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Home > Geometric algebra

Here the real numbers are used as scalars in a vector space V. From now on, a vector is something in V itself.

The outer product (the exterior product, or the wedge product) is defined such that the graded algebra ( exterior algebra of Hermann Grassmann) of multivectors is generated. The geometric algebra is the algebra generated by the geometric product (which is to be thought of as more fundamental) with (for all multivectors )

1. Associativity
2. Distributivity over the addition of multivectors: and
3. ContractionIn mathematics, contraction has several meanings: See contraction mapping. The contraction structural rule, in proof theory. Contraction of a tensor. It occurs when a pair of literal indices (one a subscript, the other a superscript) of a mixed tensor are for any "vector" (a grade-one element) is a scalarAbstract algebra Algebra Linear algebra The concept of a scalar is used in mathematics and physics. The concept used in physics is a more concrete version of the same idea that goes by that name in mathematics. In mathematics, the meaning of scalar depend ( real number)

We call this algebra a geometric algebra . The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the associative algebraAlgebra In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. They are thus special algebras. Definition An associative algebra A over.

The connection between Clifford algebras and quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, the distance between two points in three-dimensional Euclidean space is found by taking the square root of a quadratic form involving six vars come from the contraction property. This rule also gives the space a metricSee: Metrics International System of Units, colloquially called the Metric System, and also metrication. Software metric. Metric space, treating a mathematical abstraction of the notion of distance''. Metric tensor. Normed vector space. Metric and metrica defined by the naturally derived inner product. It is to be noted that in geometric algebra in all its generality there is no restriction whatsoever on the value of the scalar, it can very well be negative, even zero (in that case, the possibility of an inner product is ruled out if you require ).

The usual dot productIn mathematics, the dot product (also known as the scalar product and the inner product is a function (·) : V × V → F, where V is a vector space and F its underlying field. In other words, it maps a pair of vectors to a scalar. When the latter term i and cross productIn mathematics, the cross product is a binary operation on vectors in three dimensions. It is also known as the vector product or outer product . It differs from the dot product in that it results in a vector rather than in a scalar. Its main use lies in of traditional vector algebra (on ) find their places in geometric algebra as the inner product

(which is symmetric) and the outer product

with

(which is antisymmetric). Relevant is the distinction between axial and polar vectors in vector algebra, which is natural in geometric algebra as the mere distinction between vectors and bivectors (elements of grade two). The here is the unit pseudoscalar of Euclidean 3-space, with establishes a duality between the vectors and the bivectors, and is named so because of the expected property .

A useful example is , and to generate , an instance of geometric algebra called spacetime algebra by Hestenes. The electromagnetic field tensor, in this context, becomes just a bivector where the imaginary unit is the volume element, giving an example of the geometric reinterpretation of the traditional "tricks".

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Topics: Geometric Algebra Product Space Vector Clifford Generated...

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