The following content is provided under a Creative
Commons license.
Your support will help MIT OpenCourseWare
continue to offer high quality educational resources for free.
To make a donation, or view additional materials
from hundreds of MIT courses, visit MIT OpenCourseWare
at ocw.mit.edu.
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 meaning 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
x_1 up to x_n.
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.
It's like talking about f of x versus y equals f of x.
In a manner of speaking, it's the f of x that's important.
We won't get into that right now.
What was important, though, about y equals f of 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 of 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 equal 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 x_0 minus the magnitude of delta x and x_0
plus the magnitude of delta x.
In other words, we mark off an interval of magnitude delta x,
on either side of x_0 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 of x_0 plus delta x comma y_0,
minus f of x_0 comma y_0, and divide that by delta x.
In other words, with y_0 being treated as a constant,
notice that essentially, this is the change
in f as x varies between x_0 and x_0 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 y_0.
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 were 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 of x_0, y_0, that would have meant what?
We hold x constant at some value x_0,
let y vary between y_0 and y_0 plus delta y,
and take the limit of this expression, f of x_0 comma y_0
plus delta y minus f of x_0, y_0, 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 of x sub 1 at the point x_1
equals a_1, et cetera, x_n equals a_n.
By definition, what I do is I hold a_2 up to a_n constant.
I let x_1 vary from a_1 to a_1 plus delta x.
I compute what?
f of a_1 plus delta x comma a_2, et cetera, comma a_n,
minus f of a_1 up to a_n.
Noticing that that numerator is simply the change in f as x_1
varies from a_1 to a_1 plus delta x_1
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 x_1.
That's the average rate of change.
Then I take the limit as delta x_1 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 a_n'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 x_1.
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 of x_1, x_2, x_3, where
the specific f that I have in mind is obtained as follows.
It's x_1 times x_2 times x_3 plus e to the x_1 power.
Now, to take the derivative of this with respect to x_1,
all that this says is treat x_2 and x_3
as if they were constants.
Well, look at it.
If I treat x_2 and x_3 as if they were constants,
if I differentiate this with respect to x_1, all I have left
is what?
This particular constant, which is x_2 times x_3.
And the derivative of e to the x_1, with respect to x_1,
is e to the x_1.
Therefore, f sub x_1 in this case is x_2*x_3 plus e
to the x_1.
In particular, if I evaluate f sub x sub 1 at the three-tuple
(1, 2, 3)-- namely, I replace x_1 by 1, x_2 by 2, x_3 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 x_2,
I would've treated x_1 and x_3 as constants.
With x_1 and x_3 as constants, the derivative of this with
respect to x_2 is just x_1*x_3.
Treating x_1 as a constant, the derivative of e
to the x_1 with respect to the variable x_3 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 x_1
at a 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 of 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 x_1."
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 x_1 up
to x_n, 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 x_1 up to x_n, respectively.
In other words, this means what?
The partial derivative of f with respect to x_1.
That means what?
Take the ordinary derivative as if x_1 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 to the 3x plus 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 sine of 2x minus y is cosine 2x minus 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 to the 3x plus y times the derivative of 3x plus 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, lookit, everything
that happened for calculus of a single variable
happens for calculus of several variables.
Then all of a sudden, you'll find in our 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 as 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 as 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.
Of 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 and 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 equal 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 and 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
are 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, for x equals x_0, y equals y_0,
we can think of the point (x_0, y_0) as being in the xy-plane.
w, which is f of x_0, y_0, is just the height
to this particular surface.
In other words, if I let f of x_0, y_0 be called w_0,
notice that the function f of x, y, at the point x equals x_0,
y equals y_0, graphically corresponds to the point
whose coordinates are x_0, y_0, and w_0.
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 x_0.
Notice that in three-dimensional space, x equals x_0
is the equation of this plane.
And this plane-- see, what is it the plane x equals x_0?
It's the plane that goes through the line x equals
x_0 parallel to the xy-plane.
This plane intersects my surface in a particular curve.
That curve passes through P_0.
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 y_0.
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 P_0.
That's the same P_0 that's over here,
because after all, the point on the surface that is directly
above (x_0, y_0) is P_0.
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 (x_0, y_0),
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 (x_0, y_0).
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 our 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 x_0 was chosen so that x_0 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 x_0 is.
But that's not the crucial issue right now.
The point that I'm driving at is,
if I if I vectorize the tangent line and call that v_1,
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 y_0.
It's the partial of w with respect to y evaluated at what?
x_0 comma y_0.
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 (x_0, y_0), times k.
And notice, by the way, that this is a number.
That once I take the partial derivative
and evaluate it at a point, I have a number.
So this is what?
A constant vector once x_0 and y_0 are fixed.
In a similar way, my vector v_2, which is for my curve parallel
to the wx-plane-- the slope of that curve at the point P_0
is not dw/dx, it's the partial of w with respect
to x, evaluated (x_0, y_0).
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 y_0.
At any rate, in an analogous manner,
v_2 turns out to be what vector?
It's i plus the partial of w with respect
to x evaluated (x_0, y_0) 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
the cross product here, is that I can now
take the cross product of v_1 and v_2.
Sparing you the details, remember what I do here.
I just write down i, j, k. v_1 has these components,
v_2 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 (x_0, y_0).
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 (x_0, y_0, w_0), 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 minus w_0 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 ore 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.
Help OCW continue to provide free and open access to MIT
courses by making a donation at ocw.mit.edu/donate.