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Geometric Algebra For Physicists

Some members of the geometric algebra represent geometric objects in Rn. Other members represent geometric operations on the geometric objects. Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas,including linear algebra, multivariable calculus, real analysis, complex analysis, and euclidean, noneuclidean, and projective geometry. They provide a unified mathematical language for physics (classical and quantum mechanics, electrodynamics, relativity), the geometrical aspects of computer science (e.g., graphics, robotics, computer vision), and engineering.

Geometric Algebra for Physicists


This textbook for the first undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering a majority of the usual linear algebra topics. The link is to the book's web page.

I have created a six video YouTube playlistGeometric Algebra, about 72 minutes in all, taken from the book. Some knowledge of linear algebra is a prerequisite for the videos, but no knowledge of geometric algebra is assumed.The book assumes no previous knowledge of linear algebra.

This textbook for the first undergraduate vector calculus course presents a unified treatment of vector calculus and geometric calculus, while covering a majority of the usual vector calculus topics. The link is to the book's web page.

I have created a five video YouTube playlistGeometric Calculus, about 53 minutes in all, taken from the book. It is a sequel to my Geometric Algebra playlist. Some knowledge of vector calculus is a prerequisite for the videos, but no knowledge of geometric calculus is assumed.The book assumes no knowledge of vector calculus.

The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.

Abstract: Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the vector derivative exists and is Lebesgue integrable. We also give sufficient conditions that the vector derivative exists.

"My purpose here is to provide, with a minimum of mathematical machinery and in the fewest possible pages, a clear and careful explanation of the physical principles and applications of classical general relativity. The prerequisites are single variable calculus, a few basic facts about partial derivatives and line integrals, a little matrix algebra, and some basic physics. Only a bit of the algebra of tensors is used; it is developed in about a page of the text. The book is for those seeking a conceptual understanding of the theory, not computational prowess. Despite it's brevity and modest prerequisites, it is a serious introduction to the physics and mathematics of general relativity which demands careful study. The book can stand alone as an introduction to general relativity or it can be used as an adjunct to standard texts."

In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions.

The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities in vector algebra such as oriented area, oriented angle of rotation, torque, angular momentum and the electromagnetic field. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of V \displaystyle V and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity.

Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes[4] and Chris Doran,[5] as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity.[6] GA has also found use as a computational tool in computer graphics[7] and robotics.

For vectors a \displaystyle a and b \displaystyle b , we may write the geometric product of any two vectors a \displaystyle a and b \displaystyle b as the sum of a symmetric product and an antisymmetric product:

The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically, a \displaystyle a and b \displaystyle b are parallel if their geometric product is equal to their inner product, whereas a \displaystyle a and b \displaystyle b are perpendicular if their geometric product is equal to their exterior product. In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with the dot product of standard vector algebra. The exterior product of two vectors can be identified with the signed area enclosed by a parallelogram the sides of which are the vectors. The cross product of two vectors in 3 \displaystyle 3 dimensions with positive-definite quadratic form is closely related to their exterior product.

Most instances of geometric algebras of interest have a nondegenerate quadratic form. If the quadratic form is fully degenerate, the inner product of any two vectors is always zero, and the geometric algebra is then simply an exterior algebra. Unless otherwise stated, this article will treat only nondegenerate geometric algebras.

A multivector that is the exterior product of r \displaystyle r linearly independent vectors is called a blade, and is said to be of grade r \displaystyle r .[e] A multivector that is the sum of blades of grade r \displaystyle r is called a (homogeneous) multivector of grade r \displaystyle r . From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.

Therefore, every blade of grade r \displaystyle r can be written as a geometric product of r \displaystyle r vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by a block matrix that is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions. If the new vectors of the nondegenerate subspace are normalized according to

The set of all possible products of n \displaystyle n orthogonal basis vectors with indices in increasing order, including 1 \displaystyle 1 as the empty product, forms a basis for the entire geometric algebra (an analogue of the PBW theorem). For example, the following is a basis for the geometric algebra G ( 3 , 0 ) \displaystyle \mathcal G(3,0) :

This is a grading as a vector space, but not as an algebra. Because the product of an r \displaystyle r -blade and an s \displaystyle s -blade is contained in the span of 0 \displaystyle 0 through r + s \displaystyle r+s -blades, the geometric algebra is a filtered algebra.

The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper (Dorst 2002) gives a full treatment of several different inner products developed for geometric algebras and their interrelationships, and the notation is taken from there. Many authors use the same symbol as for the inner product of vectors for their chosen extension (e.g. Hestenes and Perwass). No consistent notation has emerged.

Dorst (2002) makes an argument for the use of contractions in preference to Hestenes's inner product; they are algebraically more regular and have cleaner geometric interpretations. A number of identities incorporating the contractions are valid without restriction of their inputs.For example,

A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra.[16] For compactness, we'll use a single capital letter to represent an ordered set of vector indices. I.e., writing

Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup of linear functions on multivectors, which can still be used when necessary. The geometric algebra of an n \displaystyle n -dimensional vector space is spanned by a basis of 2 n \displaystyle 2^n elements. If a multivector is represented by a 2 n 1 \displaystyle 2^n\times 1 real column matrix of coefficients of a basis of the algebra, then all linear transformations of the multivector can be expressed as the matrix multiplication by a 2 n 2 n \displaystyle 2^n\times 2^n real matrix. However, such a general linear transformation allows arbitrary exchanges among grades, such as a "rotation" of a scalar into a vector, which has no evident geometric interpretation.

A general linear transformation from vectors to vectors is of interest. With the natural restriction to preserving the induced exterior algebra, the outermorphism of the linear transformation is the unique[h] extension of the versor. If f \displaystyle f is a linear function that maps vectors to vectors, then its outermorphism is the function that obeys the rule 041b061a72


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