Multivariable Calculus: A Textbook by Jerry Shurman, Exercises of Calculus

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Multivariable Calculus

Jerry Shurman

Reed College

• 1 Results from One-Variable Calculus Preface ix
  • 1.1 The Real Number System
  • 1.2 Foundational and Basic Theorems
  • 1.3 Taylor’s Theorem
• 2 Euclidean Space Part I Multivariable Differential Calculus
  • 2.1 Algebra: Vectors
  • 2.2 Geometry: Length and Angle
  • 2.3 Analysis: Continuous Mappings
  • 2.4 Topology: Compact Sets and Continuity
  • 2.5 Review of the One-Variable Derivative
  • 2.6 Summary
• 3 Linear Mappings and Their Matrices
  • 3.1 Linear Mappings
  • 3.2 Operations on Matrices
  • 3.3 The Inverse of a Linear Mapping
  • 3.4 Inhomogeneous Linear Equations
    • Consequences 3.5 The Determinant: Characterizing Properties and Their
  • 3.6 The Determinant: Uniqueness and Existence
  • 3.7 An Explicit Formula for the Inverse
  • 3.8 Geometry of the Determinant: Volume
  • 3.9 Geometry of the Determinant: Orientation
  • 3.10 The Cross Product, Lines, and Planes in R
  • 3.11 Summary
• 4 The Derivative vi Contents - Catastrophe 4.1 Trying to Extend the Symbol-Pattern: Immediate, Irreparable
  • 4.2 New Environment: the Bachmann–Landau Notation
  • 4.3 One-Variable Revisionism; the Derivative Redefined
  • 4.4 Basic Results and the Chain Rule
  • 4.5 Calculating the Derivative
  • 4.6 Higher Order Derivatives
  • 4.7 Extreme Values
  • 4.8 Directional Derivatives and the Gradient
  • 4.9 Summary
• 5 Inverse and Implicit Functions
  • 5.1 Preliminaries
  • 5.2 The Inverse Function Theorem
  • 5.3 The Implicit Function Theorem
    • Examples 5.4 Lagrange Multipliers: Geometric Motivation and Specific
  • 5.5 Lagrange Multipliers: Analytic Proof and General Examples
  • 5.6 Summary
• 6 Integration Part II Multivariable Integral Calculus
  • 6.1 Machinery: Boxes, Partitions, and Sums
  • 6.2 Definition of the Integral
  • 6.3 Continuity and Integrability
  • 6.4 Integration of Functions of One Variable
  • 6.5 Integration Over Nonboxes
  • 6.6 Fubini’s Theorem
  • 6.7 Change of Variable
  • 6.8 Topological Preliminaries for the Change of Variable Theorem
  • 6.9 Proof of the Change of Variable Theorem
  • 6.10 Summary
• 7 Approximation by Smooth Functions
  • 7.1 Spaces of Functions
  • 7.2 Pulse Functions
  • 7.3 Convolution
  • 7.4 Test Approximate Identity and Convolution
  • 7.5 Known-Integrable Functions
  • 7.6 Summary
• 8 Parametrized Curves Contents vii
  • 8.1 Euclidean Constructions and Two Curves
  • 8.2 Parametrized Curves
  • 8.3 Parametrization by Arc Length
  • 8.4 Plane Curves: Curvature
  • 8.5 Space Curves: Curvature and Torsion
  • 8.6 General Frenet Frames and Curvatures
  • 8.7 Summary
• 9 Integration of Differential Forms
  • 9.1 Integration of Functions Over Surfaces
  • 9.2 Flow and Flux Integrals
  • 9.3 Differential Forms Syntactically and Operationally
  • 9.4 Examples: 1-forms
  • 9.5 Examples: 2-forms on R
  • 9.6 Algebra of Forms: Basic Properties
  • 9.7 Algebra of Forms: Multiplication
  • 9.8 Algebra of Forms: Differentiation
  • 9.9 Algebra of Forms: the Pullback
  • 9.10 Change of Variable for Differential Forms
  • 9.11 Closed Forms, Exact Forms, and Homotopy
  • 9.12 Cubes and Chains
  • 9.13 Geometry of Chains: the Boundary Operator
  • 9.14 The General Fundamental Theorem of Integral Calculus
  • 9.15 Classical Change of Variable Revisited
  • 9.16 The Classical Theorems
  • 9.17 Divergence and Curl in Polar Coordinates
  • 9.18 Summary
• Index

