Author | Price, G. Baley. author |
---|---|
Title | Multivariable Analysis [electronic resource] / by G. Baley Price |
Imprint | New York, NY : Springer New York, 1984 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-5228-3 |
Descript | XIV, 656 p. online resource |
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 -- Spernerโs Lemma and the Intermediate-Value Theorem -- 22. Introduction -- 23. Sperner Functions; Spernerโs 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. Spernerโs 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. Stokesโ 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. Integrals of 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 CauchyโRiemann Equations -- 65. The Stolz Condition -- 66. Integrals -- 67. A Special Case of Cauchyโs Integral Theorem -- 68. Cauchyโs Integral Formula -- 69. Taylor Series for a Differentiable Function -- 70. Complex-Valued Functions of Real Variables -- 71. Cauchyโs Integral Theorem -- II: Functions of Several Complex Variables -- 72. Derivatives -- 73. The CauchyโRiemann Equations and Differentiability -- 74. Cauchyโs 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. Sylvesterโs Theorem of 1839 and 1851 -- 82. The SylvesterโFranke Theorem -- 83. The BazinโReissโPicquet Theorem -- 84. Inner Products -- 85. Linearly Independent and Dependent Vectors; Rank of a Matrix -- 86. Schwarzโs Inequality -- 87. Hadamardโs Determinant Theorem -- Real Numbers, Euclidean Spaces, and Functions -- 88. Some Properties of the Real Numbers -- 93. The Nested Interval Theorem -- 94. The BolzanoโWeierstrass Theorem -- 95. The HeineโBorel Theorem -- 96. Functions -- 97. Cauchy Sequences -- References and Notes -- Index of Symbols