I'm almost through with my (Mech.) engineering and am trying to touch some advanced concepts in Computational Science which requires me to study Calculus and Linear Algebra in a more theoretical standpoint. I have been through both courses with a strong applied flavor but this does not allow me to read SIAM journals. What I want is a very quick and concise review of Calculus and Linear Algebra to a level to understand and follow rigor easily.
I have been through other questions dealing with this but none of them have brevity as a constraint.
What I need to reach:
Advanced Numerical Methods (Iterative Methods)
Partial Differential Equations (Fluid Mechanics)
Vector Calculus (Div, Grad, Curl etc)
(Will I need anything else other than LA and Calculus for this?)
Regarding my current math background:
I have finished Kenneth Ross' Book in my current process of re-learning. ( I enjoyed it thoroughly) I had taken a LA course (long back) which used Strang but I don't remember anything other than the stuff we use regularly in Engineering (like inverses, eigen vectors and iterative methods). I can differentiate, integrate and solve matrices quite easily. The problem I face is not in engineering mathematics but when I read papers.
Stuff I have reviewed:
I have Apostol's both books but they are long and dry.
I have tried reading Spivak's Calculus but the "juice" was in the exercises which I don't have time for.
I have tried reading Spivak's Calculus on Manifolds. I like it. I can follow it. But does it serve my purpose?
Will I need Real Analysis? If yes, any recommendations on the same constraints?
EDIT: I think my question has become too specific. I would appreciate it if someone provided me concise books on the above topics. I think I'll find time to solve exercises.