Preface

This is the text for a two-semester multivariable calculus course. The setting is n-dimensional Euclidean space, with the material on differentiation culminat- ing in the Inverse Function Theorem and its consequences, and the material on integration culminating in the Generalized Fundamental Theorem of Inte- gral Calculus (often called Stokes’s Theorem) and some of its consequences in turn. The prerequisite is a proof-based course in one-variable calculus. Some familiarity with the complex number system and complex mappings is occa- sionally assumed as well, but the reader can get by without it. The book’s aim is to use multivariable calculus to teach mathematics as a blend of reasoning, computing, and problem-solving, doing justice to the structure, the details, and the scope of the ideas. To this end, I have tried to write in a style that communicates intent early in the discussion of each topic rather than proceeding coyly from opaque definitions. Also, I have tried occasionally to speak to the pedagogy of mathematics and its effect on the process of learning the subject. Most importantly, I have tried to spread the weight of exposition among figures, formulas, and words. The premise is that the reader is ready to do mathematics resourcefully by marshaling the skills of

• geometric intuition (the visual cortex being quickly instinctive),
• algebraic manipulation (symbol-patterns being precise and robust),
• and incisive use of natural language (slogans that encapsulate central ideas enabling a large-scale grasp of the subject).

Thinking in these ways renders mathematics coherent, inevitable, and fluent. In my own student days I learned this material from books by Apostol, Buck, Rudin, and Spivak, books that thrilled me. My debt to those sources pervades these pages. There are many other fine books on the subject as well, such as the more recent one by Hubbard and Hubbard. Indeed, nothing in this book is claimed as new, not even its neuroses. Whatever improvement the exposition here has come to show over the years is due to innumerable ideas and comments from my students in turn.

x Preface

By way of a warm-up, chapter 1 reviews some ideas from one-variable calculus, and then covers the one-variable Taylor’s Theorem in detail. Chapters 2 and 3 cover what might be called multivariable pre-calculus, in- troducing the requisite algebra, geometry, analysis, and topology of Euclidean space, and the requisite linear algebra, for the calculus to follow. A pedagogical theme of these chapters is that mathematical objects can be better understood from their characterizations than from their constructions. Vector geometry follows from the intrinsic (coordinate-free) algebraic properties of the vector inner product, with no reference to the inner product formula. The fact that passing a closed and bounded subset of Euclidean space through a continuous mapping gives another such set is clear once such sets are characterized in terms of sequences. The multiplicativity of the determinant and the fact that the determinant indicates whether a linear mapping is invertible are conse- quences of the determinant’s characterizing properties. The geometry of the cross product follows from its intrinsic algebraic characterization. Further- more, the only possible formula for the (suitably normalized) inner product, or for the determinant, or for the cross product, is dictated by the relevant properties. As far as the theory is concerned, the only role of the formula is to show that an object with the desired properties exists at all. The intent here is that the student who is introduced to mathematical objects via their characterizations will see quickly how the objects work, and that how they work makes their constructions inevitable. In the same vein, chapter 4 characterizes the multivariable derivative as a well approximating linear mapping. The chapter then solves some multivari- able problems that have one-variable counterparts. Specifically, the multivari- able chain rule helps with change of variable in partial differential equations, a multivariable analogue of the max/min test helps with optimization, and the multivariable derivative of a scalar-valued function helps to find tangent planes and trajectories. Chapter 5 uses the results of the three chapters preceding it to prove the Inverse Function Theorem, then the Implicit Function Theorem as a corollary, and finally the Lagrange Multiplier Criterion as a consequence of the Implicit Function Theorem. Lagrange multipliers help with a type of multivariable optimization problem that has no one-variable analogue, optimization with constraints. For example, given two curves in space, what pair of points— one on each curve—is closest to each other? Not only does this problem have six variables (the three coordinates of each point), but furthermore they are not fully independent: the first three variables must specify a point on the first curve, and similarly for the second three. In this problem, x 1 through x 6 vary though a subset of six-dimensional space, conceptually a two-dimensional subset (one degree of freedom for each curve) that is bending around in the ambient six dimensions, and we seek points of this subset where a certain function of x 1 through x 6 is optimized. That is, optimization with constraints can be viewed as a beginning example of calculus on curved spaces.

