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Differential forms and cohomology: course

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Contents

1 Description

$$\int_ σ dω = \int_{∂σ} ω$$
Derivative vs boundary

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.

2 Prerequisites

Just linear algebra, in the sense of theory of vector spaces.

3 Contents

1. Introduction

  1. Topology in real life
  2. Topology via Calculus
  3. Why do we need differential forms?

2. Continuous differential forms

  1. Differentials
  2. Examples of differential forms
  3. Algebra of differential forms
  4. Wedge product of continuous forms
  5. Exterior derivative
  6. Properties of the exterior derivative

3. de Rham cohomology

  1. Calculus and algebra vs topology
  2. Closed and exact forms
  3. Quotients of vector spaces
  4. Closed and exact forms continued
  5. de Rham cohomology
  6. Change of variables for differential forms

4. Cubical differential forms

  1. Cubical complexes
  2. Discrete differential forms
  3. Algebra of discrete differential forms
  4. Calculus of discrete differential forms

5. Cubical cohomology

  1. Cochain complexes and cohomology
  2. Cohomology of figure 8

6. Manifolds and differential forms

  1. Manifolds model a curved universe
  2. More about manifolds
  3. Tangent bundle
  4. Tangent bundles and differential forms

7. Integration of differential forms

  1. Orientation
  2. Integration of differential forms of degree 0 and 1
  3. Orientation of manifolds
  4. Integral theorems of vector calculus
  5. Integration of differential forms of degree 2
  6. Properties of integrals of differential forms
  7. General Stokes Theorem
  8. Continuous vs discrete differential forms

8. Maps

  1. Examples of maps
  2. Commutative diagrams
  3. Cell maps
  4. Chain operators
  5. Cohomology operators

9. From vector calculus to exterior calculus

  1. Fundamental correspondence and Hodge duality
  2. Dual cells and dual forms
  3. Identities of vector calculus


4 Notes

Reading: