Differential forms: course

Redirect to:

• Differential forms and cohomology: course

Book

Description

We take a more advanced view of calculus. The approach allows us to study calculus on manifolds as well as discrete exterior calculus. The course may be called: "Advanced (Advanced calculus)".

Stokes theorem

$$\int_ σ dω = \int_{∂σ} ω$$

Prerequisites

Calculus 3 (or preferably vector calculus), proofs, linear algebra

Lectures

The lectures now are a part of a draft of a book called Applied Topology and Geometry.

The calculus of differential forms...

• Where do differential forms come from?
• Why do we need differential forms?

• Examples of differential forms
• Algebra of differential forms
• Exterior derivative
• Closed and exact forms
• Closed and exact forms continued
• Algebraic operations with forms and cohomology

• Discrete differential forms
• Algebra of discrete differential forms
• Calculus of discrete differential forms
• Chains vs cochains

• Fundamental correspondence and Hodge duality: part 1
• Fundamental correspondence and Hodge duality: part 2
• Applications of discrete forms

• Manifolds model a curved universe
• More about manifolds
• Tangents and differential forms
• Tangent bundles and differential forms

• Integration of forms on manifolds: part 1
• Integration of forms on manifolds: part 2

• Orientation of manifolds
• Differential forms as multilinear functions

Exercises

• Differential forms: homework 1 -- linearity of exterior derivative
• Differential forms: homework 2 -- linearity of integral on intervals
• Differential forms: homework 3 -- path => step-path
• Differential forms: homework 4 -- details of #2
• Differential forms: homework 5 -- $dd=0$ in dim 3, discrete
• Differential forms: homework 6 -- cohomology of figure 8
• Differential forms: homework 7 -- lemma about products
• Differential forms: homework 8 -- cubical complex = surface
• Differential forms: homework 9 -- separation of variables
• Differential forms: homework 10 -- change of variables - general case

• Differential forms: exams

Notes

• The content came from these handwritten lecture notes.
• Differential Forms: A Complement to Vector Calculus by Weintraub was used as the official textbook.

More reading:

• Discrete exterior calculus
• A Geometric Approach to Differential Forms by David Bachman
• Vector Calculus, Linear Algebra, and Differential Forms by Hubbard
• Discrete Exterior Calculus by Hirani
• Discrete Differential Forms for Computational Modeling by Desbrun, Kanso, Tong
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni