subsonicman said:
So Marnemath, you're saying if I get Courant volume 2 to study multivariable then I should get Courant volume 1 rather than spivak to review one variable calculus?
And I'm unsure how far I want to go into differential equations. I want to have enough background to solve any physics problem involving them, but since I've barely even looked at differential equations before, save for very simple ones, I don't know how interesting the topic would be for me. Is it required to have a solid grasp of differential equations to study further in other math classes like analysis and topology?
No. I'm saying if you picked Apostol, Spivak, and or Courant, you'll have be doing fine. Apostol, has an interesting approach and a geometric approach that is valued by some people. Spivak is easy to read, and helps you out with proofs (ie by breaking them down into smaller easier problems or giving big hints.) Courant is fun to read and very similar to Spivak (to the point I think Spivak read Courant and then put it in his own words.)
I tend NOT to recommend Courant for self studying just because it has a lot less exercises and even though a lot of problems are the same as Spivak, the lack of hints can make them inaccessible to a student not prepared. Still, if you can find a cheap copy, it's a a good read by a great mathematician.
As for multivariable calculus, I tend to recommend to have linear algebra before taking it. Although it's extremely common to learn the subject without LA, I've found that if you know it already, then it's easier to generalize and see why certain things are done in a multivariable book. With that said, it's by no means required, and a lot of books will teach you the LA that is required.*
As for Differential Equations, it's a very interesting and beautiful subject and to this day remains my favorite field in mathematics. In fact, if I wasn't addicted to having money, I would've done my PhD relating to Differential Equations. Anyway, at the introduction level, it isn't hard. You'll need some baisc LA, ie the concept of linear indepedence. Nevertheless, it isn't hard to learn it at the Boyce level, and that will get you started to solve a good number of equations you'll need to solve in Physics. So Analysis and Topology do not depend on Diffy Q. However, for more advance treatment of Diffy Q, you'll need Analysis and Topology (and this can be said about nearly every advacement treatment of any subject, including Probability!)
*Although if you do Spivak's manifolds, follow WannabeNewton's advice and know LA!
