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
the book New Foundations for Classical Mechanics,
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
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.
The book Clifford Algebra to Geometric
Calculus is the first and still the most complete exposition of Geometric
Calculus (GC). But it is more of a reference book than a
textbook, so can it be a difficult read for beginners. The Tutorial on
Geometric Calculus is a guide for serious students who want to dig
deeply into the subject. It presents helpful background and aims to
clarify objectives, important results and methods in the book. It is
supplemented by a hot-linked annotated bibliography of papers
elaborating on various aspects of Geometric Algebra and Calculus.
An interactive online GA Primer
is available in Beta
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