Multivariable Calculus Pdf Notes 171112

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MULTIVARIABLE CALCULUS NOTES
JAMESMCIVOR
0. Introduction
ThesearemynotesforthecourseMath53: MultivariableCalculus,atUCBerkeley,inthesummer
of 2011. This document is a sketch of what occurs in lecture. In the ﬁrst third of the course (i.e.,
until the ﬁrst midterm), my presentation will diﬀer somewhat from that of Stewart. The material
is the same, but the order of topics is diﬀerent. As a result, I ask you in the beginning few weeks
to read these notes especially carefully. Later on, as I begin to follow Stewart more closely, these
notes will be more brief. Please advise me of typos or other mistakes. I will update this document
throughout the course.
1. Overview
Today I won’t cover anything in great detail, but merely give you a feel for what we will do in
this course. Yet in many ways this lecture is the most important, since one of the hardest things
about a course like this is keeping all the ideas and techniques organized in your mind.
So, the broad outline of the course is:
(1) Geometry: Curves and Surfaces in Two and Three Dimensions
(2) Calculus of Scalar Functions
(3) Calculus of Vector Functions
Here’s a rough idea of what the terms in the above mean:
(1) Curve: Something like a bent line. Such things can be described with one free variable.
(2) Surface: Something like a bent plane. Can be described by two free variables.
(3) Dimension: the number of variables it takes to describe all the points on an object. Lines
and other curves are one-dimensional, planes and other surfaces are two-dimensional.
(4) Scalar: A quantity that can be described by a single number. Physical examples: Temper-
ature, speed.
(5) Vector: A quantity which has both magnitude and direction. Physical examples: Velocity,
force.
- Deﬁne R2 and R3. Right-hand convention.
So far, you have probably primarily studied functions of one variable, usually written f(x). Here
f is the name of the function, and x is the name of the independent variable. We usually have to
say at some point what values x may take. For instance, x may be allowed to be any real number,
or just a number between 0 and 1, etc. The possible values for x form what is called the domain
of f. The important thing to notice about your previous calculus experience is that the domain was
always either R or a subset of R, such as [0,1].
Can we have a function whose domain is R2 instead of R? Certainly, and since the points of R2
consist of two real numbers, the input to such a function will be two real numbers, usually written
f(x,y). For example, the function f(x,y) = x2 + y2 is a function whose domain is all of R2. The
function g(x,y) = 1 is a function which takes two inputs x and y, but its domain is not all of R2,
xy
since it is not deﬁned when either x or y is zero. This expression really deﬁnes a function whose
2
domain is all of R except the x- and y-axes. In similar fashion we can describe functions whose
domain is R3, for instance h(x,y,z) = xyz2 +ysin(xz).
Thus we have extended the domain of functions from R to R2 and R3. That is, we have given
ourselves greater ﬂexibility with the “inputs” of our functions. But what about the outputs? All
the functions above output just a number, or scalar. We would like functions which output points
in R2 or R3. We will see that points of R2 or R3 are the same thing as vectors, and so we will
call a function whose outputs are in R2 or R3 a vector function, or vector-valued function, or
sometimes a vector ﬁeld. Often we use capital letters to name vector ﬁelds, or sometimes bold-face
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type. For example, F(x) = (2x,x ) is a vector ﬁeld in R whose domain is R. The vector ﬁeld
H(x,y,z) = (x2y,sinyz,ey) is a vector ﬁeld in R3 whose domain is also R3.
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2 JAMESMCIVOR
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So our goals will be to ﬁrst understand what various objects in R and R look like, and then
to study scalar functions on those objects, and ﬁnally vector functions. Here’s the most important
thing to remember about this class: if you can keep the diﬀerent functions straight, the calculus
part will be easy; but if you get the functions muddled, and get mixed up between scalar and vector
functions, then all the calculus we do can seem much more complicated than it needs to be.
