Functions of several variables

144 Pages·1981·5.089 MB·English

by  B. D. Craven (auth.)

Functions of
several variables
Functions of
several variables
B.D. CRAVEN
Reader in Mathematics,
University 0/ Melbourne
LONDON AND NEW YORK
CHAPMAN AND HALL
First published 1981
by Chapman and Hall Ltd
11 New Fetter Lane, London EC4P 4EE
Published in the USA by
Chapman and Hall
in association with Methuen, Inc.
733 Third Avenue, New York NY 10017
© 1981 B.D. Craven
ISBN-13: 978-0-412-23340-1 e-ISBN -13: 978-94-010-9347-7
DOT: 10.1007/978-94-010-9347-7
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British Library Cataloguing in Publication Data
Craven, B.D.
Functions of several variables.
1. Functions of several real variables
I. Title
515.8'4 QA331.5 80-42132
Contents
PREFACE vii
1. DIFFERENTIABLE FUNCTIONS 1
1.1 Introduction 1
1.2 Linear part of a function 3
1.3 Vector viewpoint 5
1.4 Directional derivative 6
1.5 Tangent plane to a surface 7
1.6 Vector functions 8
1.7 Functions of functions 10
2. CHAIN RULE AND INVERSE FUNCTION THEOREM 13
2.1 Norms 13
2.2 Frechet derivatives 17
2.3 Chain rule 22
2.4 Inverse function theorem 25
2.5 Implicit functions 29
2.6 Functional dependence 34
2.7 Higher derivatives 35
3. MAXIMA AND MINIMA 41
3.1 Extrema and stationary points 41
3.2 Constrained minima and Lagrange multipliers 49
v
vi Contents
3.3 Discriminating constrained stationary points 53
3.4 Inequality constraints 56
3.5 Discriminating maxima and minima with inequality
constraints 62
Further reading 66
4. INTEGRATING FUNCTIONS OF SEVERAL VARIABLES 67
4.1 Basic ideas of integration 67
4.2 Double integrals 71
4.3 Length, area and volume 79
4.4 Integrals over curves and surfaces 84
4.5 Differential forms 95
4.6 Stokes's theorem 101
Further reading 111
APPENDICES 112
A. Background required in linear algebra and elementary
calculus 112
B. Compact sets, continuous functions and partitions of unity 117
C. Answers to selected exercises 120
INDEX (including table of some special symbols) 135
Preface
This book is aimed at mathematics students, typically in the second
year of a university course. The first chapter, however, is suitable for
first-year students. Differentiable functions are treated initially from
the standpoint of approximating a curved surface locally by a fiat
surface. This enables both geometric intuition, and some elementary
matrix algebra, to be put to effective use. In Chapter 2, the required
theorems - chain rule, inverse and implicit function theorems, etc. -
are stated, and proved (for n variables), concisely and rigorously.
Chapter 3 deals with maxima and minima, including problems with
equality and inequality constraints. The chapter includes criteria for
discriminating between maxima, minima and saddlepoints for
constrained problems; this material is relevant for applications, but
most textbooks omit it. In Chapter 4, integration over areas, volumes,
curves and surfaces is developed, and both the change-of-variable
formula, and the Gauss-Green-Stokes set of theorems are
obtained. The integrals are defined with approximative sums (ex
pressed concisely by using step-functions); this preserves some
geometrical (and physical) concept of what is happening. Consequent
on this, the main ideas of the 'differential form' approach are
presented, in a simple form which avoids much of the usual length
and complexity. Many examples and exercises are included.
The background assumed is elementary calculus of functions of
one real variable, and some matrix algebra. In modern syllabuses,
vii
viii Preface
this material is taught in schools, or at the beginning of a university
course, and so the students will already know it. However, the
essential material is summarized in an appendix, for those who need
it.
Why is another textbook written? Existing textbooks on 'functions
of several variables' are often too advanced for a student beginning
