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Week 4

8/2 Week 8: Literature Review

We’ve read works of some import; Numerical Methods by Leveque, Differentiable Manifolds by Boothby, Differential Forms by Flanders, Shockwaves and Reaction Diffusion Equations by Smoller and a number of papers by Professor Young and his collaborators.

Numerical Methods (The Green Book) was discussed in week one, but we include it here for completeness. It was our written introduction to the subject matter.

Differentiable Manifolds (The Blue Book) is very well written. In the beginning it relies heavily on a working knowledge of point-set topology. It introduces the idea of “C” and “C infinity” mappings, the rank of a function, the chain rule.

Differential Forms (The Red Book) is an introductory book for scientists and engineers. It is highly compact. It is highly formal. It contains very little geometry that a typical undergraduate would be prepared to understand. It defines ideas such as bases of differential forms, the wedge product, the hodge star, the tangent space at a point, the dual of this tangent space and exterior derivatives.

Shockwaves and Reaction Diffusion Equations (The Yellow Book) If our work this Summer were a class, Chapters 15-20 of SRDE could serve as a textbook. It develops the proofs of conservation laws, the method of characteristics, the Rankine Hugoniot condition, the method of finite differences, simple waves, the importance of the Eigensystem, Riemann Invariants, and analysis using bounded variation and total variation. Chapter 20 culminates in the “Glimm Scheme”.

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