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| ISBN-13: | 9780486131955 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 09/12/2012 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 480 |
| File size: | 38 MB |
| Note: | This product may take a few minutes to download. |
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Advanced Calculus of Several Variables
By C. H. EDWARDS JR.
Dover Publications, Inc.
Copyright © 1973 Academic Press, Inc.All rights reserved.
ISBN: 978-0-486-13195-5
CHAPTER 1
Euclidean Space and Linear Mappings
Introductory calculus deals mainly with real-valued functions of a single variable, that is, with functions from the real line R to itself. Multivariable calculus deals in general, and in a somewhat similar way, with mappings from one Euclidean space to another. However a number of new and interesting phenomena appear, resulting from the rich geometric structure of n-dimensional Euclidean space Rn.
In this chapter we discuss Rn in some detail, as preparation for the development in subsequent chapters of the calculus of functions of an arbitrary number of variables. This generality will provide more clear-cut formulations of theoretical results, and is also of practical importance for applications. For example, an economist may wish to study a problem in which the variables are the prices, production costs, and demands for a large number of different commodities; a physicist may study a problem in which the variables are the coordinates of a large number of different particles. Thus a "real-life" problem may lead to a high-dimensional mathematical model. Fortunately, modern techniques of automatic computation render feasible the numerical solution of many high-dimensional problems, whose manual solution would require an inordinate amount of tedious computation.
1 THE VECTOR SPACE Rn
As a set, Rn is simply the collection of all ordered n-tuples of real numbers. That is,
Rn - {(x1, x2, ..., xn): each xi [member of] R}.
Recalling that the Cartesian product A × B of the sets A and B is by definition the set of all pairs (a, b) such that a [member of] A and b [member of] B, we see that Rn can be regarded as the Cartesian product set R × · · · × R (n times), and this is of course the reason for the symbol Rn.
The geometric representation of R3, obtained by identifying the triple (x1, x2, x3) of numbers with that point in space whose coordinates with respect to three fixed, mutually perpendicular "coordinate axes" are x1, x2, x3 respectively, is familiar to the reader (although we frequently write (x, y, z) instead of (x1, x2, x3) in three dimensions). By analogy one can imagine a similar geometric representation of Rn in terms of n mutually perpendicular coordinate axes in higher dimensions (however there is a valid question as to what "perpendicular" means in this general context; we will deal with this in Section 3).
The elements of Rn are frequently referred to as vectors. Thus a vector is simply an n-tuple of real numbers, and not a directed line segment, or equivalence class of them (as sometimes defined in introductory texts).
The set Rn is endowed with two algebraic operations, called vector addition and scalar multiplication (numbers are sometimes called scalars for emphasis). Given two vectors x = (x1, ..., xn) and y = (y1, ..., yn) in Rn, their sumx + y is defined by
x + y = (x1 + y1, ..., xn + yn)
that is, by coordinatewise addition. Given a [member of] R, the scalar multiple ax is defined by
ax = (ax1, ..., axn)
For example, if x = (1, 0, -2, 3) and y = (-2, 1, 4, -5) then x + y = (-1, 1, 2, -2) and 2x = (2, 0, -4, 6). Finally we write 0 = (0,...., 0) and -x = (-1)x, and use x - y as an abbreviation for x + (-y).
The familiar associative, commutative, and distributive laws for the real numbers imply the following basic properties of vector addition and scalar multiplication:
V1 x + (y + z) = (x + y) + z
V2 x + y = y + z
V3 x + 0 = x
V4 x + (-x) = 0
V5(ab)x = a(bx)
V6(a + b)x = ax + bx
V7a(x + y) = ax + ay
V8 lx = x
(Here x, y, z are arbitrary vectors in Rn, and a and b are real numbers.) V1–V8 are all immediate consequences of our definitions and the properties of R. For example, to prove V6, let x = (x1, ..., xn). Then
e1 = (1, 0, 0, ..., 0),
e2 = (0, 1, 0, ..., 0),
en = (0, 0, 0, ..., 0, 1),
The remaining verifications are left as exercises for the student.
A vector space is a set V together with two mappings V × V -> V and R × V -> V, called vector addition and scalar multiplication respectively, such that V1–V8 above hold for all x, y, z [member of] V and a, b, [member of] R (V3 asserts that there exists such 0 [member of] V that x + 0 = x for all x [member of] V, and V4 that, given x [member of] V, there exists -x [member of] V such that x + (-x) = 0). Thus V1–V8 may be summarized by saying that Rn is a vector space. For the most part, all vector spaces that we consider will be either Euclidean spaces, or subspaces of Euclidean spaces.
