Calculus on Manifolds

The most important calculus is calculus on manifolds. It explains how to make sense of computations, and to make the computations themselves easy. The main concepts and ideas in this theory are intrinsic, i.e., they are independent of the choice of coordinates. That is why modern math speaks the language of manifolds. The course leads directly to de Rham cohomology and the Stokes formula, which is one of the cornerstones of the theory.

Prerequisites: Multivariable calculus, one-variable calculus (differentiability, derivative, and integration) and linear algebra (vectors and linear forms, linear transformations and determinant).


Curriculum:

  1. Definition and examples of smooth manifolds.
  2. Orientability and orientation.
  3. Tangent vectors and tangent space to a manifold at a point. Tangent bundles. Vector fields.
  4. Skew-symmetric forms on linear spaces. Wedge product.
  5. Differential forms on manifolds. Exterior differential.
  6. Smooth maps of manifolds. Diffeomorphisms. The transformation rule under coordinate change for functions, vector fields and differential forms.
  7. Integration in . Coordinate change in the integral. Integration of differential forms. Stokes theorem for a cube in .
  8. Integration on manifolds.
  9. Manifold with boundary. Induced orientation of the boundary.
  10. General Stokes theorem. Green’s formula, Gauss-Ostrogradskii divergence theorem, Stokes formula for a surface in .
  11. Closed and exact forms. The Poincare lemma. De Rham cohomology.

Textbooks

  • J. Munkres, Analysis on Manifolds, Cambridge, MA: Perseus Publishing, 1991.