x approaches 0
of f (x0 plus delta x, y0) minus f (x0, y0), and divide
that by delta x.
In other words, with y0 being treated as a constant, notice
that essentially, this is the change in f as x varies
between x0 and x0 plus delta x, divided by the change in x.
In other words, it's the limit of an average rate of change
of f of x with respect to x for a fixed value of y0.
And to indicate that, we write this analogously, I would say,
to how we write functions of a single real variable.
In other words, we write f prime.
Instead of f prime, we indicate--
let me put it this way, when you had only one real
variable, was there any danger of misinterpreting the prime?
After all, with only one independent variable, when you
said derivative, it was obvious what variable you with
differentiating with respect to.
Here, there are two variables, x and y.
So instead of the prime as a subscript, we write down the
variable with respect to which we're taking the derivative.
And then we put in the point at which the derivative is
being evaluated.
You see, in a similar way, to have defined f sub y (x0, y0),
that would have meant-- what?
We hold x constant at some value x0, let y vary between
y0 and y0 zero plus delta y, and take the limit of this
expression, f (x0, y0 plus delta y) minus f (x0, y0),
over delta y.
Take the limit of that expression as delta y
approaches 0, you see.
And we're going to show this pictorially in a while.
After all, that's why we chose the case n equals 2, was so
that we could eventually illustrate this thing
pictorially.
But for the time being, I prefer not to introduce the
picture because I want you to see that even in the case
where my input consists of an n-tuple where n is greater
than 2, or 3, or 4--
whatever you want--
that this thing still makes sense.
In other words, in general, suppose w is a function of the
n independent variables x sub 1 up to x sub n.
And I want to compute f (x sub 1) at the point x1 equals a1--
et cetera-- xn equals an.
By definition, what I do is I hold a2 up to an constant.
I let x1 vary from a1 to a1 plus delta x.
I compute-- what?
f of a1 plus delta x, a2--
et cetera-- comma an, minus f of a1 up to an.
Noticing that that numerator is simply the change in f as
x1 varies from a1 to a1 plus delta x1 while all the other
variables are being held constant.
Which of course I can do if the variables are independent.
I then divide that by delta x1.
That's the average rate of change.
Then I take the limit as delta x1 approaches 0.
And this I can do for any number of variables, provided
that they're independent.
Keep that in mind, if x sub n could be expressed in terms of
the remaining x's, how can I hold a sub n constant?
Or how can I vary a sub n while I hold
the other ones constant?
In other words, if the an's--
if one of variables depends on the others, they are related
in terms of their motion.
Keep in mind here, I just chose delta x1.
Analogous definition is held for f sub x sub 2, f sub x sub
3, et cetera.
OK?
I think again that once you see a few examples, the
mystery vanishes.
What this thing says in plain English is that when you have
a function of several variables and you want to take
the derivative with respect to one of those variables,
pretend that every one of the other variables
was being held constant.
Let me give an example.
Let w be f (x1, x2, x3), where the specific f that I have in
mind is obtained as follows.
It's x1 times x2 times x3 plus e to the x1 power.
Now, to take the derivative of this with respect to x1, all
that this says is treat x2 and x3 as if they were constants.
Well, look at it.
If I treat x2 and x3 as if they were constants--
if I differentiate this with respect to x1--
all I have left is-- what?
This particular constant, which is x2 times x3.
And the derivative of e to the x1, with respect to
x1, is e to the x1.
Therefore, f sub x1 in this case is x2 x3,
plus e to the x1.
In particular, if I evaluate f sub x sub 1 at the three-tuple
(1, 2, 3)--
namely, I replace x1 by 1, x2 by 2, x3 by 3--
I obtain-- what?
2 times 3 plus e to the first power.
In other words, this is just a number, 6 plus e.
On the other hand, if I had decided to differentiate this
with respect to x2, I would've treated
x1 and x3 as constants.
With x1 and x3 as constants, the derivative of this with
respect to x2 is just x1 x3.
