Books various maths subjects

Question

I am a Civil Engineering student and i am planning on following physics in my career.I want to be ready for the advanced undergraduate courses that i will attend to,so i need to learn Differential Equations,Multivariable Calculus and Linear Algebra by my own first.I don't want to learn these subjects via the traditional physics way(which is via courses on Mathematical Methods for Physicists) because i will be missing the mathematical beauty of these subjects(most of it comes out from knowing/learning or figuring out a proof of a theorem).
So,specifically i want books(individual book per subject or a book for all of them,i just want the source with the most detail) that have a great amount of proofs,that have problems that verge from easy(to get the grips with something) to very challenging(to test intuition and how good the reader is in manipulating the maths) but also offer a great deal of intuition and have graphs were it is necessary.
Having all these in mind,feel free to inform me if there is any other subject that you think that a physicists must know in detail and i will then edit the question in order to include it in it.
Also,i have heard a more general book called Calculus by James Stewart and another one which is called Calculus:Early Transcendentals by Anton.Are these books good enough and cover everything that i want from a book?

Answer 1

What about all in one ? :-) this book covers almost all what you want.

(Sorry I don't have enough reputation for comment)

Answer 2

I think Stewart's or Anton's Calculus are not the right choice for you...unless you have not yet completed a course in university calculus. So, the answer here necessarily depends on your background. If I assume you have take differential and integral calculus of one-variable then I recommend: Susan Colley's Vector Calculus (even the first or second edition is also fine, I see the latest edition is rather pricey) This text is unusual in that it is somewhere between texts like Stewart or Anton and the higher level math literature. It has both easy and challenging problems and it makes and effort to use vector notation (rather than just introducing it and doing everything with graphs like so many other multivariate calculus texts)

I tend to like texts in the Springer undergraduate text series like Vector Calculus by Paul Matthews. Since you want proofs, for linear algebra I recommend Linear Algebra by Insel, Spence and Friedberg. However, Damiano and Little is also very good and has a bit more on the elementary linear algebra material. You might want to start with a text like Lay's Linear Algebra to start your study if you have not taken other linear algebra in your education. You might like James Hefferon's Linear Algebra text which has a complete solution guide written by the author (both freely available as well as the source code at this official website)

All of this said, there are many folks who are writing open source calculus, differential equations, linear algebra... you name it, and it is probably worth an hour of your time to search out if any of those open source texts fit your needs. See my website for what I've posted and you might also search based on the recommendations at this open source math text link site (there are other meta sites, I have not exhaustively searched to find the optimal one)

Answer 3

If you are really interested in proofs and exploring calculus from first principles, I will highly recommend Michael Spivak's Calculus textbook (available here). It explains things very well in a very thorough and methodical approach. It also has easy as well as very challenging questions that would stretch the minds of even the brightest mathematicians. However, I must warn you that its really more of an introduction to real analysis (i.e. more for mathematicians really) so if you're interested in more applied mathematics (of which most Engineers are), Engineering Mathematics textbooks would be best for you. Alternatively, you could just use both and get the best of both worlds. Best wishes.