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In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors ...
Geometric algebra (GA) is an extension or completion of vector algebra (VA). [1] The reader is herein assumed to be familiar with the basic concepts and operations of VA and this article will mainly concern itself with operations in the GA of 3D space (nor is this article intended to be mathematically rigorous).
Geometric Algebra is a book written by Emil Artin and published by Interscience Publishers, New York, in 1957. It was republished in 1988 in the Wiley Classics series ( ISBN 0-471-60839-4 ). In 1962 Algèbre Géométrique , a translation into French by Michel Lazard , was published by Gauthier-Villars, and reprinted in 1996.
The universal geometric algebra (n, n) of order 2 2n is defined as the Clifford algebra of 2n-dimensional pseudo-Euclidean space R n, n. [1] This algebra is also called the "mother algebra". It has a nondegenerate signature. The vectors in this space generate the algebra through the geometric product.
In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl 1,3 (R), or equivalently the geometric algebra G(M 4) to physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and General Relativity" and "reduces the mathematical divide between classical, quantum and ...
In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors. [1] It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function.
In exterior algebra and geometric algebra the exterior product of two vectors is a bivector, while the exterior product of three vectors is a trivector. A bivector is an oriented plane element and a trivector is an oriented volume element, in the same way that a vector is an oriented line element. Given vectors a, b and c, the product
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these ...