Geometric algebra

B. R. McDonald, Geometric Algebra Over Local Rings (1976)... 

Basis Representation for Three-Dimensional Geometric Algebra... 

We demonstrate the elegance of geometric algebra by deriving Eq. (29) as follows ... 

Geometric algebra approach offers some advantages over other methods presented in the literature. First of all, atomic position vectors themselves are manipulated instead of their components, and hence all expressions are simple at each stage of derivation. This is not the case when Cartesian components and back substitutions are used to obtain contravariant measuring vectors [57]. As a... 

Shape Coordinates. The gradients of any explicitly defined curvilinear shape coordinate can be found by the methods of geometric algebra. The results given in Table I are useful in this context. We solve the gradients of a bond angle 0paT by direct vectorial differentiation as a simple example ... 

To put it briefly, geometric algebra [32-35] is an extension of the real number system to incorporate the geometric concept of direction that is, it is a system of directed numbers. Geometric algebra integrates the well-established branches... 

We present the sum and product rules of geometric algebra in this section. There is only one kind of addition, but in contrast there are several products. However, geometric product is in a special position, because all other products (such as the inner and cross products) can be derived from it. 

Such a simple expression does not exist in the ordinary vector algebra, where supplementary algebraic structures in the form of rotation matrices are needed. Geometric algebra offers an effective way of describing rotations. For example, the spinor cl /2 describing the net rotation of two successive rotations, first e 1/2, then c 2 2, is found by multiplying... 

To find more on the use of geometric algebra in spherical trigonometry, the reader should consult Ref. 50, pages 661-667. 

The machinery of geometric algebra makes it possible to differentiate and integrate functions of vector variables in a coordinate-free manner. The conventionally separated concepts of the gradient, divergence, and curl are... 

The right-hand side in Eq. (A 134) may appear at first sight peculiar to the reader unfamiliar with geometric algebra, because such an expression does not exist in ordinary vector algebra. Eq. (A 134) can be derived by substituting A = Va to the general rule Vaa A = A (where A is any multivector independent of a). As a hopefully useful example,... 

J. Pesonen, Vibrational coordinates and their gradients A geometric algebra approach. J. Chem. Phys. 112, 3121-3132 (2000). 

J. Pesonen, Application of geometric algebra to theoretical molecular spectroscopy (http // ethesis.helsinki.fi/julkaisut/mat/kemia/vk/pesonen/). PhD thesis, Helsinki, 2001. 

J. Pesonen, Exact kinetic energy operators for polyatomic molecules, in Applications of Geometric Algebra in Computer Science and Engineering, L. Dorst, C. Doran, and J. Lasenby, eds., Birkhauser, Boston, 2002, p. 261-270. 

T. Havel and I. Najfeld, Applications of geometric algebra to the theory of molecular conformation. Part 1. The optimum alignment problem. J. Mol. Struct. (Theochem.) 308, 241-262 (1994). 

G. Sobczyk, Simplicial calculus with geometric algebra (http //modelingnts. la. asu. edu/pdf/SIMP CAL. pdf), in Clifford Algebras and Their Applications in Mathematical Physics, A. Micali, R. Boudet, and J. Helmstetter, eds., Kluwer, Dordrecht, 1992, p. 279-292. 

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