Rigorous Text in Multivariable Calculus and Linear Algebra

Solution 1:

Ted Shifrin's book Multivariable Mathematics is actually one of the best books of this type while not being very well known. Unfortunately, it's very expensive, so unless you can find it in your library, I would choose something else.

Otherwise I would just recommend Spivak's Calculus on Manifolds together with some linear algebra book. For linear algebra I would recommend either Axler's Linear Algebra Done Right or Linear Algebra by Fiedberg, Insel and Spence

Solution 2:

The best book on this for beginners is John and Barbara Hubbard's Vector Calculus,Linear Algebra And Differential Forms: A Unified Approach, 3rd edition. It is an incredibly rich treasure trove of results, all presented with full rigor and with a striking amount of originality. What I love best about this book is that it balances rigorous mathematics with applications better then any textbook I've ever seen. It is a bit pricey, but now as a gift for Christmas, it's perfect since it'll be a lifetime investment for a serious student of mathematics.

It's simply awesome, you're wasting your time buying anything else. (I'm not saying there's not any other good books on rigorous multivariable calculus-there certainly are. I just think any other book for a beginner is waste of money to buy. )