Treating x1 as a constant, the derivative e to the x1 with
respect to the variable x3, is 0.
Because the derivative of any constant is 0.
Notice that this is not really a constant.
It's a constant once I've fixed x1 of the particular
value, which is what this particular definition says.
In particular, f sub x sub 2 of (1, 2, 3) is
1 times 3, or 3.
I should also mention that sometimes instead of writing f
(x sub 1), we write something that looks like the regular
derivative, only we make a funny kind of script d instead
of an ordinary d.
This is read the partial derivative of w
with respect to x1.
And the relationship between these two notations is rather
similar to the relationship that exists between the
notations dy/dx and f prime of x.
At any rate, if we have the n independent variables x1 up to
xn, and f is a function of those
variables, f sub x sub 1--
et cetera-- f sub x sub n are called the partial derivatives
of f with respect to x1 up to xn, respectively.
In other words, this means-- what?
The partial derivative of f with respect to x1.
That means-- what?
Take the ordinary derivative as if x1
were the only variable.
This is the partial of f with respect to x sub n.
That means take the ordinary derivative of f as if x sub n
were the only variable.
And you can always do this, provided that your variables
are independent.
Because our definitions so closely parallel the structure
for what happened in the case of one independent variable--
including the limit theorems, the distance formulas, and
what have you--
it turns out that the usual derivative
properties still hold.
For example, if w happens to be the function of the two
independent variables e to the 3x plus y times sine 2x minus
y, and I want to take the derivative of w
with respect to x--
meaning differentiate this treating y as a constant--
notice the treating y as a constant gives me two
functions of--
what?
x.
In other words, my function is a product of two functions,
each of which depends on x.
So I use the ordinary product rule to differentiate this.
Namely, how do I use the product rule to take the
partial of w with respect to x?
I treat y as a constant and I differentiate as if x is the
only variable.
I say-- what?
It's the first factor e ^ (3x + y) times the derivative of
the second with respect to x, treating y as a constant.
If I treat y as a constant, the derivative of sin (2x - y)
is cos (2x - y) times the derivative of what's inside
with respect to x.
And that's just 2 plus--
what?
The second term times the derivative of the first.
And the derivative of e to the 3x plus y with respect to x,
treating y as a constant, is e ^ (3x + y) times the
derivative of 3x + y, with respect to x, which is just 3.
And so I have the partial of w with respect to x.
In a similar way, I could have found the partial of w with
respect to y, et cetera.
I should now introduce one little problem that will
bother you, maybe even after I help you with this.
In fact, I guess what really crushes me is the student who
comes up in a live class-- after class-- and says, I
understood it until I heard you lecture on it.
I hope this doesn't cause that problem.
But the danger is--
I tell you look it--
everything that happened for calculus of a single variable
happens for calculus of several variables.
Then all of a sudden, you'll find an odd textbook.
In every textbook, just about, they'll say things like, if
you take the partial of the variable u with respect to x,
you do not get the reciprocal of the partial of x with
respect to u.
In other words, remember, for one variable, dy/dx, right?,
was the reciprocal of dx/dy.
They say it's not true in several variables.
I have written in an accentuated question mark here
because I want to explain something very
important to you.
And by the way, if you understand this, you're 90% of
the way home free as far as understanding how calculus of
several variables is used in most important applications.
Let me take a very simple case.
Let u equal x plus y, and let v equal x minus y, Now
obviously, if u equals x plus y, the partial of u with
respect to x, If we hold y constant-- we treat x as the
only variable--
is 1.
Right?
The partial of u with respect to x is 1.
On the other hand, notice that if I add these two equations,
the y term drops out.
I get-- what?
2x equals u plus v, divide through by 2.
That says x is equal to 1/2 u plus 1/2 v.
Let me now take the partial of x with respect to u, holding v
constant, treating u as the only variable.
It's easy for me to see that the partial of x with respect
to u is 1/2.
