Recommended book for advanced calculus

[torito_verdejo]

I'm looking for recommendations about advanced calculus books. I'm interested in going further and deeper than nth-order linear differential equations, but overall as a Physics student I'm deeply interested in being very, very comfortable dealing with line, surface and volume integration.

Specifically, my biggest concerns at the moment are two:

1. Applying methods to solve linear and surface integrals without really understanding why they work.
2. I have the feeling that when it comes to this kind of integrals, the hardest part is parametrizing the geometrical object, so I want to be "fluent" at that.

Let me also tell you that I might be a physics student, but I don't like seeing mathematics as a mere tool. Maths without proofs or foundation is like putting an end to the hunger without eating, so I want my books to tell me why we do things this or that way, not just giving me solution recipes.

Thank you very much. :)

 

[WWGD]

Do you have access to a college or otherwise Math/Engineering library? You can maybe browse through books on that section see which feel right. I suggest as a rule of thumb to consider books that include a notations index and some solved problems , as a sign the author(s) have made an effort to write the book carefully.

 

[s00mb]

If you really want to understand why the integrals work you need to take Real Analysis (which some colleges call advanced calculus) or grab a book on it. That course will give you the understanding and tools you need to understand your integrals, but it is a lengthy process. I think it's worth it though. If you want to be fluent in surface and line integrals I'd recommend the Khan Academy videos on surface integrals or line integrals (just google it). I wouldn't underestimate his videos they are actually pretty good. As far as a book alone goes on calc 3 I'd suggest what the earlier person said, going to the library and just finding a feel for the teaching style you like the best.

 

[Zexuo]

For your listed concerns I can recommend the following:

1. Introduction to Analysis by Maxwell Rosenlicht (https://store.doverpublications.com/0486650383.html). I had this one on hand while working through a first year calculus textbook so I can read some of the proofs omitted by the latter.

2. A Catalog of Special Plane Curves by J Dennis Lawrence (https://store.doverpublications.com/0486602885.html). This book contains descriptions and equations for a large number of curves in various coordinate systems, including their parametric forms.

 

[torito_verdejo]

For your listed concerns I can recommend the following:

1. Introduction to Analysis by Maxwell Rosenlicht (https://store.doverpublications.com/0486650383.html). I had this one on hand while working through a first year calculus textbook so I can read some of the proofs omitted by the latter.

2. A Catalog of Special Plane Curves by J Dennis Lawrence (https://store.doverpublications.com/0486602885.html). This book contains descriptions and equations for a large number of curves in various coordinate systems, including their parametric forms.

Thank you for the references!