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An introduction to analysis

Author: W R Wade
Publisher: Upper Saddle River, NJ : Prentice Hall, ©2000.
Edition/Format:   Print book : English : 2nd edView all editions and formats
Summary:

Provides a bridge from sophomore calculus to graduate courses which use analytic ideas such as real and complex analysis, partial and ordinary differential equations, numerical analysis, fluid  Read more...

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Document Type: Book
All Authors / Contributors: W R Wade
ISBN: 0130144096 9780130144096
OCLC Number: 41165400
Description: xii, 611 pages : illustrations ; 25 cm
Contents: Part I. One-Dimensional Theory --
1 The Real Number System 1 --
1.1 Ordered field axioms 1 --
1.2 The Well-Ordering Principle 13 --
1.3 The Completeness Axiom 18 --
1.4 Functions, countability, and the algebra of sets 24 --
2 Sequences in R 34 --
2.1 Limits of sequences 34 --
2.2 Limit theorems 38 --
2.3 The Bolzano-Weierstrass Theorem 44 --
2.4 Cauchy sequences 48 --
2.5 Limits supremum and infimum 51 --
3 Continuity on R 57 --
3.1 Two-sided limits 57 --
3.2 One-sided limits and limits at infinity 65 --
3.3 Continuity 70 --
3.4 Uniform continuity 79 --
4 Differentiability on R 84 --
4.1 The derivative 84 --
4.2 Differentiability theorems 91 --
4.3 The Mean Value Theorem 93 --
4.4 Monotone functions and the Inverse Function Theorem 101 --
5 Integrability on R 106 --
5.1 The Riemann integral 106 --
5.2 Riemann sums 115 --
5.3 The Fundamental Theorem of Calculus 125 --
5.4 Improper Riemann integration 134 --
5.5 Functions of bounded variation 140 --
5.6 Convex functions 145 --
6 Infinite Series of Real Numbers 152 --
6.2 Series with nonnegative terms 158 --
6.3 Absolute convergence 163 --
6.4 Alternating series 171 --
6.5 Estimation of series 175 --
6.6 Additional tests 179 --
7 Infinite Series of Functions 182 --
7.1 Uniform convergence of sequences 182 --
7.2 Uniform convergence of series 190 --
7.3 Power series 195 --
7.4 Analytic functions 205 --
7.5 Applications 217 --
Part II. Multidimensional Theory --
8 Euclidean Spaces 223 --
8.1 Algebraic structure 223 --
8.2 Limits of sequences 233 --
8.3 Limits of functions 236 --
8.4 The total derivative 245 --
9 Topology of Euclidean Spaces 253 --
9.1 Interior, closure, boundary 253 --
9.2 Compact sets 261 --
9.3 Connected sets 265 --
9.4 Continuous functions 268 --
9.5 Applications 274 --
10 Metric Spaces 284 --
10.2 Limits of functions 290 --
10.3 Interior, closure, boundary 295 --
10.4 Compact sets 300 --
10.5 Connected sets 306 --
10.6 Continuous functions 310 --
11 Differentiability on R[superscript n] 315 --
11.1 Partial derivatives and partial integrals 315 --
11.2 The definition of differentiability 325 --
11.3 Differentiability theorems 335 --
11.4 The Mean Value Theorem and Taylor's Formula 341 --
11.5 The Inverse Function Theorem 350 --
11.6 Optimization 360 --
12 Integration on R[superscript n] 372 --
12.1 Jordan regions 372 --
12.2 Riemann integration on Jordan regions 382 --
12.3 Iterated integrals 393 --
12.4 Change of variables 406 --
12.5 Partitions of unity 419 --
12.6 The gamma function and volume 429 --
13 Fundamental Theorems of Vector Calculus 437 --
13.1 Curves 437 --
13.2 Oriented curves 449 --
13.3 Surfaces 456 --
13.4 Oriented surfaces 467 --
13.5 Theorems of Green and Gauss 475 --
13.6 Stokes's Theorem 484 --
14 Fourier Series 493 --
14.2 Summability of Fourier series 499 --
14.3 Growth of Fourier coefficients 506 --
14.4 Convergence of Fourier series 513 --
14.5 Uniqueness 519 --
15 Differentiable Manifolds 525 --
15.1 Differential forms on R[superscript n] 525 --
15.2 Differentiable manifolds 537 --
15.3 Stokes's Theorem on manifolds 548 --
A. Algebraic laws 557 --
B. Trigonometry 560 --
C. Matrices and determinants 564 --
D. Quadric surfaces 570 --
E. Vector calculus and physics 574 --
F. Equivalence relations 577.
Responsibility: William R. Wade.

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