this topic. Any 'advanced calculus' textbook has a 'chapter n' on
functions of several variables; but this is only accessible to students
who have worked through the preceding many chapters, and thus
only if the same book has been prescribed for various earlier courses.
To fill this gap, a concise, and inexpensive, text is offered, specifically
on functions of several variables.
1. Differentiable functions
1.1 INTRODUCTION
Letfbe a real function of two real variables, x and y say. This means
that, to each pair x, y in some region in the plane, there corresponds
a real number f(x, y). This number may, but need not, be given by
a formula, e.g.
f(x,y) = x2-3xy_y3 or f(x,y) = cos(2x+3y).
If x, y, z are Cartesian coordinates in three-dimensional space, then
the equation z = f(x, y) represents, geometrically, a surface. Some
examples are as follows.
The unit sphere, namely the sphere with centre (0, 0, 0) and radius
1, has equation x2+ y2 + Z2 = 1. Solving this equation for z gives
two values: z = ±(1-x2_y2)1/2. If we pick out the positive square
root, the equation
z = +(l-x2_y2)1/2,
of which the right side is a function of x and y, represents the hemi
sphere which lies above the x, y coordinate plane. The region in the
x, y plane for which this function is defined is the disc {(x, y) : x2 +
y2 ~ I}. (Note that here a circle means a curve, the plane region
inside it is a disc. Similarly a sphere means the surface; the 'solid
sphere' is called a ball.)
2 Functions of Several Variables
Consider the parabola in the z, x plane, given by the equation
z = x2• If this parabola is rotated about the z-axis, it traces out a
surface, called a paraboloid of revolution. The equation of this para
boloid is obtained by replacing X2, which is the squared distance
from the z-axis of a point in the z, x plane, by x2+ y2, which is the
squared distance from the z-axis of a point (x, y, z) in three
dimensional space. So the paraboloid has the equation
Z=X2+y2.
The function of x and y occurring here is defined for all values of
x and y. The paraboloid is a bowl-shaped surface (see Fig. 1.1).
z
z
--------~~------~x
(8) (b)
Figure 1.1 Parabola and paraboloid
Consider the sphere X2 + y2 + Z2 = 1, and stretch it in the direction
of the x-axis by the ratio a, in the direction of the y-axis by the ratio
b, and in the direction of the z-axis by the ratio c; thus the point
(x, y, z) moves to (ax, by, cz). The sphere is thus stretched to a
surface whose equation is
This surface is called an ellipsoid with semiaxes a, b, c (see Fig. 1.2).
If this equation is solved for z, and the positive square root taken,
this gives
Differentiable functions 3
The right side of this equation defines a function of x and y, for the
region in the x, y plane inside the ellipse (x/a) 2 +( y/b)2 = 1.
z
Figure 1.2 Ellipsoid
These examples have some properties in common. An equation
z = f(x, y) usually describes only part of the geometrical surface,
such as a sphere or ellipsoid. The surface consists of one untorn
piece; this means that the points (x, y,f(x, y» and (xo, yo,f(xo, Yo))
on the surface are close together if the points (x, y) and (xo, Yo) are
close together in the plane; this expresses the concept that I is a
continuous function of its variables x, y. Any small part of the surface
(near one point) is in some sense 'nearly flat', even though the sur
face is actually curved. Thus a man standing on the (spherical)
earth sees his local neighbourhood as roughly flat.
These ideas must now be made precise. We start with the concept
of 'nearly flat'.
1.2 LINEAR PART OF A FUNCTION
Consider the function J, where
lex, y) = x2-3xy-y3;
and try to approximate I(x, y) near the point (x, y) = (2, - 1) by a
simpler function. To do this, set x = 2+g and y = -I +1/. Then
I(x, y) = f(2+g, -1 +1/) = (4+4g+g2)-3( -2-g+2,.,+g1/)
-( -I +371-31/2+1/3)
= 11 +(7g-91/)+(g2_3g1/+31/2+1/3). (1.2.1)