It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In algebra, numbers are often represented by symbols called variables such as a, n, x, y or z.
Mathematically, a geometric algebra may be defined as the Clifford algebra of a vector space with a quadratic form. Clifford's contribution was to define a new product, the geometric product, that united the Grassmann and Hamilton algebras into a single structure.
Adding the dual of the Grassmann exterior product the "meet" allows the use of the Grassmann—Cayley algebraand a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra CGA providing a framework for classical geometries.
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. 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. Specific examples of geometric algebras applied in physics include the spacetime algebra or the less common alternative formulation, the algebra of physical space and the conformal geometric algebra.
Geometric calculusan extension of GA that incorporates differentiation and integrationcan be used to formulate other theories such as complex analysisdifferential geometrye.
Geometric algebra has been advocated, most notably by David Hestenes  and Chris Doran as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanicselectromagnetic theory and relativity.
The geometric product was first briefly mentioned by Hermann Grassmann who was chiefly interested in developing the closely related exterior algebra. InWilliam Kingdon 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 in the s by Hestenes, who advocated its importance to relativistic physics.Learn more about our Ivy-League research paper writing service.
Frequently asked questions about custom term papers, essays, research papers, and projects are answered here. A Time-line for the History of Mathematics (Many of the early dates are approximates) This work is under constant revision, so come back later.
Please report any errors to me at [email protected] Undergraduate Research Ideas. He had written a paper called the "Trustworthy Jackknife" in which he tried to figure out when the jackknife method gave dependable variance estimates.
you will use ideas in abstract algebra and Fourier analysis to write an efficient computer program that is part of the JPEG image compression .
The Journal of Pure and Applied Algebra concentrates on that part of algebra likely to be of general mathematical interest: algebraic results with immediate applications, and the development of algebraic theories of sufficiently general relevance to allow for future applications..
Contact Details for Submission Manuscripts may be submitted to [email protected] but preferably directly to. The geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which is a superset of both the scalars and the vector srmvision.comatically, a geometric algebra may be defined as the Clifford algebra of a vector space with a quadratic form.
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