Mathematical Methods and Physical Insights: An Integrated Approach [Schramm] 9781107156418

Mathematics instruction is often more effective when presented in a physical context. Schramm uses this insight to help develop students' physical intuition as he guides them through the mathematical methods required to study upper-level physics. Based on the undergraduate Math Methods course he has taught for many years at Occidental College, the text encourages a symbiosis through which the physics illuminates the math, which in turn informs the physics. Appropriate for both classroom and self-study use, the text begins with a review of useful techniques to ensure students are comfortable with prerequisite material. It then moves on to cover vector fields, analytic functions, linear algebra, function spaces, and differential equations. Written in an informal and engaging style, it also includes short supplementary digressions ('By the Ways') as optional boxes showcasing directions in which the math or physics may be explored further. Extensive problems are included throughout, many taking advantage of Mathematica, to test and deepen comprehension.

頁數：792
版次：第1版
年份：2022年
規格：精裝/單色
ISBN：9781107156418

Part I. Things You Just Gotta' Know:
1. Prelude: symbiosis
2. Coordinating coordinates
3. Complex numbers
4. Index algebra
5. Brandishing binomials
6. Infinite series
7. Interlude: orbits in a central potential
8. Ten integration techniques and tricks
9. The Dirac delta function
10. Coda: statistical mechanics
Part II. The Calculus of Vector Fields:
11. Prelude: visualizing vector fields
12. grad, div & curl
13. Interlude: irrotational and incompressible
14. Integrating scalar & vector fields
15. The theorems of Gauss & Stokes
16. Mostly Maxwell
17. Coda: Simply connected regions
Part III. Calculus in the Complex Plane:
18. Prelude: path independence in the complex plane
19. Series, singularities & branches
20. Interlude: conformal mapping
21. The calculus of residues
22. Coda: analyticity & causality
Part IV. Linear Algebra:
23. Prelude: superposition
24. Vector space
25. The inner product
26. Interlude: rotations
27. The Eigenvalue problem
28. Coda: normal modes
Entr'acte: Tensors
29. Cartesian tensors
30. Beyond cartesian
Part V. Orthogonal Functions:
31. Prelude: 1 2 3 . . . infinity
32. Eponymous polynomials
33. Fourier series
34. Convergence and completeness
35. Interlude: beyond the straight & narrow
36. Fourier transforms
37. Coda: of time intervals and frequency bands
Part VI. Differential Equations:
38. Prelude: first order first
39. Second-order ODEs
40. Interlude: the Sturm-Liouville Eigenvalue problem
41. Partial differential equations
42. Green's functions
43. Coda: quantum scattering
Appendix A. Curvilinear coordinates
Appendix B. Rotations in R3
Appendix C. The Bessel family of functions