Geometric algebra

The geometric algebra (GA) of a vector space is an algebraic structure, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which is a superset of both the scalars ${\displaystyle \mathbf {R} }$ and the vector space ${\displaystyle V}$. Mathematically, a geometric algebra is defined as the Clifford algebra of a real vector space with a quadratic form. In practice, several derived operations are generally defined, and together these allow a correspondence of elements, subspaces and operations of the algebra with physical interpretations. A derived operation, the exterior product, defines an exterior algebra on the same space and provides many of the interpretations of its elements, as well as demonstrating that a GA is basis-independent.

The scalars and vectors have their usual interpretation, and make up distinct subspaces of a GA. Bivectors provide a more natural representation of pseudovector quantities in vector algebra such as oriented area, oriented angle of rotation, torque, angular momentum, electromagnetic field and the Poynting vector. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of ${\displaystyle V}$ and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike vector algebra, a GA naturally accommodates any number of dimensions and any quadratic form such as in relativity. However, a direct representation within the algebra has not been found for all physical and geometric operations, including general linear transforms of ${\displaystyle V}$, which are represented as a class of linear function named outermorphisms, as well as other quantities such as the stress–energy tensor or Ricci curvature, in contrast with the more universal tensor algebra.

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