And certainly, if I now compare these two, it should
be clear that this is not the reciprocal of
this, or vice versa.
The point that I wanted to mention,
though, was the following.
That when here--
we said, take the partial of u with respect to x--
we were assuming that the independent
variables were x and y.
And when we said, here, take the partial of x with respect
to u, what did we hold constant?
We held v constant.
In other words, I'm not going to keep this
habit up very long.
But just for the time being, I would like you to get used to
the idea of putting in a subscript the variables that
are being held constant.
You see, in the textbook when they say this is the case,
what they really mean is, if you differentiate u with
respect to x, holding y constant, that in general will
not be the reciprocal of the derivative of x with respect
to u, holding v constant.
So I can cross out the question mark now.
But the point is, notice that the variables that you're
holding constant here are different.
You see, suppose instead I took the partial of x with
respect to u, holding y constant.
In other words, suppose I solve for x in
terms of u and y.
That happens to be very easy to do here.
Given that u equals x plus y, it follows immediately that x
is equal to u minus y.
Now, if I take the partial of x with respect to u, treating
y is a constant--
so let me do that--
what do I get for an answer?
I get 1 for an answer.
In other words, the reciprocal of the partial of x with
respect to u, treating y as a constant--
as the other independent variable--
is equal to the partial of u with respect to x when it's
also y that's being considered as the other variable.
When the variables outside here match
up, the recipes work.
Course, that leads to the question, then why do the
textbooks tell you that this isn't true?
Why do they pick this particular representation
rather than this one?
And I have a combination example--
an explanation here.
And it's simply this.
Let's suppose I was given that w equals e to the x plus y
times cosine x minus y.
It seems that since the only way the variable appears in
the first factor is as x plus y, and the only way it appears
in the second factor is in the form x minus y, a very natural
substitution might be to say that u equals x plus y and v
equal x minus y.
In other words, I could either visualize w as being a
function of x and y--
in which case, it would be e to the x plus y times
cosine x minus y--
or, I could visualize w as being a simpler function of
the two new variables u and v, where w is then just-- what? e
to the u times cosine v.
Notice again, that this formula here is much simpler
than this one.
This one says-- what?
Take e to of the sum of these two, multiplied by the cosine
of the difference of these two.
And this one just says-- what?
Take e to the first times the cosine of the second.
But the point is, that you might make this change of
variables to simplify your computations.
The point is, that either we would treat w as a function of
x and y, or we would have treated w as a function of u
would v.
And by the way, notice here my use of f and g.
Notice that w is a different function of x and y than it is
of u and v. But this notation says-- what?
We can consider w either in terms of x and y, or in terms
of u and v. Hardly ever would we consider w as a
function of u and y.
In other words, either we would change the
variables or we don't.
Or, for example, in terms of polar versus Cartesian
coordinates, either we use x and y or we use r and theta.
We don't usually use r and x.
In other words-- going back to the textbook example again--
If we look at this particular situation here, all the book
is saying is, when you're differentiating with respect
to n, you usually mean that y is the
variable being held constant.
When we differentiate with respect to u, we usually means
that v is the variable that's being held constant.
Again, more of this is said in our exercises.
But for the time being, I want you to see how important it is
when you have several variables to keep track of
which are the dependent variables, which of the
independent variables, and how they're coupled.
I said earlier that one of the nice things about picking n
equals 2 is that you can draw a nice
picture of the situation.
And what I thought you might like to see is the following.
Let's suppose you have that w is some function of x and y.
What that means is, we can locate-- for a given value of
x and a given value of y.
In other words, the for x equals x0, y equals y0, we can
think of the point (x0, y0) as being in the xy plane.
w, which is f (x0, y0), is just the height to this
particular surface.
In other words, if I let f (x0, y0) be called w0 , notice
that the function f (x,y) at the point x equals x0, y
equals y0, graphically corresponds to the point whose
coordinates are x0, y0, and w0.
OK?
This is now a surface.
