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PAUL F.A. BARTHA, PhD, is Associate Professor in the Department of Philosophy at The University of British Columbia, Canada. He has authored or coauthored journal articles on topics such as probability and symmetry, probabilistic paradoxes, and the general philosophy of science.
DZUNG MINH HA, PhD, is Associate Professor in the Department of Mathematics at Ryerson University, Canada. Dr. Ha focuses his research in the areas of ergodic theory and operator theory.
Preface | p. ix |
Background Material | |
Sets and Functions | p. 3 |
Sets in General | p. 3 |
Sets of Numbers | p. 5 |
Functions | p. 9 |
Real Numbers | p. 13 |
Review of the Order Relations | p. 13 |
Completeness of Real Numbers | p. 15 |
Sequences of Real Numbers | p. 16 |
Subsequences | p. 17 |
Series of Real Numbers | p. 21 |
Intervals and Connected Sets | p. 24 |
Vector Functions | p. 27 |
Vector Spaces: The Basics | p. 27 |
Bilinear Functions | p. 33 |
Multilinear functions | p. 35 |
Inner Products | p. 38 |
Orthogonal Projections | p. 40 |
Spectral Theorem | p. 42 |
Differentiation | |
Normed Vector Spaces | p. 45 |
Preliminaries | p. 45 |
Convergence in Normed Spaces | p. 47 |
Norms of Linear and Multilinear Transformations | p. 50 |
Continuity in Normed Spaces | p. 51 |
Topology of Normed Spaces | p. 54 |
Derivatives | p. 63 |
Functions of a Real Variable | p. 63 |
Differentiable Functions | p. 70 |
Existence of Derivatives | p. 73 |
Partial Derivatives | p. 74 |
Rules of Differentiation | p. 78 |
Differentiation of Products | p. 80 |
Diffeomorphisms and Manifolds | p. 83 |
The Inverse Function Theorem | p. 83 |
Graphs | p. 86 |
Manifolds in Parametric Representations | p. 87 |
Manifolds in Implicit Representations | p. 89 |
Differentiation on Manifolds | p. 91 |
Higher-Order Derivatives | p. 95 |
Definitions | p. 95 |
Change of Order in Differentiation | p. 95 |
Sequences of Polynomials | p. 95 |
Local Extremal Values | p. 95 |
Integration | |
Multiple Integrals | p. 99 |
Jordan Sets and Volume | p. 99 |
Integrals | p. 104 |
Images of Jordan Sets | p. 109 |
Change of Variables | p. 111 |
Integration on Manifolds | p. 115 |
Euclidean Volumes | p. 115 |
Integration on Manifolds | p. 116 |
Oriented Manifolds | p. 122 |
Integrals of Vector Fields | p. 124 |
Integrals of Tensor Fields | p. 128 |
Stokes' Theorem | p. 131 |
Basic Stokes' Theorem | p. 131 |
Flows | p. 132 |
Flux and Change of Volume in a Flow | p. 136 |
Exterior Derivatives | p. 138 |
Regular and Almost Regular Sets | p. 141 |
Stokes' theorem on Manifolds | p. 148 |
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