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Differential forms and cohomology: course
Contents
Description
Differential forms provide a modern view of calculus. They also give you a start with algebraic topology in the sense that one can extract topological information about a manifold from its space of differential forms. It's called cohomology.
Prerequisites
Just linear algebra, in the sense of theory of vector spaces.
Contents
1. Introduction
2. Continuous differential forms
- Differentials
- Examples of differential forms
- Algebra of differential forms
- Wedge product of continuous forms
- Exterior derivative
- Properties of the exterior derivative
3. de Rham cohomology
- Calculus and algebra vs topology
- Closed and exact forms
- Quotients of vector spaces
- Closed and exact forms continued
- de Rham cohomology
- Change of variables for differential forms
4. Cubical differential forms
- Cubical complexes
- Discrete differential forms
- Algebra of discrete differential forms
- Calculus of discrete differential forms
5. Cubical cohomology
6. Manifolds and differential forms
- Manifolds model a curved universe
- More about manifolds
- Tangent bundle
- Tangent bundles and differential forms
7. Integration of differential forms
- Orientation
- Integration of differential forms of degree 0 and 1
- Orientation of manifolds
- Integral theorems of vector calculus
- Integration of differential forms of degree 2
- Properties of integrals of differential forms
- General Stokes Theorem
- Continuous vs discrete differential forms
8. Maps
9. From vector calculus to exterior calculus
- Fundamental correspondence and Hodge duality
- Dual cells and dual forms
- Identities of vector calculus
Notes
Reading: