Multivariable Calculus Lectures, Lecture notes of Vector Analysis

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Multivariable Calculus

Lectures

Richard J. Brown

ii CONTENTS

Curves in R

n

• Lecture 7. Directional Derivatives
  • The Directional Derivative.
  • 7.0.0.1. Vector form of a partial derivative.
• Lecture 8. Implicit and Inverse Function Theorems
  • 8.1. The Implicit Function Theorem.
  • 8.1.1. In three variables.
  • 8.2. The Inverse Function Theorem.
• Lecture 9. Curves in Euclidean Space
  • Implicit differentiation.
  • Via parameterization.
• Lecture 10. Vector Fields
  • Vector Fields.
• Lecture 11. Differentials and Taylor Series
  • The differential of a function.
  • The Taylor series.
• Lecture 12. Extrema
  • Local extrema.
• Lecture 13. Optimization
  • One variable optimization.
• Lecture 14. The Definite Integral
  • Volumes of regions.
• Lecture 15. The Definite Triple Integral
  • Volumes of higher dimensional regions.
• Lecture 16. Changing Variables in Integration
  • Parameterization.
• Lecture 17. The Line Integral
  • 17.0.1. Return to notation.
  • 17.0.2. Line Integrals.
  • 17.0.2.1. Real-valued, scalar functions.
  • 17.0.2.2. Real-valued, vector functions (vector fields).
• Lecture 18. The Theorem of Green
  • 18.0.1. Green’s Theorem.
  • 18.0.2. Conservative Vector Fields.
• Lecture 19. Surface Parameterizations
  • 19.0.1. Coordinates on a surface.
  • 19.0.2. Surface area.
• Lecture 20. Surface Integration CONTENTS iii
  • 20.0.1. Functions defined on surfaces.
• Lecture 21. The Theorem of Stokes
• Lecture 22. Thh Theorem of Gauss
  • 22.0.1. Gauss’ Theorem.
  • 22.0.2. Divergence.
  • 22.0.3. Curl.
• Lecture 23. Differential Forms
  • 23.0.1. Multilinear algebra.
• Lecture 24. More about Forms
  • 24.0.1. A covector product.
  • 24.0.2. Integrating forms.
• Lecture 25. Generalized Stokes’ Theorem
  • 25.0.1. More notation.
  • 25.0.2. In the language of forms, The Theorem of Gauss.
  • 25.0.3. In the language of forms, The Theorem of Stokes.
  • 25.0.4. In the language of forms, The Theorem of Green.
    • Calculus. 25.0.5. In the language of forms, The Fundamental Theorem of

2 1. PRELIMINARIES

call R

2 = R × R the direct product of the two groups R. (A direct product is

a Cartesian product on the underlying sets with whatever added structure

the individual sets have and give to the product.) We can also multiply

elements of R

2 by real numbers (scalars multiplication), where

c ⋅ (a, b) = (ca, cb), for a, b, c ∈ R,

and these two notions behave well together (meaning that they satisfy cer-

tain conditions that facilitate additional structure and study on R

2 ).

1.1.2. Linear algebra. Now R is also a field, but R

2 is not: One cannot

construct a good notion of multiplication in R

2 that satisfies all of the field

axioms. However, with the notion of addition of ordered pairs, along with

scalar multiplication, we can give R

2 the structure of a vector space over R.

Definition 1.1 (Intuitive). A linear or vector space over a field is a set

V of objects together with two operations which can be added together and

multiplied by field elements in a “compatible” way.

It is common, in a linear space, to call the individual set elements “vec-

tors”. We also say that R

2 is a vector space over R. But it will be a good

idea to make a very important distinction:

Using Figure 2 as a guide, we will distinguish between points in R

2 ,

given by all 2-tuples of numbers written as

R

2 = ™p = (x, y) T x, y ∈ Rž ,

and vectors in R

2 , denoted as the set of all possible 2 × 1-matrices, or 2-

vectors

R

2 = œp = 

x

y

W x, y ∈ R¡.

Figure 2. Points versus vectors, as elements of R 2 .

Some notes:

● Technically speaking, these two descriptions of the plane are quite

different, even as there are “equivalent”. Note that I am using

quotes here because we have not yet defined this (mathematical)

term. But intuitively we do see these two descriptions of the plane

as the same. For now we will leave it as is.

