Global analysis : differential forms in analysis, geometry and physics / Ilka Agricola, Thomas Friedrich ; translated by Andreas Nestke.

Format:

Book

Author/Creator:

Agricola, Ilka, 1973-

Contributor:

Friedrich, Thomas, 1949-

Series:

Graduate studies in mathematics 1065-7339 ; v. 52.

Graduate studies in mathematics, 1065-7339 ; v. 52

Standardized Title:

Globale Analysis. English

Language:

English

German

Subjects (All):

Differential forms.

Mathematical physics.

Physical Description:

xiii, 343 pages : illustrations ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2002]

Contents:

Chapter 1. Elements of Multilinear Algebra 1

Chapter 2. Differential Forms in R[superscript n] 11

2.1. Vector Fields and Differential Forms 11

2.2. Closed and Exact Differential Forms 18

2.3. Gradient, Divergence and Curl 23

2.4. Singular Cubes and Chains 26

2.5. Integration of Differential Forms and Stokes' Theorem 30

2.6. The Classical Formulas of Green and Stokes 35

2.7. Complex Differential Forms and Holomorphic Functions 36

2.8. Brouwer's Fixed Point Theorem 38

Chapter 3. Vector Analysis on Manifolds 47

3.1. Submanifolds of R[superscript n] 47

3.2. Differential Calculus on Manifolds 54

3.3. Differential Forms on Manifolds 67

3.4. Orientable Manifolds 69

3.5. Integration of Differential Forms over Manifolds 76

3.6. Stokes' Theorem for Manifolds 79

3.7. The Hedgehog Theorem (Hairy Sphere Theorem) 81

3.8. The Classical Integral Formulas 82

3.9. The Lie Derivative and the Interpretation of the Divergence 87

3.10. Harmonic Functions 94

3.11. The Laplacian on Differential Forms 100

Chapter 4. Pfaffian Systems 111

4.1. Geometric Distributions 111

4.2. The Proof of Frobenius' Theorem 116

4.3. Some Applications of Frobenius' Theorem 120

Chapter 5. Curves and Surfaces in Euclidean 3-Space 129

5.1. Curves in Euclidean 3-Space 129

5.2. The Structural Equations of a Surface 141

5.3. The First and Second Fundamental Forms of a Surface 147

5.4. Gaussian and Mean Curvature 155

5.5. Curves on Surfaces and Geodesic Lines 172

5.6. Maps between Surfaces 180

5.7. Higher-Dimensional Riemannian Manifolds 183

Chapter 6. Lie Groups and Homogeneous Spaces 207

6.1. Lie Groups and Lie Algebras 207

6.2. Closed Subgroups and Homogeneous Spaces 215

6.3. The Adjoint Representation 221

Chapter 7. Symplectic Geometry and Mechanics 229

7.1. Symplectic Manifolds 229

7.2. The Darboux Theorem 236

7.3. First Integrals and the Moment Map 238

7.4. Completely Integrable Hamiltonian Systems 241

7.5. Formulations of Mechanics 252

Chapter 8. Elements of Statistical Mechanics and Thermodynamics 271

8.1. Statistical States of a Hamiltonian System 271

8.2. Thermodynamical Systems in Equilibrium 283

Chapter 9. Elements of Electrodynamics 295

9.1. The Maxwell Equations 295

9.2. The Static Electromagnetic Field 299

9.3. Electromagnetic Waves 304

9.4. The Relativistic Formulation of the Maxwell Equations 311

9.5. The Lorentz Force 317.

Notes:

Includes bibliographical references (pages 333-335) and index.

ISBN:

0821829513

OCLC:

50143357