Mathematical Methods and Physical Insights: An Integrated Approach (1版)

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. Initial chapters review essential mathematical concepts and techniques that students need to know, helping students to fill gaps in their background knowledge and enabling instructors to focus on the main course material Each Part has sections designed to highlight and apply the techniques and concepts within, so that physics motivates the math, which deepens understanding of physics Boxed asides throughout the book called 'By the Ways' (BTWs) provide optional digressions into the math or physics, to help to lift motivated students to the next level Flexible paths through the text allow it to be used for both one- and two-semester courses, depending on the course goals and instructor's preference 本書目錄： 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 會計 經濟 國際貿易 財務 企業管理 數學 機率與統計 資訊與資工 工業工程 社會與心理 醫務管理 物理 化學與化工 電子與電機 機械工程 航空工程 土木工程 環境工程 生命科學 護理 休閒觀光與餐飲 化妝品科學 其他 帳號 密碼 驗證碼