1.1. REAL EUCLIDEAN SPACE R n

. 3

● In time, we will need to be able to define vectors based at arbitrary

points in R

2

. Noticing a difference between points and vectors

(with the same entries) as descriptions of the elements of the plane

will help greatly in this course when, for example, we define and

understand vector fields.

● We can add still more structure to the vector space R

2

. There is a

multiplication of vectors in R

2 where the product is not a vector,

but a real number (a scalar): a scalar product, sometimes called a

dot product or an inner product on vectors (equivalently points):

a

b

c

d

= ac + bd ∈ R.

With this new structure, the plane becomes an example of an inner

product space. This is very useful for vector spaces, since with this

new structure, we can define notions of a distance between vectors,

a vector’s size, the angle between vectors, etc. And with these

notions of measurement, the plane R

2 , as an inner product space,

becomes a place where we can do Euclidean geometry. Hence,

with this additional structure, we call the plane an example of a

Euclidean Space.

1.1.3. The vector space R

n

. All of this still works if we generalize

properly to ordered n-tuples of numbers: Define, for n ∈ N,

R

n = R × R ×... × R ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n-terms

= ™x = (x 1 , x 2 ,... , xn) T xi ∈ R, for i = 1 ,... , nž

x =

x 1

x 2

xn

R

R

RR

R

RR

R

R

RR

R

R

RR

R

RR

xi ∈ R, for i = 1 ,... , n

Now, a set of k n-vectors v 1 , v 2 ,... , vk ∈ R

n are called linearly indepen-

dent if for real scalars ci, i = 1 ,... , k,

(1.1.1) c 1 v 1 + c 2 v 2 +... + ckvk = 0

is only solved by c 1 = c 2 =... = ck = 0. If this is true, then none of the vectors

can be written as a linear combination of the others.

Example 1.1. v 1 =

, v 2 =

, and v 3 =

are linearly

dependent since 3v 1 − v 2 − v 3 = 0. Thus, for instance, one can write v 3 as a

linear combination of the others;

3 v 1 − v 2 = v 3.

1.2. LINEAR SPACES INSIDE R n 5

Figure 3. The standard bases in R 2 and R 3 .

all of R

n , how it sits inside R

n will be important. We call a vector space

generated by a subset of vectors in a vector space a vector subspace:

Definition 1.2. A linear or vector subspace W of a vector space V is a

subset of the elements of V that satisfy

(1) 0 ∈ W ⊂ V ,

(2) If w 1 , w 2 ∈ W , then w 1 + w 2 ∈ W , and

(3) if w ∈ W , then for all c ∈ R, cw ∈ W.

Figure 4. The xy-plane in R 3 .

It is good to note here that ALL vector

subspaces pass through the origin (contain

the zero-vector).

And going back to Equation 1.1.1, note

that for any k ∈ N, the set of n-vectors

span {v 1 , v 2 ,... , vk} is ALWAYS a linear

subspace of R

n

. How big it is as a subspace

depends on the number of vis that are linearly independent.

Example 1.3. The set span

is commonly referred

to as the xy-plane in R

3 , thinking of the standard coordinates in R

3

. The

span of these three vectors only makes a plane in three space since the third

vector is simply twice the first plus 3/2 times the second. A basis for the

span of these three 3-vectors can readily be the first two vectors in the

standard basis of R

3

. Note that one can also call this linear subspace the

(z = 0 )-plane. In this way, the xy-plane is a version of R

2 sitting inside R

3

as a subspace of all 3-vectors with 0 in the last component. See Figure 4.

Example 1.4. span

is a line passing through the

origin in R

3 .

6 1. PRELIMINARIES

Example 1.5. Let V = span

a =

, b =

. Then V ⊂ R

3

is a 2-dimensional subspace, since a and b are linearly independent (recall

that the dimension of a (finite-dimensional) vector space is the number of

elements in any basis), and V ⊂ R

3 will look like a plane passing through

the origin (See Figure 5, with a and b in red). The two 3-vectors

c =

, d =

differ in that c ∈ V (shown in blue in Figure 5, while d ~∈ V (shown in green

in the figure). Indeed, c = −a + b, but there are not constants ca, cb ∈ R,

where caa + cbb = d. We would say that d is linearly independent from V.

Figure 5. V = span {a, b}.

Further, by Example 1.5, we can view the lines

passing through a and b as coordinate axes for V.

