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| Preface | |
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| The Algebra and Topology of R[superscript n] | |
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| Review of Linear Algebra | |
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| Matrix Inversion and Determinants | |
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| Review of Topology in R[superscript n] | |
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| Compact Subspaces and Connected Subspaces of R[superscript n] | |
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| Differentiation | |
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| The Derivative | |
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| Continuously Differentiable Functions | |
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| The Chain Rule | |
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| The Inverse Function Theorem | |
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| The Implicit Function Theorem | |
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| Integration | |
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| The Integral over a Rectangle | |
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| Existence of the Integral | |
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| Evaluation of the Integral | |
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| The Integral over a Bounded Set | |
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| Rectifiable Sets | |
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| Improper Integrals | |
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| Change of Variables | |
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| Partitions of Unity | |
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| The Change of Variables Theorem | |
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| Diffeomorphisms in R[superscript n] | |
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| Proof of the Change of Variables Theorem | |
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| Applications of Change of Variables | |
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| Manifolds | |
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| The Volume of a Parallelopiped | |
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| The Volume of a Parametrized-Manifold | |
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| Manifolds in R[superscript n] | |
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| The Boundary of a Manifold | |
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| Integrating a Scalar Function over a Manifold | |
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| Differential Forms | |
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| Multilinear Algebra | |
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| Alternating Tensors | |
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| The Wedge Product | |
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| Tangent Vectors and Differential Forms | |
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| The Differential Operator | |
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| Application to Vector and Scalar Fields | |
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| The Action of a Differentiable Map | |
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| Stokes' Theorem | |
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| Integrating Forms over Parametrized-Manifolds | |
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| Orientable Manifolds | |
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| Integrating Forms over Oriented Manifolds | |
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| A Geometric Interpretation of Forms and Integrals | |
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| The Generalized Stokes' Theorem | |
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| Applications to Vector Analysis | |
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| Closed Forms and Exact Forms | |
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| The Poincare Lemma | |
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| The deRham Groups of Punctured Euclidean Space | |
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| Epilogue--Life Outside R[superscript n] | |
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| Differentiable Manifolds and Riemannian Manifolds | |
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| Bibliography | |
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| Index | |