You are previewing A Course in Mathematics for Students of Physics.
O'Reilly logo
A Course in Mathematics for Students of Physics

Book Description

This textbook, available in two volumes, has been developed from a course taught at Harvard over the last decade. The course covers principally the theory and physical applications of linear algebra and of the calculus of several variables, particularly the exterior calculus. The authors adopt the 'spiral method' of teaching, covering the same topic several times at increasing levels of sophistication and range of application. Thus the reader develops a deep, intuitive understanding of the subject as a whole, and an appreciation of the natural progression of ideas. Topics covered include many items previously dealt with at a much more advanced level, such as algebraic topology (introduced via the analysis of electrical networks), exterior calculus, Lie derivatives, and star operators (which are applied to Maxwell's equations and optics). This then is a text which breaks new ground in presenting and applying sophisticated mathematics in an elementary setting. Any student, interpreted in the widest sense, with an interest in physics and mathematics, will gain from its study.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Contents of Volume 2
  7. Preface
  8. 1. Linear transformations of the plane
    1. 1.1 Affine planes and vector spaces
    2. 1.2 Vector spaces and their affine spaces
    3. 1.3 Functions and affine functions
    4. 1.4 Euclidean and affine transformations
    5. 1.5 Linear transformations
    6. 1.6 The matrix of a linear transformation
    7. 1.7 Matrix multiplication
    8. 1.8 Matrix algebra
    9. 1.9 Areas and determinants
    10. 1.10 Inverses
    11. 1.11 Singular matrices
    12. 1.12 Two-dimensional vector spaces
    13. Appendix: the fundamental theorem of affine geometry
    14. Summary
    15. Exercises
  9. 2. Eigenvectors and eigenvalues
    1. 2.1 Conformal linear transformations
    2. 2.2 Eigenvectors and eigenvalues
    3. 2.3 Markov processes
    4. Summary
    5. Exercises
  10. 3. Linear differential equations in the plane
    1. 3.1 Functions of matrices
    2. 3.2 The exponential of a matrix
    3. 3.3 Computing the exponential of a matrix
    4. 3.4 Differential equations and phase portraits
    5. 3.5 Applications of differential equations
    6. Summary
    7. Exercises
  11. 4. Scalar products
    1. 4.1 The euclidean scalar product
    2. 4.2 The Gram–Schmidt process
    3. 4.3 Quadratic forms and symmetric matrices
    4. 4.4 Normal modes
    5. 4.5 Normal modes in higher dimensions
    6. 4.6 Special relativity
    7. 4.7 The Poincaré group and the Galilean group
    8. 4.8 Momentum, energy and mass
    9. 4.9 Antisymmetric forms
    10. Summary
    11. Exercises
  12. 5. Calculus in the plane
    1. Introduction
    2. 5.1 Big ‘oh’ and little ‘oh’
    3. 5.2 The differential calculus
    4. 5.3 More examples of the chain rule
    5. 5.4 Partial derivatives and differential forms
    6. 5.5 Directional derivatives
    7. 5.6 The pullback notation
    8. Summary
    9. Exercises
  13. 6. Theorems of the differential calculus
    1. 6.1 The mean-value theorem
    2. 6.2 Higher derivatives and Taylor’s formula
    3. 6.3 The inverse function theorem
    4. 6.4 Behavior near a critical point
    5. Summary
    6. Exercises
  14. 7. Differential forms and line integrals
    1. Introduction
    2. 7.1 Paths and line integrals
    3. 7.2 Arc length
    4. Summary
    5. Exercises
  15. 8. Double integrals
    1. 8.1 Exterior derivative
    2. 8.2 Two-forms
    3. 8.3 Integrating two-forms
    4. 8.4 Orientation
    5. 8.5 Pullback and integration for two-forms
    6. 8.6 Two-forms in three-space
    7. 8.7 The difference between two-forms and densities
    8. 8.8 Green’s theorem in the plane
    9. Summary
    10. Exercises
  16. 9. Gaussian optics
    1. 9.1 Theories of optics
    2. 9.2 Matrix methods
    3. 9.3 Hamilton’s method in Gaussian optics
    4. 9.4 Fermat’s principle
    5. 9.5 From Gaussian optics to linear optics
    6. Summary
    7. Exercises
  17. 10. Vector spaces and linear transformations
    1. Introduction
    2. 10.1 Properties of vector spaces
    3. 10.2 The dual space
    4. 10.3 Subspaces
    5. 10.4 Dimension and basis
    6. 10.5 The dual basis
    7. 10.6 Quotient spaces
    8. 10.7 Linear transformations
    9. 10.8 Row reduction
    10. 10.9 The constant rank theorem
    11. 10.10 The adjoint transformation
    12. Summary
    13. Exercises
  18. 11. Determinants
    1. Introduction
    2. 11.1 Axioms for determinants
    3. 11.2 The multiplication law and other consequences of the axioms
    4. 11.3 The existence of determinants
    5. Summary
    6. Exercises
  19. Further reading
  20. Index