A. 綱要

1.向量/張量分析導引: Classical divergence/Stokes theorems, generalized coordinates, calculus in G.C., vector calculus, tensor calculus, covariant differentiation, covariant way of expressing divergence/stokes theorems,
Riemann’s counting arguments and Riemann curvature tensor
2.PDE and the ideas of S. Lie for Lie group: PDE of first order, Lagrange,
Jacobi, Poisson and Pfaff’s ideas; Lie’s motivation from Galois theory,
Symmetries simplifies solving PDE, analogy with Galois theory, Lie’s
Three fundamental theorems, differential forms as a tool and language
3. Differential forms and algebraic topology: Cartan’s invention of differential
Forms for solving Pfaff’s problem, Poincare’s invariant integrals and differential
Forms, Poincare’s 3-body problem and topology, Poincare’s “Analysis situs” as a beginning of algebraic topology, difficulties in earliy development of topology