Preface 

ix  
Part I. ONEDIMENSIONAL THEORY 



1  (33) 


1  (12) 

The WellOrdering Principle 


13  (5) 


18  (6) 

Functions, countability, and the algebra of sets 


24  (10) 


34  (23) 


34  (4) 


38  (6) 

The BolzanoWeierstrass Theorem 


44  (4) 


48  (3) 

Limits supremum and infimum 


51  (6) 


57  (27) 


57  (8) 

Onesided limits and limits at infinity 


65  (5) 


70  (9) 


79  (5) 


84  (22) 


84  (7) 

Differentiability theorems 


91  (2) 


93  (8) 

Monotone functions and the Inverse Function Theorem 


101  (5) 


106  (46) 


106  (9) 


115  (10) 

The Fundamental Theorem of Calculus 


125  (9) 

Improper Riemann integration 


134  (6) 

Functions of bounded variation 


140  (5) 


145  (7) 

Infinite Series of Real Numbers 


152  (30) 


152  (6) 

Series with nonnegative terms 


158  (5) 


163  (8) 


171  (4) 


175  (4) 


179  (3) 

Infinite Series of Functions 


182  (41) 

Uniform convergence of sequences 


182  (8) 

Uniform convergence of series 


190  (5) 


195  (10) 


205  (12) 


217  (6) 
Part II. MULTIDIMENSIONAL THEORY 



223  (30) 


223  (10) 


233  (3) 


236  (9) 


245  (8) 

Topology of Euclidean Spaces 


253  (31) 

Interior, closure, boundary 


253  (8) 


261  (4) 


265  (3) 


268  (6) 


274  (10) 


284  (31) 


284  (6) 


290  (5) 

Interior, closure, boundary 


295  (5) 


300  (6) 


306  (4) 


310  (5) 


315  (57) 

Partial derivatives and partial integrals 


315  (10) 

The definition of differentiability 


325  (10) 

Differentiability theorems 


335  (6) 

The Mean Value Theorem and Taylor's Formula 


341  (9) 

The Inverse Function Theorem 


350  (10) 


360  (12) 


372  (65) 


372  (10) 

Riemann integration on Jordan regions 


382  (11) 


393  (13) 


406  (13) 


419  (10) 

The gamma function and volume 


429  (8) 

Fundamental Theorems of Vector Calculus 


437  (56) 


437  (12) 


449  (7) 


456  (11) 


467  (8) 

Theorems of Green and Gauss 


475  (9) 


484  (9) 


493  (32) 


493  (6) 

Summability of Fourier series 


499  (7) 

Growth of Fourier coefficients 


506  (7) 

Convergence of Fourier series 


513  (6) 


519  (6) 


525  (32) 


525  (12) 


537  (11) 

Stokes's Theorem on manifolds 


548  (9) 
Appendices 

557  (22) 


557  (3) 


560  (4) 

C. Matrices and determinants 


564  (6) 


570  (4) 

E. Vector calculus and physics 


574  (3) 


577  (2) 
References 

579  (1) 
Answers and Hints to Exercises 

580  (17) 
Subject Index 

597  (13) 
Notation Index 

610  