Once you are comfortable with the idea of functions having many variables, you can ask how to
diﬀerentiate them. Once you get used to it, this is very easy. Since you have more than one variable,
you have more than one derivative: you can diﬀerentiate with respect to each diﬀerent variable. Say
we have a function f(x,y) of two variables. To diﬀerentiate with respect to x we pretend that y is
a constant (sometimes I just replace y by 17 or some other obscure number to prevent confusion).
Then take a normal derivative with respect to x. This is called the partial derivative of f with
respect to x, and denoted ∂f.
∂x
2 y ∂f y ∂f 2 y
Example 1.1. If f(x,y) = x y +x+xe , then ∂x = 2xy +1+e and ∂y = x +xe .
The concept of integration is slightly more subtle. Let’s say we’re taking a deﬁnite integral of
a single-variable real-valued function. Then the information we need is not just a function, but
an interval, or region, over which to integrate it. When we allow functions with more than one
variable, we must also allow diﬀerent regions to integrate over. The reason some students ﬁnd the
later parts of this course confusing is that there seem to be many types of integrals. But there are
really only three types: line integrals, surface integrals, and volume integrals. The diﬀerent types
just depend on the dimension of the region over which we integrate. Line integrals are integrals over
a one-dimensional region, such as the x-axis, or some curve. Surface integrals are integrals over a
two-dimensional region, maybe the xy-plane, or a rectangle in the plane, or even a strange curved
surface, like part of an ellipse, for instance. Both types of integrals involve scalar functions - in fact,
it only makes sense to integrate scalar functions. Whenever you see an integral involving a vector
ﬁeld (and you will see plenty of these at the end of the course), notice that we always do something
to reduce the vector ﬁeld to a scalar before performing the integration. Usually this involves vector
operations like the dot product.
SUMMARY:
- We’ll study functions of several variables.
- These are functions whose domains are curves or surfaces in R2 and R3
- If the output is just a number, such a function is called a scalar function.
- If the output is a point in R2 or R3 (i.e., a vector), such a function is called a vector function.
- Partial derivatives are taken by treating all the other variables as constants.
- Integrals only make sense for scalar functions, but there are ways of turning vector functions into
scalar functions, and then integrating these.
- There are diﬀerent types of integrals for scalar functions (line, surface, or volume integals), de-
pending on the dimension of the region you want to integrate over.
2. “Cutting Out” Curves and Surfaces by Equations
2.1. The Idea of Cutting Out. There are two fundamentally diﬀerent ways to describe a curve
or a surface. One is by giving equations, and this will be our focus today. Tomorrow we will discuss
the other method, using parameters. I like to call the ﬁrst method “cutting out” curves and surfaces,
for reasons that will become clear shortly. Let’s work in R2 for the moment. If I go through and
2
pick out all the points (x,y), with no equations, then what I get is all of R . But if I instead go
through and pick out only those points (x,y) which satisfy the equation y −x = 0, then what I get
is a line, namely the line y = x. So by picking out only the points satisfying a certain equation,
2 2
I have focused my attention on a smaller part of R . While R itself is two-dimensional (it’s a
plane), the line I picked out by my equation y − x = 0 is a one-dimensional object. We will see
that as a general rule, each equation you impose cuts down the dimension by one. What if I instead
use the equation y + x = 0? I just get a diﬀerent line, this one with slope -1 instead of 1. But
still with this one new equation I have gone from the two-dimensional R2 to a one-dimensional object.
Nowsuppose I play a new game. Starting with R2 as above, I now go through and pick out only
those points (x,y) which satisfy both equations y − x = 0 and y + x = 0. What sort of object have
I picked out now? To ﬁnd out, we simply solve both equations simultaneously. The ﬁrst equation
forces y = x, while the second equation forces y = −x. The only choices of x and y that work are
2
(0,0), often called simply the origin. Thus by starting with all of R and imposing two equations,
MULTIVARIABLE CALCULUS NOTES 3
we are reduced to just one point. A point is regarded as a zero-dimensional object, which is consis-
tent with the trend that each equation reduces the dimension by one (please remember that this is
not always so - we’ll see a counterexample shortly).
