Summary
Offering readability, practicality and flexibility, Wade presents Fundamental Theorems from a practical viewpoint.Introduces central ideas of analysis in a onedimensional setting, then covers multidimensional theory. Offers separate coverage of topology and analysis. Numbers theorems, definitions and remarks consecutively. Uniform writing style and notation. Practical focus on analysis.For those interested in learning more about analysis.
Table of Contents
Preface 

xi  
Part I. ONEDIMENSIONAL THEORY 



1  (34) 


1  (12) 

1.2 WellOrdering Principle 


13  (5) 


18  (6) 

1.4 Functions, countability, and the algebra of sets 


24  (11) 


35  (23) 


35  (4) 


39  (6) 

2.3 BolzanoWeierstrass Theorem 


45  (4) 


49  (3) 

2.5 Limits supremum and infimum 


52  (6) 


58  (27) 


58  (8) 

3.2 Onesided limits and limits at infinity 


66  (5) 


71  (8) 


79  (6) 


85  (22) 


85  (7) 

4.2 Differentiability theorems 


92  (2) 


94  (8) 

4.4 Monotone functions and Inverse Function Theorem 


102  (5) 


107  (47) 


107  (10) 


117  (10) 

5.3 Fundamental Theorem of Calculus 


127  (9) 

5.4 Improper Riemann integration 


136  (6) 

5.5 Functions of bounded variation 


142  (5) 


147  (7) 

6 Infinite Series of Real Numbers 


154  (30) 


154  (6) 

6.2 Series with nonnegative terms 


160  (5) 


165  (8) 


173  (4) 


177  (4) 


181  (3) 

7 Infinite Series of Functions 


184  (41) 

7.1 Uniform convergence of sequences 


184  (8) 

7.2 Uniform convergence of series 


192  (5) 


197  (10) 


207  (12) 


219  (6) 
Part II. MULTIDIMENSIONAL THEORY 



225  (31) 


225  (9) 

8.2 Planes and linear transformations 


234  (8) 


242  (7) 

8.4 Interior, closure and boundary 


249  (7) 


256  (34) 


256  (7) 


263  (7) 


270  (7) 


277  (3) 


280  (10) 


290  (31) 


290  (6) 


296  (5) 

10.3 Interior, closure and boundary 


301  (5) 


306  (6) 


312  (4) 

10.6 Continuous functions 


316  (5) 

11 Differentiability on Rn 


321  (60) 

11.1 Partial derivatives and partial integrals 


321  (11) 

11.2 Definition of differentiability 


332  (7) 

11.3 Derivatives, differentials, and tangent planes 


339  (9) 


348  (4) 

11.5 Mean Value Theorem and Taylor's Formula 


352  (6) 

11.6 Inverse Function Theorem 


358  (11) 


369  (12) 


381  (68) 


381  (13) 

12.2 Riemann integration on Jordan regions 


394  (13) 


407  (13) 


420  (12) 


432  (9) 

12.6 Gamma function and volume 


441  (8) 

13 Fundamental Theorems of Vector Calculus 


449  (57) 


449  (12) 


461  (7) 


468  (11) 


479  (9) 

13.5 Theorems of Green and Gauss 


488  (8) 


496  (10) 


506  (32) 


506  (6) 

14.2 Summability of Fourier series 


512  (7) 

14.3 Growth of Fourier coefficients 


519  (7) 

14.4 Convergence of Fourier series 


526  (6) 


532  (6) 

15 Differentiable Manifolds 


538  (32) 

15.1 Differential forms on Rn 


538  (12) 

15.2 Differentiable manifolds 


550  (11) 

15.3 Stokes's Theorem on manifolds 


561  
Appendices 



570  (3) 


573  (4) 

C. Matrices and determinants 


577  (6) 


583  (4) 

E. Vector calculus and physics 


587  (3) 


590  (2) 
References 

592  (1) 
Answers and Hints to Exercises 

593  (18) 
Subject Index 

611  (13) 
Notation Index 

624  
Excerpts
This text provides a bridge from "sophomore" calculus to graduate courses that use analytic ideas, e.g., real and complex analysis, partial and ordinary differential equations, numerical analysis, fluid mechanics, and differential geometry. For a twosemester course, the first semester should end with Chapter 8. For a threequarter course, the second quarter should begin in Chapter 6 and end somewhere in the middle of Chapter 11. Our presentation is divided into two parts. The first half, Chapters 1 through 7 together with Appendices A and B, gradually introduces the central ideas of analysis in a onedimensional setting. The second half, Chapters 8 through 15 together with Appendices C through F, covers multidimensional theory. More specifically, Chapter 1 introduces the real number system as a complete, ordered field, Chapters 2 through 5 cover calculus on the real line; and Chapters 6 and 7 discuss infinite series, including uniform and absolute convergence. The first two sections of Chapter 8 give a short introduction to the algebraic, structure ofR n , including the connection between linear functions and matrices. At that point instructors have two options. They can continue covering Chapters 8 and 9 to explore topology and convergence in the concrete Euclidean space setting, or they can cover these same concepts in the abstract metric space setting (Chapter 10). Since either of these options provides the necessary foundation for the rest of the book, instructors are free to choose the approach that they feel best suits their aims. With this background material out of the way, Chapters 11 through 13 develop the, machinery and theory of vector calculus. Chapter 14 gives a short introduction to Fourier series, including summability and convergence bf Fourier series, growth of Fourier coefficients, and uniqueness of trigonometric series. Chapter 15 gives a short introduction to differentiable manifolds which culminates in a proof of Stokes's Theorem on differentiable manifolds. Separating the onedimensional from thendimensional material is not the most efficient way to present the material, but it does have two advantages. The more abstract, geometric concepts can be postponed until students have been given a thorough introduction to analysis on the real line. Students have two chances to master some of the deeper ideas of analysis (e.g., convergence of sequences, limits of functions, and uniform continuity). We have made this text flexible in another way by including core material and enrichment material. The core material, occupying fewer than 384 pages, can be covered easily in a oneyear course. The enrichment material is included for two reasons: Curious students can use it to delve deeper into the core material or as a jumping off place to pursue more general topics, and instructors can use it to supplement their course or to add variety from year to year. Enrichment material appears in enrichment sections, marked with a superscripte, or in core sections, where it is marked with an asterisk. Exercises that use enrichment material are also marked with an asterisk, and the material needed to solve them is cited in the Answers and Hints section. To make course planning easier, each enrichment section begins with a statement which indicates whether that section uses material from any other enrichment section. Since no core material depends on enrichment material, any of the latter can be skipped without loss in the integrity of the course. Most enrichment sections (5.5, 5.6, 6.5, 6.6, 7.5, 9.4, 11.6, 12.6, 14.1, 15.1) are independent and can be covered in any order after the core material that precedes them has been dealt with. Sections 9.5, 12.5, and 15.2 require 9.4, Section 15.3 requires 12.5, and Section 14.3 requires 5.5 only to establish Lemma 14.25. This result can easily be proved for continuously differentiable functions, th