What book gives a rigorous but elementary exposition of "div, grad, curl, and all that"? Conventional second-year calculus books are as far from rigorous as anything ever gets.
Have you taken a look at Fleming's Functions of Several Variables? Relative to most of books that I've looked at, it provides the least amount of machinery necessary to prove (rigorously) the basic theorems of integral vector calculus. I find it to be very readable and easy to understand.
This question that I posed awhile back which, inexplicably, was closed as being off-topic, might also be relevant.