Reddit mentions of DIV, Grad, Curl, and All That: An Informal Text on Vector Calculus

Grad, Div, Curl and all that by H. M. Schey, recommended here. There's also A Student's Guide to Vectors and Tensors by Daniel Fleisch if you want to cover tensors as well.

Here are some suggestions :

https://www.coursera.org/course/maththink

https://www.coursera.org/course/intrologic

Also, this is a great book :

http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&qid=1346855198&sr=8-5&keywords=history+of+mathematics

It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.


EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.

For you, I would suggest :

http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&qid=1346860077&sr=8-1&keywords=rudin

http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&qid=1346860052&sr=8-4&keywords=from+matrix+to+bounded+linear+operators

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&qid=1346860077&sr=8-5&keywords=rudin

http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975

http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1346860356&sr=1-2&keywords=chaos+and+dynamics

http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&ie=UTF8&qid=1346860179&sr=1-5&keywords=numerical+analysis

This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.

One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.

Also depends on what level of mathematics you're coming from: [Div Grad Curl] (http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975) is great for learning the multivariable calc.

When it comes to an introduction to quantum, [this] (http://www.amazon.com/Quantum-Mechanics-Introduction-Robert-Scherrer/dp/0805387161/ref=sr_1_14?s=books&ie=UTF8&qid=1341515102&sr=1-14&keywords=quantum+mechanics) is probably one of the best textbooks I've used.

I do highly recommend Genome by Matt Ridley and A History of God by Karen Armstrong. It looks like Before the Big Bang might be a great idea too.

However, I'm noticing a bit of redundancy in your stacks and don't want you to get bored! In the presence of the other books, I would recommend Dawkins' The Ancestor's Tale in lieu of The Greatest Show on Earth. (Although, if you're actually not going to read all the other books, I would actually go the other way.) Similarly, I would probably choose either to read the God Delusion or a few of the other books there.

Other recommendations: how about The Red Queen by Matt Ridley, and The Seven Daughters of Eve by Bryan Sykes? These occupy niches not covered by the others.

The popular expositions on cosmology all look supremely awesome, but you should probably choose half of them. Another idea: read just The Fabric of the Cosmos by Greene, and if you love it, go ahead and learn mechanics, vector calculus, Electrodynamics, linear algebra, and Quantum Mechanics! Hmm...on second thought, that might actually take longer than just reading those books :)

Yeah. Part of me feels like I've just been lucky in finding easy problems that the "real" scientists in my field hadn't bothered to try yet.

I still don't really understand linear algebra or vector calculus, for instance. I have Linear Algebra Done Right, Div, Grad, Curl, and all that, and the Princeton Companion to Mathematics on my wish list, which may help.