Exterior Calculus: Theory and Cases

Exterior calculus is a branch of mathematics which involves differential geometry. In Exterior calculus the concept of differentiations is generalized to antisymmetric exterior derivatives and the notions of ordinary integration to differentiable manifolds of arbitrary dimensions. It therefore generalizes the fundamental theorem of calculus to Stokes' theorem. This textbook covers the fundamental requirements of exterior calculus in curricula for college students in mathematics and engineering programs. Chapters start from Heaviside-Gibbs algebra, and progress to different concepts in Grassman algebra. The final section of the book covers applications of exterior calculus with solutions. Readers will find a concise and clear study of vector calculus and differential geometry, along with several examples and exercises. The solutions to the exercises are also included at the end of the book. This is an ideal book for students with a basic background in mathematics who wish to learn about exterior calculus as part of their college curriculum and equip themselves with the knowledge to apply relevant theoretical concepts in practical situations. Cover Title Copyright End User License Agreement Contents Foreword Preface Acknowledgements Dedication List of Credits List of Symbols Part I. Heaviside-Gibbs Algebra Chapter 1. Vector Algebra on R2 and R3 1.1. Normed Vector Space: V(F) 1.2. Basic operators 1.3. Vector-Valued Functions 1.4. Vector Theorems 1.5. Remarks 1.6. Exercises Part II. Grassmann Algebra Chapter 2. Geometric Algebra on G2 2.1. Geometric Algebra on G2 2.2. Properties on G2 2.3. Reflections and Rotations on a Plane 2.4. Geometric Representation of a Line on R2 2.5. Geometric Representation of a Plane on R2 2.6. Remarks 2.7. Exercises Chapter 3. Geometric Algebra on G3 3.1. Geometric Algebra on G3 3.2. Properties on G3 3.3. Reflections and Rotations in Space 3.4. Geometric Representation of a Line on R3 3.5. Geometric Representation of a Plane on R3 3.6. Remarks 3.7. Exercises Chapter 4. Geometric Algebra on Gn 4.1. Preliminaries 4.2. Geometric Algebra on Gn 4.3. Properties on Gn 4.4. Reflections and Rotations on Gn 4.5. Analytical Representation of a Line in Rn 4.6. Analytical Representation of a Plane in Rn 4.7. Remarks 4.8. Exercises Chapter 5. Differentiation 5.1. Differential of a Function 5.2. Differential Forms 5.3. Differentiation of Forms 5.4. Remarks 5.5. Exercises Chapter 6. Integration 6.1. Preliminaries 6.2. Integration of 0−Forms 6.3. Integration of 1−Forms 6.4. Integration of 2−Forms 6.5. Integration of 3−Forms 6.6. Integration of k−Forms 6.7. Remarks 6.8. Exercises Chapter 7. Fundamental Theorem of Calculus 7.1. Preliminaries 7.2. Green Theorem 7.3. Stokes’ Theorem 7.4. Gauss’ Theorem 7.5. Fundamental Theorem of Calculus 7.6. Remarks 7.7. Exercises Part III. Applications Chapter 8. Applications 8.1. Mathematical Epidemiology 8.2. Structural Proteomics 8.3. Ampere’s Law 8.4. Remarks Chapter 9. Solutions Solution 1.1. Solution 1.5. Solution 2.1. Solution 2.17. Solution 3.1. Solution 3.17. Solution 4.1. Solution 4.13. Solution 5.1. Solution 5.8. Solution 5.10. Solution 6.1. Solution 6.11. Solution 7.1. Solution 7.5. Solution 7.6. Solution 7.8. Solution 7.9. Solution 7.10. References [20] Subject Index G-Z