This text breaks new ground in presenting and applying sophisticated mathematics in an elementary setting. Aimed at physics students, it covers the theory and physical applications of linear algebra and of the calculus of several variables, particularly the exterior calculus. The exterior differential calculus is now being recognized by mathematicians and physicists as the best method of formulating the geometrical laws of physics, and the frontiers of physics have already begun to reopen fundamental questions about the geometry of space and time. Covering the basics of differential and integral calculus, the authors then apply the theory to interesting problems in optics, electronics (networks), electrostatics, wave dynamics, and finally to classical thermodynamics. The authors adopt the 'spiral method' of teaching (rather than rectilinear), covering the same topic several times at increasing levels of sophistication and range of application.