Exterior analysis using applications of differential forms
Exterior analysis uses differential forms (a mathematical technique) to analyze curves, surfaces, and structures. Exterior Analysis is a first-of-its-kind resource that uses applications of differential forms, offering a mathematical approach to solve problems in defining a precise measurement to e...
| Autor principal: | |
|---|---|
| Formato: | Libro electrónico |
| Idioma: | Inglés |
| Publicado: |
Waltham, MA :
Academic Press
2013.
|
| Edición: | 1st edition |
| Colección: | Gale eBooks
|
| Materias: | |
| Ver en Biblioteca Universitat Ramon Llull: | https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009627715606719 |
Tabla de Contenidos:
- Front Cover; Exterior Analysis: Using Applications of Differential Forms; Copyright Page; Table of Contents; Preface; Chapter I: Exterior Algebra; 1.1. Scope of the Chapter; 1.2. Linear Vector Spaces; 1.3. Multilinear Functionals; 1.4. Alternating k-Linear Functionals; 1.5. Exterior Algebra; 1.6. Rank of an Exterior Form; I. Exercises; Chapter II: Differentiable Manifolds; 2.1. Scope of the Chapter; 2.2. Differentiable Manifolds; 2.3. Differentiable Mappings; 2.4. Submanifolds; 2.5. Differentiable Curves; 2.6. Vectors. Tangent Spaces; 2.7. Differential of a Map Between Manifolds
- 2.8. Vector Fields. Tangent Bundle2.9. Flows Over Manifolds; 2.10. Lie Derivative; 2.11. Distributions. the Frobenius Theorem; II. Exercises; Chapter III: Lie Groups; 3.1. Scope of the Chapter; 3.2. Lie Groups; 3.3. Lie Algebras; 3.4. Lie Group Homomorphisms; 3.5. One-Parameter Subgroups; 3.6. Adjoint Representation; 3.7. Lie Transformation Groups; III. Exercises; Chapter IV: Tensor Fields on Manifolds; 4.1. Scope of the Chapter; 4.2. Cotangent Bundle; 4.3. Tensor Fields; IV. Exercises; Chapter V: Exterior Differential Forms; 5.1. Scope of the Chapter; 5.2. Exterior Differential Forms
- 5.3. Some Algebraic Properties5.4. Interior Product; 5.5. Bases Induced By the Volume Form; 5.6. Ideals of the Exterior Algebra Λ(M); 5.7. Exterior Forms Under Mappings; 5.8. Exterior Derivative; 5.9. Riemannian Manifolds. Hodge Dual; 5.10. Closed Ideals; 5.11. Lie Derivatives of Exterior Forms; 5.12. Isovector Fields of Ideals; 5.13. Exterior Systems and Their Solutions; 5.14. Forms Defined on a Lie Group; V. Exercises; Chapter VI: Homotopy Operator; 6.1. Scope of the Chapter; 6.2. Star-Shaped Regions; 6.3. Homotopy Operator; 6.4. Exact and Antiexact Forms; 6.5. Change of Centre
- 6.6. Canonical Forms of 1-Forms, Closed 2- Forms6.7. An Exterior Differential Equation; 6.8. A System of Exterior Differential Equations; VI. Exercises; Chapter VII: Linear Connections; 7.1. Scope of the Chapter; 7.2. Connections on Manifolds; 7.3. Cartan Connection; 7.4. Levi-Civita Connection; 7.5. Differential Operators; VII. Exercises; Chapter VIII: Integration of Exterior Forms; 8.1. Scope of the Chapter; 8.2. Orientable Manifolds; 8.3. Integration of Forms in the Euclidean Space; 8.4. Simplices and Chains; 8.5. Integration of Forms on Manifolds; 8.6. The Stokes Theorem
- 8.7. Conservation Laws8.8. The Cohomology of De Rham; 8.9. Harmonic Forms. Theory of Hodge-De Rham; 8.10. Poincare Duality; VIII. Exercises; Chapter IX: Partial Differential Equations; 9.1. Scope of the Chapter; 9.2. Ideals Formed By Differential Equations; 9.3. Isovector Fields of the Contact Ideal; 9.4. Isovector Fields of Balance Ideals; 9.5. Similarity Solutions; 9.6. The Method of Generalised Characteristics; 9.7. Horizontal Ideals and Their Solutions; 9.8. Equivalence Transformations; IX. Exercises; Chapter X: Calculus of Variations; 10.1. Scope of the Chapter
- 10.2. Stationary Functionals
