Math 6111 Calculus on manifolds
The course begins with a review of some aspects of multivariable calculus, focusing on the inverse and implicit function theorems. It then defines differentiable manifolds (from both the abstract and embedding points of view) making use of the implicit function theorem to construct examples of manifolds. Next it defines scalar and vector fields and discusses integral curves, Lie brackets and Lie derivatives. From there it moves on to differential forms and considers exterior derivatives, wedge products, the Hodge star operator and the Frobenius theorem. Finally it considers volume forms, orientability and integration and finishes up with the all important Stokes theorem.
Depending on time and class interests it may consider applications of the exterior calculus to differential geometry, topology (deRham cohomology), differential equations or theoretical physics.
An undergraduate background in analysis and linear algebra is expected.
Generally course notes but the following are also good references.
- Calculus on Manifolds, Michael Spivak
- Differential Topology, Victor Guillemin and Alan Pollack
- Analysis on Manifolds, James R. Munkres
- Differential forms with applications to the physical sciences, Harley Flanders
- Geometrical Methods of Mathematical Physics, Bernard Schutz