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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.

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You know that you can only comprehend material properly by making enough exercises right? –  sxd Jan 24 '12 at 15:57
    
I know but time is not on my side. Also, in books like Spivak's Calculus, the examples are of the form "Calculate int((sin(x)^2)" or similar. I know how to do that! Maybe quixotic, but I want a book to "magically" supplement my applied knowledge with rigor so that I can go back to Navier-Stokes and Conjugate Gradients :) –  Inquest Jan 24 '12 at 16:02
    
Especially when picking up new fields:) –  sxd Jan 24 '12 at 16:02
    
I don't think what you are trying to achieve is possible, at least for most people. Speaking for myself, I only learn through the process of trying to prove theorems and solve problems. Math, unfortunately, can't be learned through osmosis. The old cliche that "mathematics is not a spectator sport" is in fact true. –  ItsNotObvious Jan 24 '12 at 16:53
    
If you've already taken these courses, I would hesitate to suggest you go back and hit the books in some thorough manner. It would be a waste of time. Instead, try to read an article and write down the first thing you don't understand. Then go back to the book or post here to clarify the concept. You are in the perfect situation to learn math, as you have some comp sci phenomenon you want to learn to model mathematically which you may already understand at the intuitive level. –  dls Jan 24 '12 at 17:18
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2 Answers 2

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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 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 (Springer Undergraduate Mathematics Series) Bryan Rynne, M.A. Youngson

or Introductory Functional Analysis with Applications, Erwin Kreyszig

For a review of multivariable calculus I liked Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, H.M. Schey

And for numerical methods in linear algebra take a look at, Numerical linear algebra, Trefethen. This should get you started.

You will also need to know about numerical ODE's like Numerical Analysis: An Introduction, by W Gautschi

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Wow. Thank you! I wasn't expecting an answer anymore. Actually, I need SV and MV Calculus from more of a review point of view. Functional and Real Analysis I'll be starting from scratch. –  Inquest Feb 16 '12 at 18:50
    
In addition to Edison's suggestions (of which I especially recommend Kreyszig's and Schey's books), you may want to consider Gilbert Strang's Introduction to Applied Mathematics, which for some reason doesn't seem to be as well known as it should be. amazon.com/dp/0961408804 –  Dave L. Renfro Feb 16 '12 at 19:08
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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

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This is fairly basic and moreover, I am looking at something in book form. I'm sick of lectures. –  Inquest Jan 24 '12 at 17:45
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