Question: 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](http://rads.stackoverflow.com/amzn/click/1441928111) 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.

Answer: 
I myself am also an engineer interested in fluid mechanics. As you pointed out, the constraint on brevity is impossible, unless it is only to review the material. It seems to me that to understand SIAM publications you will need mathematics at the graduate level. 

You will need real analysis. Ross's book is a start. Take a look at something like Royden's [Real Analysis](http://rads.stackoverflow.com/amzn/click/013143747X) book. You should at least read the first part of the book, and possibly the second part.

Linear algebra is also a start. Know you will want to study this further. Take a look at 

[Linear Functional Analysis](http://rads.stackoverflow.com/amzn/click/1848000049) (Springer Undergraduate Mathematics Series)
Bryan Rynne, M.A. Youngson

or [Introductory Functional Analysis with Applications](http://rads.stackoverflow.com/amzn/click/0471504599), Erwin Kreyszig 

For a review of multivariable calculus I liked [Div, Grad, Curl, and All That: An Informal Text on Vector Calculus](http://rads.stackoverflow.com/amzn/click/0393925161), H.M. Schey

And for numerical methods in linear algebra take a look at, [Numerical linear algebra](http://rads.stackoverflow.com/amzn/click/0898713617), Trefethen. This should get you started.

You will also need to know about numerical ODE's like [Numerical Analysis: An Introduction](http://rads.stackoverflow.com/amzn/click/0817638954), by W Gautschi
Have you considered trying KhanAcademy? 
The learning curve should be pretty good and there's the advantage that the lessons are presented in a modular fashion. 
Here's a link, should you not be familiar with it: <http://www.khanacademy.org/#linear-algebra>