2.2. Intersections. Now let’s work in R3. Points of R3 are usually denoted (x,y,z), where x, y,
and z can be any real numbers. As before, imposing no equations at all gives us all of R3. If we
impose one equation, say x+y+z = 0, we get a plane (plot some points and see for yourself). This
is a two dimensional object, sitting inside the three-dimensional space R3. What if now we impose
two equations: the original one x+y+z = 0, and a new one, say x2+z2 = 1 (this equation doesn’t
involve y at all, but that doesn’t matter). The ﬁrst equation by itself, as we saw, cuts out a plane
in R3. The second equation, as we will see in a few lectures, describes a cylinder directed along the
3
y-axis. Imposing both equations at once turns out to give us an ellipse in R . It can be visualized
by imagining that we “sliced” the cylinder with the plane, giving a cross-section which is the ellipse.
Once again, imposing two equations reduces the original dimension (three) by two, since an ellipse
is a one-dimensional object.
So we have seen that you can start with R2 or R3, and “cut out” a nice curve or surface by giving
one or more equations: here “cutting” means throwing away all those points that don’t satisfy your
equations. This is like a sculptor, who starts with a huge block of marble (R3), and chisels away the
parts of the stone that he/she does NOT want in the ﬁnal sculpture. An equation is like a chisel,
and what it removes is just the points that are NOT satisﬁed by the equation. Count yourselves
lucky you only have to wield an equation and not a chisel!
The other thing we have seen is that imposing more than one equation results in an object which
is the intersection of the objects deﬁned by each equation. Every equation by itself corresponds to
some shape, and when you impose the equations simultaneaously you just select those points that
lie in all of those shapes at once. This makes sense in terms of the chisel analogy - more equations
means more chisels, which means more cutting, resulting in a smaller object, and intersecting things
gives smaller objects.
2.3. Possible Sources of Confusion. One thing you need to be careful of is that sometimes an
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equation can be ambiguous, and you need to take care whether you are working in R or R . This is
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related to situations such as the above (the surface x +z = 1), where an equation does not involve
one of the variables. Here’s an example demonstrating this ambiguity:
Example 2.1. What shape would you say the equation x2 +y2 = 1 deﬁnes?
Most of you would probably say a circle of radius 1. That’s correct if we are working in R2,
but false in R3. Why false in R3? For one thing, we said that usually each equation reduces the
dimension by one. Well, a circle is a type of curve, hence has dimension one, so we expect to need
3 3
two equations, not one, in order to cut out a circle in R . When working in R , there is a variable
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z which does not appear in the equation x + y = 1. This is an imporant point: when a variable
does not appear, all values of that variable are allowed. So what x2 + y2 = 1 cuts out is a circle
in the xy-plane, which is extended along the z direction; in other words, a cylinder of radius one in
the z-direction. For another example of this ambiguity, the equation x+y = 2 cuts out a line in R2,
but a plane in R3. As before, this plane is easy to describe: you just draw the original line in the
xy-plane, and extend it inﬁnitely in the z-direction.
Another possible source of confusion is that sometimes equations may cut out something that
is really two or more curves, even though we’ll still call it a curve. For instance, what does the
2
equation xy = 0 cut out in R ? The equation has solutions whenever either x or y is zero. This
happens along the y- and x-axes, respectively. So the “curve” cut out by xy = 0 is really a union
of two lines. This is not really a problem, just slightly confusing terminology: “curve” for what you
might normally call “two curves” or “two lines”. Most of the equations we’ll run into will just cut
out one curve, but don’t be alarmed if it looks like two.
2.4. When a new equation does not reduce the dimension of the ﬁgure. There remains
one loose end to tie up: we have used heavily the “dimension-drop” phenomenon, whereby each
extra equation reduces the dimension of the object under consideration. I mentioned that this does
not always hold, and I must now show you why, so you know when to use it and when not. The
4 JAMESMCIVOR
only time when the dimension drop does not work is when the extra equation imposed is redundant.
