Can anyone recommend a textbook for studying vector calculus (vector analysis) only, that focuses on the theoretical mathematics behind vector calculus?
Currently, I am using vector analysis by Snider. I have also taken a look at vector calculus by Marsden. Both of these books skip a large amount of the theory behind what we are doing and why it matters.
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I'm not sure what level you're looking for (your question could be more precisely worded), but I recently liked the look of
If you're not interested in the "Cartesian Tensors" part of the title then don't worry, that's left until the end; just focus on the first six chapters.
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A really excellent book about that argument is:
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Colley's Vector Calculus is very good for a more-formal-than-normal treatment of multivariable calculus. The focus is on things like partial derivatives, div, grad, curl, and multiple integrals. It could be used as a first exposure to these topics, and proofs are supplied.
For texts on a subject better described as multivariable analysis, I highly recommend Munkres's Analysis on Manifolds and/or Spivak's Calculus on Manifolds. These focus on topological and analytical aspects (unlike Colley) and are intended to extend the ideas of multivariable analysis to manifolds.
Rigorous yet accessible text by J. H. Hubbard and B. B. Hubbard. Vector calculus, linear algebra, and differential forms: a unified approach with Maple 10 VP. Pearson Education, Limited, Mar. 2006.