And on each axis, we can use the length of the con-

tained vector as the unit length along that axis,

marking, for example, 0 at the origin and 1 at the

head of a. This provides a coordinate system di-

rectly on V , using the ordered pair (ca, cb) as the

coordinates in V. Thus the vector c ∈ V ⊂ R

3 cor-

responds to the vector

∈ R

3 (or the point

( 1 , − 4 , 0 ) ∈ R

3 ), but in the coordinates defined di-

rectly on V by the basis {a, b}, c = 

∈ V (or the point (− 1 , 1 ) ∈ V )

in the parameterization of V given by the basis. The idea of placing coordi-

nates directly on a subspace instead of using the ambient coordinates of the

larger space is an important one. We will spend much time on this.

Figure 6. A line in R 3

as the intersection of 2

planes.

1.2.1. Planes and Lines in R

3

. One way to de-

scribe a subspace like V ∈ R

3 is through another form

of multiplication of vectors, this one where the product

of two 3-vectors is again a 3-vector. (Note that this is

extremely rare and for now is limited to R

3 .) The cross

product of two vectors a × b = n is a vector normal (as

in zero dot product) to both a and b. Hence, for any

vector n, the set of all vectors normal to n is a two

dimensional subspace V ∈ R

3

. And, if n is given as the

cross product of two linearly independent vectors a and

8 1. PRELIMINARIES

In this fashion, we can now describe any plane in R

3 : Given any vector na,

based at a, one can describe a plane in R

3 passing through a and normal to

na via the equation

na ●^ va =

nx

ny

nz

●

x − a 1

y − a 2

z − a 3

= 0 = nx(x − a 1 ) + ny(y − a 2 ) + nz (z − a 3 ).

Note that this plane is not a linear subspace of R

3 as a vector space (the

vector x = y = z = 0 is not in the plane, for example, when any of the

three ai’s are nonzero). But it is a vector space, has a basis, and will play

a very important role in understanding how functions involving more than

one independent variable behave, as we will see.

One conclusion that can be drawn from this is that one can define a

plane in R

3 via a single equation. But then, what is the equation of a line

in R

3 ? Here is an example:

Example 1.6. Consider the solution set for the set of equations:

x + 2 y + 3 z = 4 (eq1)

2 x − 2 y + 3 z = 1 (eq2)

¡ 2 equations in 3 unknowns.

So what does this solution set in R

3 look like? To see, solve as best as one

can:

(eq1) + (eq2) ∶ 3 x + 6 z = 5

2(eq1) − (eq2) ∶ 6 y + 3 z = 7

Then

x =

5 − 6 z

, y =

7 − 3 z

, z is free.

Better yet, we can place a single parameter t directly on this set by setting

z = t, so that x =

5 − 6 t 3

and y =

7 − 3 t 6

, along with z = t makes a parameterized

curve (a line) in R

3

. One could also write this as a function (using vector

notation):

c ∶ R → R

3 , c(t) =

5 − 6 t 3

7 − 3 t 6

t

Note that, in this parameterization, we still have 3 equations in 4 unknowns.

Do you notice a pattern between the number of equations, the number of

unknowns and the “size” of the space of solutions?

1.3. Linear functions.

So, roughly speaking, a space V is called linear if any linear combination

of two elements in V is still in V. So what, then, is a linear function?

1.3. LINEAR FUNCTIONS. 9

Definition 1.3. A function f ∶ R → R is called linear if

f (c 1 x 1 + c 2 x 2 ) = c 1 f (x 1 ) + c 2 f (x 2 ), ∀x 1 , x 2 ∈ R, c 1 , c 2 ∈ R

Notes:

(1) With appropriate changes, this works equally well for f ∶ R

n → R

m .

(2) Using this definition, then, the function f (x) = 3 x is linear, but the

function g(x) = 3 x + 1 is NOT! To see this,

g( 2 + 3 ) = g( 5 ) = 3 ( 5 ) + 1 = 16

= ~ g( 2 ) + g( 3 ) = ( 3 ( 2 ) + 1 ) + ( 3 ( 3 ) + 1 ) = 17.

The issue here is that for a function to be linear, the origin of

the domain (the input space) must be mapped to the origin of the

output space, so that f ( 0 ) = 0. But here g( 0 ) = 1. And thus, g(x)

is not linear. It is an example of an affine function, one that can

be seen as a composition of a linear function and a translation.

