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 CauchyRiemann 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 CauchyRiemann 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 SylvesterFranke Theorem. - 83. The BazinReissPicquet 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 BolzanoWeierstrass Theorem. - 95. The HeineBorel Theorem. - 96. Functions. - 97. Cauchy Sequences. - References and Notes. - Index of Symbols. Language: English