Calculus of differential forms: course

1 Description

This is a two-semester course in n-dimensional calculus. It covers the derivative, the integral, differential forms, and a variety of applications. An emphasis is made on the coordinate free, vector analysis.

2 Prerequisites

• Calculus 1-3
• Linear algebra

3 Lectures

3.1 Vector calculus

1. Introduction to vector calculus
2. Parametric curves as vector valued functions
3. Functions of several variables
4. Gradient
5. Extrema of functions of several variables
6. Vector functions
7. Derivative as a linear operator
8. Integration in dimension n
9. Vector integrals
10. Stokes theorem
11. Independence of path

3.2 Continuous differential forms

1. Examples of differential forms
2. Algebra of differential forms
3. Wedge product of continuous forms
4. Exterior derivative
5. Properties of the exterior derivative
6. Fundamental correspondence
7. Identities of vector calculus

3.3 Integration of differential forms

1. Inside vs outside: 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. The best one: General Stokes Theorem
8. Linear algebra in elementary calculus

3.4 Manifolds and differential forms

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

4 Further reading

• Algebraic topology: course