By a subspace of the vector space V is meant a subset W of V that is itself a vector space (with the same operations). It is clear that the subset W of V is a subspace if and only if it is "closed" under the operations of vector addition and scalar multiplication (that is, the sum of any two vectors in W is again in W, as is any scalar multiple of an element of W)—properties V1–V8 are then inherited by W from V. Equivalently, W is a subspace of V if and only if any linear combination of two vectors in W is also in W (why?). Recall that a linear combination of the vectors v1, ..., vk is a vector of the form a1v1 + ··· + akvk, where the v1, ..., vk [member of] Rn. The span of the vectors ai [member of] R is the set S of all linear combinations of them, and it is said that S is generated by the vectors v1, ..., vk.
Example 1 Rn is a subspace of itself, and is generated by the standard basis vectors
e1 = (1, 0, 0, ..., 0),
e2 = (0, 1, 0, ..., 0),
en = (0, 0, 0, ..., 0, 1),
since (x1, x2, ..., xn) = x1 e1 + x2 e2 + ··· + xn en. Also the subset of Rn consisting of the zero vector alone is a subspace, called the trivial subspace of Rn.
Example 2 The set of all points in Rn with last coordinate zero, that is, the set of all (x1, ..., xn-1, 0) [member of] Rn is a subspace of Rn which may be identified with Rn-1.
Example 3 Given (a1, a2 ..., an) [member of] R[n] the set of all (x1, x2 ..., xn) [member of] R[n] such that a1 x1 + ··· + an xn = 0 is a subspace of Rn (see Exercise 1.1).
Example 4 The span S of the vectors v1, ..., vk [member of] Rn is a subspace of Rn because, given elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and of S, and real numbers r and s, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Lines through the origin in R3 are (essentially by definition) those subspaces of R3 that are generated by a single nonzero vector, while planes through the origin in R3 are those subspaces of R3 that are generated by a pair of non-collinear vectors. We will see in the next section that every subspace V of Rn is generated by some finite number, at most Rn, of vectors; the dimension of the subspace V will be defined to be the minimal number of vectors required to generate V. Subspaces of Rn of all dimensions between 0 and n will then generalize lines and planes through the origin in R3.
Example 5 If V and W are subspaces of Rn, then so is their intersection V [intersection] W (the set of all vectors that lie in both V and W). See Exercise 1.2.
Although most of our attention will be confined to subspaces of Euclidean spaces, it is instructive to consider some vector spaces that are not subspaces of Euclidean spaces.
(Continues...)
Excerpted from Advanced Calculus of Several Variables by C. H. EDWARDS JR.. Copyright © 1973 Academic Press, Inc.. Excerpted by permission of Dover Publications, Inc..
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Table of Contents
Contents
Preface,I Euclidean Space and Linear Mappings,
1 The Vector Space Rn,
2 Subspaces of Rn,
3 Inner Products and Orthogonality,
4 Linear Mappings and Matrices,
5 The Kernel and Image of a Linear Mapping,
6 Determinants,
7 Limits and Continuity,
8 Elementary Topology of Rn,
II Multivariable Differential Calculus,
1 Curves in Rm,
2 Directional Derivatives and the Differential,
3 The Chain Rule,
4 Lagrange Multipliers and the Classification of Critical Points for Functions of Two Variables,
5 Maxima and Minima, Manifolds, and Lagrange Multipliers,
6 Taylor's Formula for Single-Variable Functions,
7 Taylor's Formula in Several Variables,
8 The Classification of Critical Points,
III Successive Approximations and Implicit Functions,
1 Newton's Method and Contraction Mappings,
2 The Multivariable Mean Value Theorem,
3 The Inverse and Implicit Mapping Theorems,
4 Manifolds in Rn,
5 Higher Derivatives,
IV Multiple Integrals,
1 Area and the 1-Dimensional Integral,
2 Volume and the n-Dimensional Integral,
3 Step Functions and Riemann Sums,
4 Iterated Integrals and Fubini's Theorem,
5 Change of Variables,
6 Improper Integrals and Absolutely Integrable Functions,
V Line and Surface Integrals; Differential Forms and Stokes' Theorem,
1 Pathlength and Line Integrals,
2 Green's Theorem,
3 Multilinear Functions and the Area of a Parallelepiped,
4 Surface Area,
5 Differential Forms,
6 Stokes' Theorem,
7 The Classical Theorems of Vector Analysis,
8 Closed and Exact Forms,
VI The Calculus of Variations,
1 Normed Vector Spaces and Uniform Convergence,
2 Continuous Linear Mappings and Differentials,
3 The Simplest Variational Problem,
4 The Isoperimetric Problem,
5 Multiple Integral Problems,
Appendix: The Completeness of R,
Suggested Reading,
Subject Index,