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Differential forms: course
From Mathematics Is A Science
Revision as of 14:44, 8 February 2013 by imported>WikiSysop
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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)".
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...
- 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
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
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