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# Differential forms and cohomology: course

### From Intelligent Perception

## 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

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

## 4 Notes

Reading:

- Differential Forms: A Complement to Vector Calculus by Weintraub
- Applied Differential Geometry by Burke
- From Calculus to Cohomology by Madsen
- A Geometric Approach to Differential Forms by David Bachman
- Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni

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