Multivector Functions of a Real Variable

There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. With such a range of vector formalisms in use, it thus appears that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising,...

In this paper, several differentiability criteria for real functions of multiple variables in n-dimensional Euclidean space are considered. Simple and easy-to-use Cauchy-like criterion is formulated and proven. Relaxed sufficient conditions for differentiability that do not require continuity of all partial derivatives are suggested. Generalization of the Cauchy-like criterion for functions on cross products of normed vector spaces (not necessarily Banach spaces) is discussed...

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 ...

Let $V$ be a $n$-dimensional real vector space. In this paper we introduce the concept of \emph{euclidean} Clifford algebra $\mathcal{C\ell}(V,G_{E})$ for a given euclidean structure on $V,$ i.e., a pair $(V,G_{E})$ where $G_{E}$ is a euclidean metric for $V$ (also called an euclidean scalar product). Our construction of $\mathcal{C\ell}(V,G_{E})$ has been designed to produce a powerful computational tool. We start introducing the concept of \emph{multivectors} over $V.$ Thes...

In this paper we introduce the concept of metric Clifford algebra $\mathcal{C\ell}(V,g)$ for a $n$-dimensional real vector space $V$ endowed with a metric extensor $g$ whose signature is $(p,q)$, with $p+q=n$. The metric Clifford product on $\mathcal{C\ell}(V,g)$ appears as a well-defined \emph{deformation}(induced by $g$) of an euclidean Clifford product on $\mathcal{C\ell}(V)$. Associated with the metric extensor $g,$ there is a gauge metric extensor $h$ which codifies all ...

This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f'(x)$ and of the integral $\int_a^bf(x)dx$. Then we investigate the case of the complex functions $f:\mathbb C\to\mathbb C$, and notably the holomorphic functions, and harmonic functions. Then, we discuss the multivariable functions, $f:\mathbb R^N\...

In this paper, the second in a series of eight we continue our development of the basic tools of the multivector and extensor calculus which are used in our formulation of the differential geometry of smooth manifolds of arbitrary topology . We introduce metric and gauge extensors, pseudo-orthogonal metric extensors, gauge bases, tetrad bases and prove the remarkable golden formula, which permit us to view any Clifford algebra Cl(V,G) as a deformation of the euclidean Cliffor...

Motivated by the general problem of extending the classical theory of holomorphic functions of a complex variable to the case of quater- nion functions, we give a notion of an H-derivative for functions of one quaternion variable. We show that the elementary quaternion func- tions introduced by Hamilton as well as the quaternion logarithm function possess such a derivative. We conclude by establishing rules for calculating H-derivatives.

For students and their lecturers and instructors interested in the natural problem of a possible generalization of l'Hopital's rule for functions depending on two or more variables, we offer our approach. For instructors, we discuss the technique of constructing indeterminate forms at a given point and having a given double limit.

The geometric calculus based on Clifford algebra is a very useful tool for geometry and physics. It describes a geometric structure which is much richer than the ordinary geometry of spacetime. A Clifford manifold ($C$-space) consists not only of points, but also of 1-loops, 2-loops, etc.. They are associated with multivectors which are the wedge product of the basis vectors, the generators of Clifford algebra. We assume that $C$-space is the true space in which physics takes...