For example, the equation x = 0 cuts out a line in R2, namely the y-axis. But so does the equation
5x = 0, since multiplying both sides by 5 doesn’t change the solution set to the equation. So if we
ﬁrst impose the equation x = 0, and then impose a second equation 5x = 0, the dimension does not
drop - in fact, nothing at all changes. Imposing one equation is the same thing as imposing both.
The second equation was redundant. Here is another example:
Example 2.2. Analyze the ﬁgure cut out by x2 +y2 +z2 = 25, y = 3, and x2 +z2 = 16
If we just obey the dimension-drop rule, we expect something zero-dimensional. Let’s see that
this is not so: ﬁrst impose x2 + y2 + z2 = 25 and y = 3, then impose x2 + z2 = 16. Geometrically,
the ﬁrst equation describes a sphere of radius 5 in R3, and the equation y = 3 describes a plane.
To see what they cut out together, we intersect the sphere and the plane, giving us a circle which
lives in the plane y = 3. If you do a little algebra, you can see that the radius of this circle is 4.
Thus the third equation tells us nothing new. We started with R3 (3-dimensional), cut out a sphere,
which is a type of surface (2-dimensional), then cut out a plane and intersected it with our sphere,
giving a circle (1-dimensional). But when we imposed the third equation, we did not drop down
to a 0-dimensional object, which is just a set of points, but stayed where we were, since the third
equation followed from the ﬁrst two. Notice, however, that what’s cut out by the third equation
isn’t the same thing as that cut out by the ﬁrst two, since the third by itself cuts out a cylinder,
whereas the ﬁrst two determine just a circle.
2.5. SUMMARY. - Equations cut out geometric objects in R2 and R3.
- The set of points satisfying the equation(s) are the points on the geometric object.
- Imposing more than one equation corresponds to intersecting the associated geometric objects -
more equations (usually) means smaller objects.
- Usually each extra equation reduces the dimension of the object, unless the equation is redundant.
- Keep track of whether you’re in R2 or R3, so you don’t confuse lines for planes, or circles for
cylinders, etc.
3. Describing Curves and Surfaces by Parameters
3.1. Parametrizations. Today we discuss the second of the two main approaches to describing
curves and surfaces: parametrization. Roughly speaking, a parametrization is a point in R2 or
R3 which changes as the values of the parameter(s) change. By varying the parameter, we can
pick out all the points on the curve or surface in which we are interested. As usual, we should
specify at the outset whether we are working in R2 or R3. First let’s consider only curves in R2.
Curves are one-dimensional, and require only one parameter (for us, this is basically the deﬁnition
of “one-dimensional”). Since we’re in R2, we will write down a point (x(t),y(t)) in R2, where the
x- and y- coordinates depend on a parameter t. For example, (t,t2) is a parametrization of a curve.
So is (cost,sint). We think of t as being a time parameter, so as time changes, the point in R2
varies. It is useful to imagine the coordinates (x(t),y(t)) describing the location of a moving particle
at time t. Sometimes a parametrization is just given by expressing x and y as functions of t, for
instance, x(t) = 2t,y(t) = 5t describes a line of slope 5/2 through the origin. This is equivalent
to putting the two functions together as an ordered pair and referring to the parametrization (2t,5t).
3.2. Going From Parameters to Cutting Out. If you’re given a parametrization and want to
know what it looks like, you can just plot some points for various values of t. This is often good
enough to give you an idea of what the curve looks like. Usually you should put a little arrow on the
curve to indicate the direction of increasing t. Having spent an entire lecture on describing curves
(and surfaces) by equations, it would also be nice to know how to translate between a parametrized
curve and an equation. This is easy to do. The parametrization involves a new variable, t, namely
x(t) = ... and y(t) = ..., whereas the equation which cuts out the curve will just be an equation
involving x and y. So to get from the parametrization to the equation involving x and y only, one
just eliminates t from the two equations. When performing the algebra, it’s handy to just write x
for x(t) and y for y(t).
For instance, to express the parametrization x(t) = 2t and y(t) = 5t of the line from above as an
equation, we must remove t from the system x = 2t,y = 5t. The ﬁrst equation say that t = x/t,

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