Looking for some decent math books and solutions

Question

I am a graduate student in engineering pursuing computational mechanics. I have learned very quickly that the math required to study in this area is very sophisticated and complicated. I need some good self-study books in

1. Partial Differential Equations
2. Functional Analysis
3. Tensor Calculus

I would greatly appreciate any input to help me learn these topics. I have already taken calculus 1,2,3, ODE, Linear Algebra, Vector Analysis and Numerical Linear Algebra.

I would greatly appreciate any pointers to books that I can get solution manuals to. The math department will not offer these courses for the remainder of my graduate program, so I will not be taking any courses related to this.

Thank you.

Answer 1

1) For PDEs, get "Partial Differential Equations" by Nash (from AMS).

2) For functional analysis, it's best to start with Kreyszig's "Introductory Functional Analysis with Applications".

3) For tensors / diff geom. start with Synge and Schild's "Tensor Calculus", which follows a "physics" style of presentation, followed by Bishop and Goldberg's "Tensor Analysis on Manifolds" and Lee's "Introduction to Smooth Manifolds".

Keep Halmos's "Finite Dimensional Vector Spaces", Hoffmann and Kunze's "Linear Algebra", both Rudins, and Mendelson's "Topology" by your side during your course of study - you'll need to refer to them numerous times.

Answer 2

I know a good reference and classic source about PDE that is Elements of Partial Differential Equations by Ian N. Sneddon. However, it needs you to have a good background in Calculus III, I suggest you to try it.