(3) Let f ∶ R

n → R

m be linear. Then, given a basis {v 1 ,... , vn} for the

domain R

n , we can write any x ∈ R

n as

x = c 1 v 1 +... + cnvn.

Then, since f is linear, we have

f (x) = f (c 1 v 1 +... + cnvn) = c 1 f (v 1 ) +... + cnf (vn)

= m

S S

f (v 1 )... f (vn)

S S

n

c 1

cn

= Am×nx.

Hence, any linear map between vector spaces can always be repre-

sented by a matrix.

LECTURE 2

Functions of Several Variables.

Synopsis. Today we begin the course in earnest in Chapter 2, although,

again like in Lecture 1, we will be covering the material mostly for notation

and viewpoint. Pay close attention to why and how we visualize functions,

through parameterizations, graphs, slices and sections. These will expose

the visual clues to how we analyze functions.

Helpful Documents.

● Mathematica: CurvesInSpace,

● Mathematica: ParameterizedSurfaces,

● Mathematica: VisualizingFunctions, and

● PDF: LevelSets.

2.1. Properties of Functions.

A function f ∶ X → Y from a set X to another set Y is defined in a

manner equal to what you have already studied in single variable calculus

(and pre-calculus):

● f assigns to each x ∈ X a single element y ∈ Y , and every element

of X has an element of Y assigned to it.

● The set X is called the domain of the function, and Y is called the

codomain.

● f (X) ⊂ Y (as a set) is called the range of f , and more precisely

called the image of X in Y under f. It is defined explicitly as

f (X) = ™y ∈ Y T y = f (x) for some x ∈ Xž.

● For a subset Z ⊂ Y , the set

f

− 1 (Z) = ™x ∈ X T^ f (x) ∈ Zž

is called the inverse image of Z in X under f , or the preimage of

Z in X (under f ). Note that if y ~∈ f (X), then f

− 1 (y) = ∅ is still

well-defined. Note also that the notation does not imply that the

function f has an inverse function. The set f

− 1 (Z) ⊂ X is only a

set.

● f is called one-to-one, or injective, if

™x ∈ X T f (x) = yž ≤ 1 , ∀y ∈ Y.

● f is called onto or surjective if ∀y ∈ Y , y = f (x) for at least one

x ∈ X.

11

12 2. FUNCTIONS OF SEVERAL VARIABLES.

● f is called bijective if f is both injective and surjective.

Note that, for this class, X and Y will be subsets of Euclidean space,

although often not the same space nor the same dimension.

Here is some additional nomenclature and notation:

● Let X ⊂ R

n and Y ⊂ R

m

. If m = 1, we call f ∶ X → Y a real-

valued or scalar-valued function on X, or on n-variables (restricted

to X). If m > 1, we say f is vector-valued. As we will see, vector-

valued functions consist of expressions that are real-valued on each

coordinate of R

m .

● Where important to the discussion, we will denote scalars as x ∈ R,

and vectors as x ∈ R

n , n > 1. We will also denote a real-valued

function as f , and a vector-valued function as f. In lecture, we

will employ the vector notation x⃗ and

f , since boldface is difficult

in chalk. Note that when it is not important to the discussion, or

for general situations, it is the case that we will use boldface for

variables, and possibly write f ∶ X → Y , and f (x) = y, even if

X ∈ R

n , n > 1, and y ∈ Y ⊂ R

m , m > 1. This is common in analysis

and should be clear in context.

● a function f ∶ X → Y is often called a map (or a mapping) from X

to Y. In some contexts, a function and a map are not the same

thing, but often they are used interchangeably.

Definition 2.1. A map p ∶ X → X is called a projection if

p(p(x)) = p(x), ∀x ∈ X.

• Here, the set comprising the image p(X) ⊂ X is called the

projection of X onto p(X). When X is a linear space and

p a linear projection, then p(X) is a linear subspace. See

Example 2.1 below.

• A projection p, restricted to its image, is the identity map. We

can write this as pT p(X)

= Idp(X).

• For X = R

n , the map pi ∶ R

n → R

n , defined by

pi ((x 1 ,... , xi− 1 , xi, xi+ 1 ,... , xn)) = ( 0 ,... , 0 , xi, 0 ,... , 0 )

is called the ith projection. Sometimes, one may write pi(x) =

xi, but this is not quite correct.

