There are many great books that cover multivariable calculus/analysis, but I'm not sure a "standard" really exists.
That being said, here are a few that I like in no particular order:
Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by Hubbard and Hubbard. This text includes proofs of the major theorems of vector calculus and, as a great benefit to the self-learner, a solutions manual for many of the problems so that you can check your work. It should be approachable by anyone with a good background in basic calculus.
Wendell Fleming's Functions of Several Variables is a somewhat more sophisticated treatment of the subject but still elementary and approachable. The author works up to proving Stoke's Theorem on manifolds. It does include measure theory but really only enough for integration in a general context to make sense.
If you're really interested in the geometric aspects of advanced calculus, take a look at Callahan's recently released Advanced Calculus: A geometric View. This treatment is far from standard but is quite rigorous and the author works very hard to convey the geometric content of the material. The book has tons of illustrations and uses numerical computer experiments to aid intuition.
Another book that also focuses on geometric content is Baxandall and Liebeck's Vector Calculus. The name of this book might remind you of very elementary texts that focus on plug-and-chug, but this impression would not be accurate. This is a very solid text, one that I wish had been taught to me in "Calculus III".
A text that is more sophisticated than all of the above but extremely well-written and approachable is Shroeder's Mathematical Analysis: A Concise Introduction. This has a much more "analytical" flavor to it, more along the lines of, Say, Rudin, but was written to be self-contained and require few prerequistites. The text starts out easy and slow but builds rapidly to cover the major theorems of analysis in general contexts (i.e., Banach, Hilbert spaces).
I'll also tip my hat to a text that was mentioned in another answer: Loomis & Sternberg's Advanced Calculus. This is a wonderful book and it is a shame that it is no longer in print.