A Problem Text in Advanced Calculus
by John M. Erdman
Publisher: Portland State University 2008
Number of pages: 377
This is a text on Advanced Calculus written for a more "theoretical" course, usually taken by majors in mathematics and physical sciences (and often called elementary analysis or intermediate analysis), it concentrates on conceptual development and proofs. It is intended for students of mathematics and others who have completed (or nearly completed) a standard introductory calculus sequence and who wish to understand where all those rules and formulas come from.
Download or read it online for free here:
by Wilfred Kaplan, Donald J. Lewis - University of Michigan Library
In the second volume of Calculus and Linear Algebra, the concept of linear algebra is further developed and applied to geometry, many-variable calculus, and differential equations. This volume introduces many novel ideas and proofs.
by Horst R. Beyer
Contents: Limits and Continuous Functions; Differentiation; Riemann Integration; Improper Integrals; Series of Real Numbers; Series of Functions; Analytical Geometry; The Riemann Integral in n-dimensions; Vector Calculus; etc.
by Viktor Blasjo - Intellectual Mathematics
A concise textbook covering precalculus through vector calculus and differential equations using informal infinitesimal reasoning. Always gives the most illuminating proofs possible, while standard books obscure key ideas under pedantic formalism.
by William V. Smith - Brigham Young University
Here is a free online calculus course. This is essentially an ordinary text, but you can read it online. There are lots of exercises and examples. The text is rigorous. We do calculus in both one and two variables at the same time.