Geometric algebra

Geometric algebra is a Clifford algebra given a geometric interpretation which makes it useful in an exceptionally wide range of physics problems, particularly those that involve rotations, phases or imaginary numbers. Proponents of geometric algebra say that it more compactly and intuitively describes classical mechanics, quantum mechanics, electromagnetic theory and relativity than standard methods do.

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In mathematics, a geometric algebra mathcal{G}_n(mathcal{V}_n) is an algebra constructed over a vector space mathcal V_n in which a geometric product is defined. For all multivectors (the elements of the algebra) mathbf{A}, mathbf{B}, mathbf{C}, the geometric product has the following properties:

Related Topics:
Mathematics - Algebra - Vector space - Multivector

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• Closure
• Distributivity over the addition of multivectors:
• * $mathbf\{A\}(mathbf\{B\}\; +\; mathbf\{C\})\; =\; mathbf\{A\}mathbf\{B\}\; +\; mathbf\{A\}mathbf\{C\}$
• * $(mathbf\{A\}\; +\; mathbf\{B\})mathbf\{C\}\; =\; mathbf\{A\}mathbf\{C\}\; +\; mathbf\{B\}mathbf\{C\}$
• Associativity
• Unit (scalar) element:
• * $1\; ,\; mathbf\; A\; =\; mathbf\; A$
• Tensor contraction: for any "vector" (a grade-one element) $mathbf\{a\},\; mathbf\{a\}^2$ is a scalar (real number)
• Commutativity of the product by an scalar:
• * $lambda\; mathbf\; A\; =\; mathbf\; A\; lambda$

Note that the first two properties are needed to be an algebra. Next two make it an associative, unital algebra.

Related Topics:
Algebra - Associative - Unital

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The distinctive point of this formulation is the natural correspondence between geometric entities and the elements of the associative algebra. This comes from the fact that the geometric product is defined in terms of the dot product and the wedge product of vectors as

Related Topics:
Associative algebra - Dot product - Wedge product

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: mathbf a , mathbf b = mathbf a cdot mathbf b + mathbf a wedge mathbf b

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The original vector space mathcal V is constructed over the real numbers as scalars. From now on, a vector is something in mathcal V itself. Vectors will be represented by boldface, small case letters.

Related Topics:
Vector space - Real number

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The outer product (the exterior product, or the wedge product) wedge is defined such that the graded algebra (exterior algebra of Hermann Grassmann) wedge^nmathcal{V}_n of multivectors is generated. Multivectors are thus the direct sum of grade k elements (k-vectors), where k ranges from 0 (scalars) to n, the dimension of the original vector space mathcal V. Multivectors are represented here by boldface caps. Note that scalars and vectors become special cases of multivectors ("0-vectors" and "1-vectors", respectively).

Related Topics:
Outer product - Exterior product - Wedge product - Graded algebra - Exterior algebra - Hermann Grassmann

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Geometric algebra: The contraction rule ►