xii Preface

surfaces in n-dimensional space, and parametrized curves are the special case k = 1. Aside from being one-dimensional surfaces, parametrized curves are interesting in their own right. Chapter 9 presents the integration of differential forms. This subject poses the pedagogical dilemma that fully describing its structure requires an in- vestment in machinery untenable for students who are seeing it for the first time, whereas describing it purely operationally is unmotivated. The approach here begins with the integration of functions over k-dimensional surfaces in n-dimensional space, a natural tool to want, with a natural definition suggest- ing itself. For certain such integrals, called flow and flux integrals, the inte- grand takes a particularly workable form consisting of sums of determinants of derivatives. It is easy to see what other integrands—including integrands suitable for n-dimensional integration in the sense of chapter 6, and includ- ing functions in the usual sense—have similar features. These integrands can be uniformly described in algebraic terms as objects called differential forms. That is, differential forms comprise the smallest coherent algebraic structure encompassing the various integrands of interest to us. The fact that differen- tial forms are algebraic makes them easy to study without thinking directly about integration. The algebra leads to a general version of the Fundamental Theorem of Integral Calculus that is rich in geometry. The theorem subsumes the three classical vector integration theorems, Green’s Theorem, Stokes’s Theorem, and Gauss’s Theorem, also called the Divergence Theorem.

Comments and corrections should be sent to [email protected].

Exercises

0.0.1. (a) Consider two surfaces in space, each surface having at each of its points a tangent plane and therefore a normal line, and consider pairs of points, one on each surface. Conjecture a geometric condition, phrased in terms of tangent planes and/or normal lines, about the closest pair of points. (b) Consider a surface in space and a curve in space, the curve having at each of its points a tangent line and therefore a normal plane, and consider pairs of points, one on the surface and one on the curve. Make a conjecture about the closest pair of points. (c) Make a conjecture about the closest pair of points on two curves.

0.0.2. (a) Assume that the factorial of a half-integer makes sense, and grant the general formula for the volume of a ball in n dimensions. Explain why it follows that (1/2)! =

π/2. Further assume that the half-integral factorial function satisfies the relation

x! = x · (x − 1)! for x = 3/ 2 , 5 / 2 , 7 / 2 ,...

Subject to these assumptions, verify that the volume of the ball of radius r in three dimensions is 43 πr^3 as claimed. What is the volume of the ball of radius r in five dimensions?

Preface xiii

(b) The ball of radius r in n dimensions sits inside a circumscribing box of sides 2r. Draw pictures of this configuration for n = 1, 2 , 3. Determine what portion of the box is filled by the ball in the limit as the dimension n gets large. That is, find

lim n→∞

vol (Bn(r)) (2r)n^

Results from One-Variable Calculus

As a warmup, these notes begin with a quick review of some ideas from one- variable calculus. The material in the first two sections is assumed to be familiar. Section 3 discusses Taylor’s Theorem at greater length, not assuming that the reader has already seen it.

1.1 The Real Number System

We assume that there is a real number system, a set R that contains two distinct elements 0 and 1 and is endowed with the algebraic operations of addition,

• : R × R −→ R,

and multiplication, · : R × R −→ R.

The sum +(a, b) is written a + b, and the product ·(a, b) is written a · b or simply ab.

Theorem 1.1.1 (Field Axioms for (R, +, ·)). The real number system, with its distinct 0 and 1 and with its addition and multiplication, is assumed to satisfy the following set of axioms.

(a1) Addition is associative: (x + y) + z = x + (y + z) for all x, y, z ∈ R. (a2) 0 is an additive identity: 0 + x = x for all x ∈ R. (a3) Existence of additive inverses: For each x ∈ R there exists y ∈ R such that y + x = 0. (a4) Addition is commutative: x + y = y + x for all x, y ∈ R. (m1) Multiplication is associative: x(yz) = (xy)z for all x, y, z ∈ R. (m2) 1 is a multiplicative identity: 1 x = x for all x ∈ R. (m3) Existence of multiplicative inverses: For each nonzero x ∈ R there exists y ∈ R such that yx = 1. (m4) Multiplication is commutative: xy = yx for all x, y ∈ R.

2 1 Results from One-Variable Calculus

(d1) Multiplication distributes over addition: (x+y)z = xz+yz for all x, y, z ∈ R.

All of basic algebra follows from the field axioms. Additive and multi- plicative inverses are unique, the cancellation law holds, 0 · x = 0 for all real numbers x, and so on. Subtracting a real number from another is defined as adding the additive inverse. In symbols,

− : R × R −→ R, x − y = x + (−y) for all x, y ∈ R.