And in the same way that we use tangent lines to replace
curves when we had one independent variable, in two
independent variables, we use tangent
planes to replace surfaces.
The question that comes up is, how do you
get a tangent plane?
And what does this have to do with the partial derivatives?
And what I would like show you is the following.
First of all, one very natural way of intersecting this
surface with a plane-- in other words, when you take the
partial of w with respect to y--
you're holding x constant, you're saying, let x equal x0.
Notice that in three dimensional space, x equals x0
is the equation of this plane.
And this plane--
see, what is it the plane x equals x0?
It's the plane that goes through the line x equals x0
parallel to the wy plane.
This plane intersects my surface in a particular curve
that passes through p0.
I can talk about the slope of the curve at
that particular point.
OK?
In a similar way, I could've sliced the surface by the
plane y equals y0.
In other words, hold y constant.
Well, you see, to hold y constant, that means I, again,
draw my plane .
It intersects the surface in a different curve, but that
curve must also go through the point p0.
That's the same p0 that's over here, because after all, the
point on the surface that is directly above x0, y0, is p0.
And that point doesn't change, no matter what plane you slice
this thing with.
OK?
And so again, I can talk about the slope of the tangent to
this particular curve of intersection.
By the way, one very brief aside before I
continue on with this.
One of the things that makes functions of several variables
so difficult is--
I would like you to observe, because this will become the
backbone of our future investigations--
that these two particular planes that I drew to
intersect my surface were very special planes.
One of them was parallel--
they both went through the point x0, y0.
But one happened to be parallel to the wy plane and
the other one happened to be parallel to the wx plane.
Notice that, in general, I could have passed infinitely
many different planes through x0, y0.
Each of which would have intersected the surface in a
different curve.
And this is what leads later to the more generalized
concept of directional derivatives that we will talk
about in more detail in future lectures.
But I simply mention the word now because in the reading
material in this assignment, some mention is made of
directional derivatives.
In other words, I wanted to point out that to take partial
derivatives, we are interested in two special directions,
even though there are other directions.
The question that comes up, of course, is where do the
partials come in here?
And secondly, knowing where partials come up in here, how
do we find the equation of a tangent plane?
And all I would like you to see from here is that I'm
going to use the same old technique as always.
Given these two tangent lines, I'm going to vectorize them.
I am then going to find the equation of a plane that
passes through those two vectors.
To do that, I'm going to have to find the normal to that
plane, et cetera.
And I'm going to wind up with the equation
of a tangent plane.
In more slow motion, all I'm saying is if I now take this
picture and draw it over here--
notice the representation--
it looks as if the curve that I get is in the wy plane.
It's actually in a plane parallel to the wy plane
characterized by x equals x sub 0.
And by the way, notice even though x equals x0 was chosen
so that x0 was fixed, I hope it's clear to you that if I
let x sub 0 vary and I picked different values of x sub 0,
the curve that I get here will, in general, differ.
You see, what I'm saying is if I take slices parallel to the
wy plane for a particular surface, the slices that I
get-- the shape--
will depend on what the particular value of x0 is.
But that's not the crucial issue right now.
The point that I'm driving at is, that if I if I vectorize
the tangent line and call that v1, what does that tangent
line look like?
First of all, let's see what its slope is.
Now again, if you didn't see this over here and all I
showed was this diagram, you would say, hey, that slope is
just the derivative of w with respect to y.
But we write that the partial of w with respect to y in
deference to the fact that w is not a function of y alone--
that w depends on both x and y, and the reason that we got
this curve was that we fixed x at some particular value.
So this is really-- what?
Not the derivative of w with respect to y
evaluated at y equals y0.
It's the partial of w with respect to y evaluated at--
what?
(x0, y0).
All right, at any rate, now that we have the slope, notice
that the slope of a vector can always be
viewed as the k component.
In this case, the k component divided by the j component.
If I make the j component 1, the slope will
just be the k component.
