The most up-to-date in-depth introduction to and argument for GA is in Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics (Am. J. Phys. 71 (2), February 2003, pp. 104--121). A synopsis for GA for relativity and quantum mechanics is provided in : SpaceTime Physics (Am. J. Phys. 71 (6), June 2003, pp. 1--24). For a simpler introduction to geometic algebra, see Primer for Geometric Algebra.
The most detailed introduction with historical perspective is presented in
the book New Foundations for Classical Mechanics,
which provides a complete coordinate-free reformulation of Newtonian Mechanics in terms of GA.
This book has many applications and exercises to develop proficiency with GA. (A more compact
introduction is given in the Cambridge tutorials.)
Invariant Body Mechanics I & II
provides the quickest entry to GA in robotics and Computational Geometry.
Readers who want a more mathematical introduction may prefer to start with the lead
papers in the first two sections of Universal Geometric Calculus.
A thorough treatment of the fundamentals is given in
New Foundations for Mathematical Physics.
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