We also assume that R is an ordered field. That is, we assume that there is a subset R+^ of R (the positive elements) such that the following axioms hold.

Theorem 1.1.2 (Order Axioms).

(o1) Trichotomy Axiom: For every real number x, exactly one of the following conditions holds:

x ∈ R+, −x ∈ R+, x = 0.

(o2) Closure of positive numbers under addition: For all real numbers x and y, if x ∈ R+^ and y ∈ R+^ then also x + y ∈ R+. (o3) Closure of positive numbers under multiplication: For all real numbers x and y, if x ∈ R+^ and y ∈ R+^ then also xy ∈ R+.

For all real numbers x and y, define

x < y

to mean y − x ∈ R+.

The usual rules for inequalities then follow from the axioms.

Finally, we assume that the real number system is complete. Complete- ness can be phrased in various ways, all logically equivalent. A version of completeness that is phrased in terms of binary search is as follows.

Theorem 1.1.3 (Completeness as a Binary Search Criterion). Every binary search sequence in the real number system converges to a unique limit.

Convergence is a concept of analysis, and therefore so is completeness. Another version of completeness is phrased in terms of set-bounds.

Theorem 1.1.4 (Completeness as a Set-Bound Criterion). Every non- empty subset of R that is bounded above has a least upper bound.

4 1 Results from One-Variable Calculus

1.1.3. Use a completeness property of the real number system to show that 2 has a positive square root.

1.1.4. (a) Prove by induction that

∑^ n

i=

i^2 = n(n + 1)(2n + 1) 6

for all n ∈ Z+.

(b) (Bernoulli’s Inequality) For any real number r ≥ −1, prove that

(1 + r)n^ ≥ 1 + rn for all n ∈ N.

(c) For what positive integers n is 2n^ > n^3?

1.1.5. (a) Use the Induction Theorem to show that for any natural number m, the sum m+n and the product mn are again natural for any natural number n. Thus N is closed under addition and multiplication, and consequently so is Z. (b) Which of the field axioms continue to hold for the natural numbers? (c) Which of the field axioms continue to hold for the integers?

1.1.6. For any positive integer n, let Z/nZ denote the set { 0 , 1 ,... , n − 1 } with the usual operations of addition and multiplication carried out taking remainders. That is, add and multiply in the usual fashion but subject to the additional condition that n = 0. For example, in Z/ 5 Z we have 2 + 4 = 1 and 2 · 4 = 3. For what values of n does Z/nZ form a field?

1.2 Foundational and Basic Theorems

This section reviews the foundational theorems of one-variable calculus. The first two theorems are not theorems of calculus at all, but rather they are theorems about continuous functions and the real number system. The first theorem says that under suitable conditions, an optimization problem is guar- anteed to have a solution.

Theorem 1.2.1 (Extreme Value Theorem). Let I be a nonempty closed and bounded interval in R, and let f : I −→ R be a continuous function. Then f takes a minimum value and a maximum value on I.

The second theorem says that under suitable conditions, any value trapped between two output values of a function must itself be an output value.

Theorem 1.2.2 (Intermediate Value Theorem). Let I be a nonempty interval in R, and let f : I −→ R be a continuous function. Let y be a real number, and suppose that

f (x) < y for some x ∈ I

1.2 Foundational and Basic Theorems 5

and f (x′) > y for some x′^ ∈ I.

Then f (c) = y for some c ∈ I.

The Mean Value Theorem relates the derivative of a function to values of the function itself with no reference to the fact that the derivative is a limit, but at the cost of introducing an unknown point.

Theorem 1.2.3 (Mean Value Theorem). Let a and b be real numbers with a < b. Suppose that the function f : [a, b] −→ R is continuous and that f is differentiable on the open subinterval (a, b). Then

f (b) − f (a) b − a

= f ′(c) for some c ∈ (a, b).

The Fundamental Theorem of Integral Calculus relates the integral of the derivative to the original function, assuming that the derivative is continuous.

Theorem 1.2.4 (Fundamental Theorem of Integral Calculus). Let I be a nonempty interval in R, and let f : I −→ R be a continuous function. Suppose that the function F : I −→ R has derivative f. Then for any closed and bounded subinterval [a, b] of I,

∫ (^) b

a

f (x) dx = F (b) − F (a).

Exercises

1.2.1. Use the Intermediate Value Theorem to show that 2 has a positive square root.

1.2.2. Let f : [0, 1] −→ [0, 1] be continuous. Use the Intermediate Value Theorem to show that f (x) = x for some x ∈ [0, 1].

1.2.3. Let a and b be real numbers with a < b. Suppose that f : [a, b] −→ R is continuous and that f is differentiable on the open subinterval (a, b). Use the Mean Value Theorem to show that if f ′^ > 0 on (a, b) then f is strictly increasing on [a, b].

1.2.4. For the Extreme Value Theorem, the Intermediate Value Theorem, and the Mean Value Theorem, give examples to show that weakening the hypotheses of the theorem gives rise to examples where the conclusion of the theorem fails.

1.3 Taylor’s Theorem 7

Differentiate again to obtain

p′′′(x) = 3 · 2 a 3 + · · · + n(n − 1)(n − 2)an(x − a)n−^3 ,

so that p′′′(a) = 3 · 2 a 3. We want p′′′(a) to equal f ′′′(a), so set

a 3 =

f ′′(a) 3 · 2

Continue in this fashion to obtain a 4 = f (4)(a)/4! and so on up to an = f (n)(a)/n!. That is, the desired coefficients are

ak = f (k)(a) k!

for k = 0,... , n.

Thus the answer to the existence part of Question 1 is yes. Furthermore, since the calculation offered us no choices en route, these are the only coefficients that can work, and so the approximating polynomial is unique. It deserves a name.

Definition 1.3.1 (nth degree Taylor Polynomial). Let I ⊂ R be a nonempty open interval, and let a be a point of I. Let n be a nonnegative integer. Suppose that the function f : I −→ R has n continuous derivatives. Then the nth degree Taylor polynomial of f at a is

Tn(x) = f (a) + f ′(a)(x − a) +

f ′′(a) 2

(x − a)^2 + · · · +

f (n)(a) n!

(x − a)n.

In more concise notation,

Tn(x) =

∑^ n

k=

f (k)(a) k! (x − a)k.

For example, if f (x) = ex^ and a = 0 then it is easy to generate the following table:

k f (k)(x)

f (k)(0) k! 0 ex^1 1 ex^1

2 ex^

3 ex^

n ex^

n!

8 1 Results from One-Variable Calculus

From the table we can read off the nth degree Taylor polynomial of f at 0,

Tn(x) = 1 + x +

x^2 2

x^3 3!

xn n!

=

∑^ n

k=

xk k!

Recall that the second question is how well the polynomial Tn(x) approxi- mates f (x) for x 6 = a. Thus it is a question about the difference f (x) − Tn(x). Giving this quantity its own name is useful.

Definition 1.3.2 (nth degree Taylor Remainder). Let I ⊂ R be a nonempty open interval, and let a be a point of I. Let n be a nonnegative integer. Suppose that the function f : I −→ R has n continuous derivatives. Then the nth degree Taylor remainder of f at a is

Rn(x) = f (x) − Tn(x).

So the second question is to estimate the remainder Rn(x) for points x ∈ I. The method to be presented here for doing so proceeds very naturally, but it is perhaps a little surprising because although the Taylor polynomial Tn(x) is expressed in terms of derivatives, as is the expression to be obtained for the remainder Rn(x), we obtain the expression by using the Fundamental Theorem of Integral Calculus repeatedly. The method requires a calculation, and so, guided by hindsight, we first carry it out so that then the ideas of the method itself will be uncluttered. For any positive integer k and any x ∈ R, define a k-fold nested integral,

Ik(x) =

∫ (^) x

x 1 =a

∫ (^) x 1

x 2 =a

∫ (^) xk− 1

xk =a

dxk · · · dx 2 dx 1.

This nested integral is a function only of x because a is a constant while x 1 through xk are dummy variables of integration. That is, Ik depends only on the upper limit of integration of the outermost integral. Although Ik may appear daunting, it unwinds readily if we start from the simplest case. First,

I 1 (x) =

∫ (^) x

x 1 =a

dx 1 = x 1

x

x 1 =a

= x − a.

Move one layer out and use this result to get

I 2 (x) =

∫ (^) x

x 1 =a

∫ (^) x 1

x 2 =a

dx 2 dx 1 =

∫ (^) x

x 1 =a

I 1 (x 1 ) dx 1

∫ (^) x

x 1 =a

(x 1 − a) dx 1 =

(x 1 − a)^2

x

x 1 =a

(x − a)^2.