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But I'm interested in physical systems which are three dimensional by nature, where the solutions aren't single numbers corresponding to natural frequencies or time evolutions of single particles, but rather continuous functions (eg describing the flow characteristics of air through a jet engine or how a shock wave reflects off a large object, etc.).

I'm set to take General Relativity next year (only undergraduate >.<), which I've been told is similar to how advanced fluid dynamics works.

As far as I can make out, you find a Lagrangian on the field you're working with and work backward to find the field for given boundary conditions and get some finite difference setup to give you the solution.

Can anyone elabourate on this and the general method of solving continuum problems in Differential Geometry?

Any explicit calculation (eg flow around a rotating turbine (pretty please? ) would be greatly appresciated.

(PS I know Set Theory and I've done Special Relativity, which involves Minkowsky Space and the Einstein Summation Notation, but I never really got what the Levi Chivita connection was, just the properties of the Levi Civita Symbol)