In other words, a vector which is tangent to the curve-- and
one that's easy to write down--
is j plus the partial of w with respect to y, evaluated
at (x0, y0), times k.
And notice, by the way, that this is a number.
That once I take the partial derivative and evaluate it at
the point, I have a number.
So this is-- what?
A constant vector once x0 and y0 are fixed.
In a similar way, my vector v2, which is for my curve
parallel to wx plane--
the slope of that curve at the point p0 is not dw dx--
it's the partial of w with respect to x,
evaluated x0, y0.
Again, why?
Because even though this looks like w is a function of x
alone, here, it was only that way because we took the second
variable and, in a sense, froze it at the value y0.
At any rate, in an analogous manner, v2 turns out to be
what vector?
It's i plus the partial of w with respect to x evaluated
(x0, y0), times k.
In other words, the slope is the partial of w
with respect to x.
And it's in the ik plane.
In other words, the xw plane.
Now that I have my two vectors, to find the plane
that passes through them, all I need is-- what?
The normal vector and a point in the plane.
Well, one of the nice things about having studied
Cartesian--
the cross product here--
is that I can now take the cross product of v1 and v2.
Sparing you the details, remember what I do here.
I just write down i, j, k. v1 has these components, v2 has
these components.
I now expand this determinant in the usual way.
And I get-- what?
This particular vector.
In other words, the i component is the partial of w
with respect to x.
The j component is a partial of w with respect to y.
Both evaluated at the point (x0, y0).
And the k component is minus 1.
Now, to find the equation of a tangent plane, all
I have to do is--
what?
Take the standard form of the plane.
In other words, the coefficients will be these.
And then I take (x0, y0, w0), which is a point in my plane.
And then the equation of the tangent plane becomes, quite
simply, this expression here.
And in fact, if I rewrite that, notice I can transpose
the (w - w0) term.
That is the change in w.
But to remind us that we're in the tangent plane, we don't
write this as delta w, we write it as delta w sub tan.
And notice that the change in w to the tangent plane is just
the partial of w with respect to x times delta x, plus the
partial of w with respect to y times delta y.
And hopefully, this starts to seem familiar to you.
This should almost look like the equation of delta y tan,
only in two dimensions.
Remember when we had delta y tan in part one of the course?
See, what this really says is-- what?
This is the change in w with respect to x, multiplied by
the total change in x.
In other words, this term is-- what?
It's the change in w due to the change in x, alone.
This term is the change in w, with respect to y, times the
total change in y.
So this term tells you the change in w due to the change
in y, alone.
And if you add these two up, since x and y are independent,
this should give you the total change in w.
The reason, of course, that you don't get the total change
in w, but rather, delta w tan, is the fact that if these
expressions here are not constant numbers as delta x
and delta y of varying, these numbers, here, are varying.
These are variables.
But you see, once you fix these at these values, these
become numbers, and now you know what delta w tan is.
Now, in some of the homework problems in the text, what
we're going to do is just practice the geometry itself.
We're going to find equations of tangent planes.
The thing that's very important to us in future
lectures from a structural point of view, is that the
equation of the tangent plane-- in other words, the
equation for delta w tan--
is far more simple to handle than the equation for
the true delta w.
You see, the equation for delta w tan just has delta x
and delta y appearing to the first power.
Whereas, if you try to find delta w exactly for an
arbitrary function of x and y, this can become a very, very
messy thing.
In short, delta w tan will play for two and more
independent variables the same role that delta y sub tan
played for us in part one of this course as a differential.
And what we'll be discussing next, or at least trying to
get at the root of, is just what does play the role of a
differential when you're dealing with calculus of
several variables?
And what we're going to find is that, again, the
computations become messy enough so that even though the
structure stays fairly nice, there are some wrinkles that
come up that will take us a considerable
amount of time to untangle.
But we'll worry about that when the time comes.
And so until that time, goodbye.
Funding for the publication of this video was provided by the
Gabriella and Paul Rosenbaum Foundation.
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