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
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”.