Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics
Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.
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Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics ...
David Hestenes,Garret Sobczyk
No preview available - 1984
applied axioms biform bivector blades canonical form chain rule Chapter coderivative complex numbers computations conventional coordinates curl tensor curvature tensor curve defined definition determined developed differential and adjoint differential forms differential geometry dimension directed integral dual eigenblade eigenvalue eigenvectors equation equivalent Euclidean expressed extensor exterior differential factor formula frame ek Geometric Algebra Geometric Calculus geometric product grade Hence identity inner product integrability condition inverse isometry Lie algebra Lie bracket Lie group linear function linear transformation mathematical matrix metric tensor multiform multilinear multivector nonsingular notation obtained operator orthogonal transformation outer products outermorphism projection proof properties prove pseudoscalar r-blade r-form r-vector right side rotation rotor satisfies scalar Section shows simple spinor spinor group subspace symmetric symplectic tangent algebra tangent vector theory tion transformation f unique unit pseudoscalar vanishes variable vector field vector manifold vector space versor