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Matches in DBpedia 2014 for { ?s ?p A geometric algebra (GA) is the Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form. The term is also sometimes used as a collective term for the approach to classical, computational and relativistic geometry that applies these algebras. The distinguishing multiplication operation that defines the GA as a unital ring is the geometric product. Taking the geometric product among vectors can yield bivectors, trivectors, or general n-vectors. The addition operation combines these into general multivectors, which are the elements of the ring. This includes, among other possibilities, a well-defined sum of a scalar and a vector, an operation that is impossible by traditional vector addition. This operation may seem peculiar, but in geometric algebra it is seen as no more unusual than the representation of a complex number by the sum of its real and imaginary components.Geometric algebra is distinguished from Clifford algebra in general by its restriction to real numbers and its emphasis on its geometric interpretation and physical applications. Specific examples of geometric algebras applied in physics include the algebra of physical space, the spacetime algebra, and the conformal geometric algebra. Geometric calculus, an extension of GA that includes differentiation and integration can be further shown to incorporate other theories such as complex analysis, differential geometry, and differential forms. Because of such a broad reach with a comparatively simple algebraic structure, GA has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents argue that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. Others claim that in some cases the geometric algebra approach is able to sidestep a "proliferation of manifolds" that arises during the standard application of differential geometry.The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related but more limited exterior algebra. In 1878, William Kingdom Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized by Hestenes in the 1960s, who recognized its importance to relativistic physics. Since then, geometric algebra (GA) has also found application in computer graphics and robotics.. }

• Geometric_algebra abstract "A geometric algebra (GA) is the Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form. The term is also sometimes used as a collective term for the approach to classical, computational and relativistic geometry that applies these algebras. The distinguishing multiplication operation that defines the GA as a unital ring is the geometric product. Taking the geometric product among vectors can yield bivectors, trivectors, or general n-vectors. The addition operation combines these into general multivectors, which are the elements of the ring. This includes, among other possibilities, a well-defined sum of a scalar and a vector, an operation that is impossible by traditional vector addition. This operation may seem peculiar, but in geometric algebra it is seen as no more unusual than the representation of a complex number by the sum of its real and imaginary components.Geometric algebra is distinguished from Clifford algebra in general by its restriction to real numbers and its emphasis on its geometric interpretation and physical applications. Specific examples of geometric algebras applied in physics include the algebra of physical space, the spacetime algebra, and the conformal geometric algebra. Geometric calculus, an extension of GA that includes differentiation and integration can be further shown to incorporate other theories such as complex analysis, differential geometry, and differential forms. Because of such a broad reach with a comparatively simple algebraic structure, GA has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents argue that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. Others claim that in some cases the geometric algebra approach is able to sidestep a "proliferation of manifolds" that arises during the standard application of differential geometry.The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related but more limited exterior algebra. In 1878, William Kingdom Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized by Hestenes in the 1960s, who recognized its importance to relativistic physics. Since then, geometric algebra (GA) has also found application in computer graphics and robotics.".