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# Calculus of differential forms: course

### From Mathematics Is A Science

## Contents

## 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

## 3 Lectures

### 3.1 Vector calculus

- Introduction to vector calculus
- Parametric curves as vector valued functions
- Functions of several variables
- Gradient
- Extrema of functions of several variables
- Vector functions
- Derivative as a linear operator
- Integration in dimension n
- Vector integrals
- Stokes theorem
- Independence of path

### 3.2 Continuous differential forms

- Examples of differential forms
- Algebra of differential forms
- Wedge product of continuous forms
- Exterior derivative
- Properties of the exterior derivative
- Fundamental correspondence
- Identities of vector calculus

### 3.3 Integration of differential forms

- Inside vs outside: 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
- The best one: General Stokes Theorem
- Linear algebra in elementary calculus

### 3.4 Manifolds and differential forms

- Manifolds model a curved universe
- More about manifolds
- Tangent bundle
- Tangent bundles and differential forms