Multivariable Calculus

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CRC Press, Jan 29, 1982 - Mathematics - 546 pages
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Classroom-tested and lucidly written, Multivariable Calculus gives a thorough and rigoroustreatment of differential and integral calculus of functions of several variables. Designed as ajunior-level textbook for an advanced calculus course, this book covers a variety of notions,including continuity , differentiation, multiple integrals, line and surface integrals, differentialforms, and infinite series. Numerous exercises and examples throughout the book facilitatethe student's understanding of important concepts.The level of rigor in this textbook is high; virtually every result is accompanied by a proof. Toaccommodate teachers' individual needs, the material is organized so that proofs can be deemphasizedor even omitted. Linear algebra for n-dimensional Euclidean space is developedwhen required for the calculus; for example, linear transformations are discussed for the treatmentof derivatives.Featuring a detailed discussion of differential forms and Stokes' theorem, Multivariable Calculusis an excellent textbook for junior-level advanced calculus courses and it is also usefulfor sophomores who have a strong background in single-variable calculus. A two-year calculussequence or a one-year honor calculus course is required for the most successful use of thistextbook. Students will benefit enormously from this book's systematic approach to mathematicalanalysis, which will ultimately prepare them for more advanced topics in the field.
 

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Contents

SOME PRELIMINARIES 1 The Rudiments of Set Theory
1
Some Logic
7
Mathematical Induction
11
Inequalities and Absolute Value
12
Equivalence Relations 1 5 9 12
15
EUCLIDEAN SPACES AND LINEAR TRANSFORMATIONS
17
The Cartesian Plane
19
Euclidean Spaces
24
II
209
The Proof of Proposition 2 3
214
THE INVERSE AND IMPLICIT FUNCTION THEOREMS
218
The Inverse Function Theorem
223
The Proof of Theorem 2 2
228
I
230
II
235
INTEGRATION
240

The Norm in R
28
Linear Transformations between Euclidean Spaces
31
Matrices and Linear Transformations
35
Some Operations on Linear Transformations and Matrices
42
Composites of Linear Transformations and Matrix Multiplication
47
Matrix Multiplication
51
CONTINUOUS FUNCTIONS
58
THE DERIVATIVE
85
THE GEOMETRY OF EUCLIDEAN SPACES
110
HIGHER ORDER DERIVATIVES AND TAYLORS
138
Taylors Theorem for Functions of One Variable
145
Taylors Theorem for Functions of n Variables
154
COMPACT AND CONNECTED SETS
160
Continuous Functions on Compact Sets
169
A Characterization of Compact Sets
172
Uniform Continuity
175
Connected Sets
177
MAXIMA AND MINIMA
182
Quadratic Forms
188
Criteria for Local Maxima and Minima
194
I
199
The Method of Lagrange Multipliers
204
Properties of the Integral
249
The Integral of a Function of Two Variables
255
The Integral of a Function of n Variables
263
Properties of the Integral
271
The Proof of Theorem 6 2
280
ITERATED INTEGRALS AND THE FUBINI THEOREM
283
Integrals over Nonrectangular Regions
290
More Examples
296
The Proof of Fubinis Theorem
304
Differentiating under the Integral Sign
306
The Change of Variable Formula
309
LINE INTEGRALS
318
SURFACE INTEGRALS
351
DIFFERENTIAL FORMS
380
INTEGRATION OF DIFFERENTIAL FORMS
403
INFINITE SERIES
421
INFINITE SERIES OF FUNCTIONS
443
APPENDIX A Determinants
477
APPENDIX B The Proof of the General Inverse Function Theorem
486
SOLUTIONS TO SELECTED EXERCISES
494
INDEX
517
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