Multivariable Analysis

This book contains an introduction to the theory of functions, with emphasis on functions of several variables. The central topics are the differentiation and integration of such functions. Although many of the topics are familiar, the treatment is new; the book developed from a new approach to the theory of differentiation. Iff is a function of two real variables x and y, its deriva- tives at a point Po can be approximated and found as follows. Let PI' P2 be two points near Po such that Po, PI, P2 are not on a straight line. The linear function of x and y whose values at Po, PI' P2 are equal to those off at these points approximates f near Po; determinants can be used to find an explicit representation of this linear function (think of the equation of the plane through three points in three-dimensional space). The (partial) derivatives of this linear function are approximations to the derivatives of f at Po; each of these (partial) derivatives of the linear function is the ratio of two determinants. The derivatives off at Po are defined to be the limits of these ratios as PI and P2 approach Po (subject to an important regularity condition). This simple example is only the beginning, but it hints at a m theory of differentiation for functions which map sets in IRn into IR which is both general and powerful, and which reduces to the standard theory of differentiation in the one-dimensional case.

Table of Contents:
Differentiate Functions and Their Derivatives.- 1. Introduction.- 2. Definitions and Notation.- 3. Elementary Properties of Differentiable Functions.- 4. Derivatives of Composite Functions.- 5. Compositions with Linear Functions.- 6. Classes of Differentiable Functions.- 7. The Derivative as an Operator.- Uniform Differentiability and Approximations; Mappings.- 8. Introduction.- 9. The Mean-Value Theorem: A Generalization.- 10. Uniform Differentiability.- 11. Approximation of Increments of Functions.- 12. Applications: Theorems on Mappings.- Simplexes, Orientations, Boundaries, and Simplicial Subdivisions.- 13. Introduction.- 14. Barycentric Coordinates, Convex Sets, and Simplexes.- 15. Orientation of Simplexes.- 16. Complexes and Chains.- 17. Boundaries of Simplexes and Chains.- 18. Boundaries in a Euclidean Complex.- 19. Affine and Barycentric Transformations.- 20. Three Theorems on Determinants.- 21. Simplicial Subdivisions.- Sperner’s Lemma and the Intermediate-Value Theorem.- 22.Introduction.- 23. Sperner Functions; Sperner’s Lemma.- 24. A Special Class of Sperner Functions.- 25. Properties of the Degree of a Function.- 26. The Degree of a Curve.- 27. The Intermediate-Value Theorem.- 28. Sperner’s Lemma Generalized.- 29. Generalizations to Higher Dimensions.- The Inverse-Function Theorem.- 30. Introduction.- 31. The One-Dimensional Case.- 32. The First Step: A Neighborhood is Covered.- 33. The Inverse-Function Theorem.- Integrals and the Fundamental Theorem of the Integral Calculus.- 34. Introduction.- 35. The Riemann Integral in ?n.- 36. Surface Integrals in ?n.- 37. Integrals on an m-Simplex in ?n.- 38. The Fundamental Theorem of the Integral Calculus.- 39. The Fundamental Theorem of the Integral Calculus for Surfaces.- 40. The Fundamental Theorem on Chains.- 41. Stokes’ Theorem and Related Results.- 42. The Mean-Value Theorem.- 43. An Addition Theorem for Integrals.- 44. Integrals Which Are Independent of the Path.- 45. The Area of a Surface.- 46. Integralsof Uniformly Convergent Sequences of Functions.- Zero Integrals, Equal Integrals, and the Transformation of Integrals.- 47. Introduction.- 48. Some Integrals Which Have the Value Zero.- 49. Integrals Over Surfaces with the Same Boundary.- 50. Integrals on Affine Surfaces with the Same Boundary.- 51. The Change-of-Variable Theorem.- The Evaluation of Integrals.- 52. Introduction.- 53. Definitions.- 54. Functions and Primitives.- 55. Integrals and Evaluations.- 56. The Existence of Primitives: Derivatives of a Single Function.- 57. The Existence of Primitives: The General Case.- 58. Iterated Integrals.- The Kronecker Integral and the Sperner Degree.- 59. Preliminaries.- 60. The Area and the Volume of a Sphere.- 61. The Kronecker Integral.- 62. The Kronecker Integral and the Sperner Degree.- Differentiable Functions of Complex Variables.- 63. Introduction.- I: Functions of a Single Complex Variable.- 64. Differentiable Functions; The Cauchy—Riemann Equations.- 65. The Stolz Condition.- 66. Integrals.- 67. A Special Case of Cauchy’s Integral Theorem.- 68. Cauchy’s Integral Formula.- 69. Taylor Series for a Differentiable Function.- 70. Complex-Valued Functions of Real Variables.- 71. Cauchy’s Integral Theorem.- II: Functions of Several Complex Variables.- 72. Derivatives.- 73. The Cauchy—Riemann Equations and Differentiability.- 74. Cauchy’s Integral Theorem.- Determinants.- 75. Introduction to Determinants.- 76. Definition of the Determinant of a Matrix.- 77. Elementary Properties of Determinants.- 78. Definitions and Notation.- 79. Expansions of Determinants.- 80. The Multiplication Theorems.- 81. Sylvester’s Theorem of 1839 and 1851.- 82. The Sylvester—Franke Theorem.- 83. The Bazin—Reiss—Picquet Theorem.- 84. Inner Products.- 85. Linearly Independent and Dependent Vectors; Rank of a Matrix.- 86. Schwarz’s Inequality.- 87. Hadamard’s Determinant Theorem.- Real Numbers, Euclidean Spaces, and Functions.- 88. Some Properties of the Real Numbers.- 93. The Nested Interval Theorem.- 94. The Bolzano—Weierstrass Theorem.- 95. The Heine—Borel Theorem.- 96. Functions.- 97. Cauchy Sequences.- References and Notes.- Index of Symbols.