• There are many extensions and generalizations of the idea of

projection in various areas of mathematics, including some

that do not seem to fit the definition above. (See, for instance,

the separate document StereographicProjection.) For now,

here are a couple of examples.

14 2. FUNCTIONS OF SEVERAL VARIABLES.

variables. In this case, the graph of that function takes on a particular look;

that of a“height over a floor” schematic:

Definition 2.2. For f ∶ X ⊂ R

n → R, the graph of f is the set

graph(f ) = ™(x, f (x)) ∈ R

n × R = R

n+ 1 T xn+ 1 = f (x)ž.

Note that this is quite useful for n = 2 (so that the graph “lives” in

R

3 , but not so useful for n > 2. Also, this is the proper generalization

for the way graphs of functions were constructed in pre-calculus and single

variable calculus. And, generally speaking, the “size” of f (X) ∈ R

3 will be

the same as that of X. It should be easy to see that it is always the case

that graph(f ) ⊂ R

n+ 1 always projects to (a copy of) X ⊂ R

n × R as

(x 1 , x 2 ,... , xn, f (x)) z→ (x 1 ,... , xn, 0 ).

See Figure 9. More generally, we have:

Definition 2.3. For f ∶ X ⊂ R

n → R

m , m ≥ 1, where f (x) = y, the graph

of f is the set

graph(f ) = ™(x, f (x)) ∈ R

n × R

m = R

n+m T y = f (x)ž.

Figure 9. For f ∶ X ⊂ R n → R,

graph(f ) ⊂ R n+ 1 .

Consider the vector-valued function g ∶

X ⊂ R

2 → R

2 , defined by g(x) =

g Œ

x

y

g 1 (x, y)

g 2 (x, y)

. Here, for i = 1 , 2,

each gi ∶ X → R is a real-valued function,

called a component function or a coordinate

function. But the graph of g ⊂ R

4 is the set

graph(g) = œ(x, y, z, u) ∈ R

4 W

z = g 1 (x, y)

u = g 2 (x, y)

It is already hard to visualize!

An easier example to visualize is the

function h ∶ R → R

2 , h(t) = (cos t, sin t). Its graph lives in R

3 as a curve

graph(h) = ™(t, x, y) ∈ R

3 T x = cos t, y = sin tž.

As one can see in Figure 10, this curve can be visualized and studied, but

is still a bit tricky to analyze.

2.2.2. Parameterizations. Generalized coordinates can be placed di-

rectly on a subset of R

n through continuous functions so that points on

the subset are distinguishable via parameter values instead of ambient co-

ordinates. (One does this on a sphere when one speaks of the latitude and

longitude of a point on our Earth.) A parameterization allows one to de-

scribe a subset of R

n by a smaller number of variables; one can generally

talk of a subset having a dimension equal to the number of variables it takes

to distinguish points on the subset, although the notion of dimension for a

space is not always very well defined.

2.2. VISUALIZATION OF FUNCTIONS. 15

Figure 10. Projections in R 3 onto the xy-plane (at left), and the unit sphere

S 2 (at right).

Return to the function h ∶ R → R

2 , h(t) = (cos t, sin t) ∈ R

2 , and consider

only the image of h ⊂ R

2

. Here, we say that h parameterizes the unit circle

in the plane. In this case, t is a coordinate, defined directly (and only)

on the circle of radius 1 in R

2 , and is a 1-dimensional parameterization.

Note here that, broadly speaking, parameterizations should be one-to-one

as functions, so that points are distinguished adequately. However, this is

not true in general, and this example is telling. Here, we would say that this

parameterization is locally-injective. We caution, though, that even this is

not true in general.

Figure 11. Parameterization of S 1 ⊂ R 2 via h ∶ R → R 2 , h(t) = (cos t, sin t).

Example 2.3. Let D ⊂ R

2 be the rectangle

D = ™(θ, ψ) ∈ R

2 T θ ∈ [ 0 , 2 π], ψ ∈ [ 0 , π]ž

as a subset of R

2

. Then the function Φ ∶ D → R

3 , Φ(θ, ψ) =

(sin θ sin ψ, cos θ sin ψ, cos ψ) provides coordinates directly on the unit sphere

in three space that correspond to the azimuth angle θ and polar angle ψ of

the standard spherical coordinate system in R

3 .

Note here two things: