Functions of Two Variables

Sean Dineen

• Publisher: Chapman and Hall/CRC
• Year: 2017
• eBook ISBN: 9781351990226
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Multivariate calculus, as traditionally presented, can overwhelm students who approach it directly from a one-variable calculus background. There is another way-a highly engaging way that does not neglect readers' own intuition, experience, and excitement. One that presents the fundamentals of the subject in a two-variable context and was set forth in the popular first edition of Functions of Two Variables. The second edition goes even further toward a treatment that is at once gentle but rigorous, atypical yet logical, and ultimately an ideal introduction to a subject important to careers both within and outside of mathematics. The author's style remains informal and his approach problem-oriented. He takes care to motivate concepts prior to their introduction and to justify them afterwards, to explain the use and abuse of notation and the scope of the techniques developed. Functions of Two Variables, Second Edition includes a new section on tangent lines, more emphasis on the chain rule, a rearrangement of several chapters, refined examples, and more exercises. It maintains a balance between intuition, explanation, methodology, and justification, enhanced by diagrams, heuristic comments, examples, exercises, and proofs.

1

Functions from R² to R

Summary. We introduce the problem of finding the maximum and minimum of a real-valued function of two variables. The one-dimensional theory suggests that we discuss the problem for functions defined on open sets and their boundaries. We define open sets, consider a possible definition of the derivative and define the graph of a function of two variables.

In this course we will discuss all possible functions which involve two variables and so look at functions from R² into R, from R into R² and from R² into R². We begin by considering functions from R² into R and our objective is to obtain methods for finding maxima and minima. If the functions are arbitrarily behaved we get nowhere, so we have to make some assumptions—we will use the general term regularity conditions—on the functions considered. These regularity conditions usually relate to continuity and differentiability. First, however, we try and see, based on our one-dimensional experience, how we might proceed and then return to look more closely at what we need in order to proceed. The main one-dimensional motivation is the following fundamental result.

Theorem 1.

If f:[a, b] → R is continuous on the closed interval [a, b], then f has a maximum and a minimum on [a, b].

In other words there are two points in [a, b], x₁ and x₂, such that

f(x1)≤f(x)≤f(x2)

for all x in [a, b].

The function f has a minimum value f(x₁) which is achieved at x₁ and a maximum value f(x₂) which is achieved at x₂ (Figure 1.1). Both the maximum and minimum are finite. The maximum (or minimum) may appear in two ways:

• (i) it may occur at a point inside [a, b], i.e., in (a, b) or
• (ii) it may occur at a boundary point, i.e., at either a or b.

Figure 1.1

In Figure 1.1(a) we see that possibility (i) occurs for both maximum and minimum, while in (b) possibility (ii) occurs for the maximum and possibility (i) occurs for the minimum.

If f is differentiable on (a, b) and the maximum occurs inside, then we have f′(x₂) = 0 so our method of proceeding in this case is to look at all x in (a, b) where f′(x) = 0. We call these the critical points of f. There is usually only a small number of critical points so we can evaluate f at these points. This takes care of all possibilities inside and since there are only two other points—the end points a and b—we can find f(a) and f(b) and locate the maximum by choosing which one of this small set of possibilities gives the largest value of the function.

If we are now considering a functio...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication
• Preface
• 1 Functions from R2 to R
• 2 Partial Derivatives
• 3 Critical Points
• 4 Maxima and Minima
• 5 Saddle Points
• 6 Sufficiently Regular Functions
• 7 Linear Approximation
• 8 Tangent Lines
• 9 Method and Examples of Lagrange Multipliers
• 10 Theory and Examples of Lagrange Multipliers
• 11 Tangent Planes
• 12 The Chain Rule
• 13 Directed Curves
• 14 Curvature
• 15 Quadratic Approximation
• 16 Vector Valued Differentiation
• 17 Complex Analysis
• 18 Line Integrals
• 19 The Fundamental Theorem of Calculus
• 20 Double Integrals
• 21 Coordinate Systems
• 22 Green’s Theorem
• Solutions
• Index