Fluid Mechanics with calculus on manifolds Fluid Mechanics is a branch of physics that uses a lot of vector calculus in $\mathbb{R}^3$ to describe phenomena mathematically. Calculus on manifolds, however, is the straightforward generalization of vector calculus and has a lot of interesting and useful tools like differential forms, lie derivatives, flows of vector fields and so on.
Is there some book/article that treats fluid mechanics using calculus on manifolds? I've searched a lot for some book like this but I could only find one called "A mathematical introduction to fluid mechanics" which although uses a lot of math restricts itself to vector calculus. Is there some other book that uses the tools of the calculus on manifolds to describe fluid mechanics phenomena?
 A: This may not be quite what you're after, but "Relativistic Hydrodynamics" by Rezzolla and Zanotti covers (relativistic) hydrodynamics in the language of differential geometry.  
This is a graduate level textbook on hydrodynamics in the context of general relativity (hence the differential geometry).  It covers kinetic theory (including quantum effects), wave propagation, numerical algorithms, and analytic work in the context of astrophysics.  You should have both some relativity and hydro under your belt before you tackle it, but if you do you may find it very useful.  It is also fairly recent (2013), so is very up to date.
A: There is a rather small section on the topic in the book "Differential Forms with Applications to the Physical Sciences" by H. Flanders. I don't find this extremely satisfactory, but what I'd do instead is take any book on fluid mechanics which has some consistent vector calculus notation and start "translating" it into the corresponding geometric notions. Perhaps Flanders can be of help at least in fixing a couple of ideas on how to proceed.
A: By "manifold", if you mean "flows along non-Euclidian surfaces" then there are several scientific but applicative papers online from the CG field, e.g. this one published at ACM Siggraph'03 / Transaction on Graphics: http://www.autodeskresearch.com/pdf/surfflow.pdf
( in this field papers are often 8 pages with figures and images of results, and since the domain is applicative a fair background in CFD should be ok to follow. But physical and numerical models often correspond to simple cases.  )
