Multivector Functions of a Multivector Variable

In this paper, we give one possible definition for functions of several variables applied to endomorphisms of finite dimensional C-vector spaces. This definition is consistent with the usual notion of a function of a square matrix. Some basic rules of computation are given, as well as a formula for differentiation of it. A few examples of possible applications is also given at the end of the article.

We give in this paper which is the fifth in a series of eight a theory of covariant derivatives of multivector and extensor fields based on the geometric calculus of an arbitrary smooth manifold M, and the notion of a connection extensor field defining a parallelism structure on M. Also we give a novel and intrinsic presentation (i.e., one that does not depend on a chosen orthonormal moving frame) of the torsion and curvature fields of Cartan's theory. Two kinds of Cartan's c...

Higher order derivatives of functions are structured high dimensional objects which lend themselves to many alternative representations, with the most popular being multi-index, matrix and tensor representations. The choice between them depends on the desired analysis since each presents its own advantages and disadvantages. In this paper, we highlight a vectorized representation, in which higher order derivatives are expressed as vectors. This allows us to construct an elega...

As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford mult...

The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on tensor calculus as they are based on the materials and conventions given in those documents. They can be used as a reference for a first course on the subject or as part of a course on tensor calculus.

The motivation of this paper is to construct the theory of vector calculus of multivariate arithmetical functions. We prove analogues of integral theorems and Poincare's lemma.

This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular cases of symmetric multilinear forms; therefore, using their symmetric properties and their coordinate representation, we implement functions to calculate mixed partial derivatives. Moreover, with directional derivatives, we deduce concise ...

This article studies divergence of multivector fields on Banach manifolds with a Radon measure. The proposed definition is consistent with the classical divergence from finite-dimensional differential geometry. Certain natural properties of divergence are transferred to the case of infinite dimension.

This paper presents a thoughful review of: (a) the Clifford algebra Cl(H_{V}) of multivecfors which is naturally associated with a hyperbolic space H_{V}; (b) the study of the properties of the duality product of multivectors and multiforms; (c) the theory of k multivector and l multiform variables multivector extensors over V and (d) the use of the above mentioned structures to present a theory of the parallelism structure on an arbitrary smooth manifold introducing the conc...

In this paper we connect classical differential geometry with the concepts from geometric calculus. Moreover, we introduce and analyze a more general Laplacian for multivector-valued functions on manifolds. This allows us to formulate a higher codimensional analog of Jacobi`s field equation.