Related papers: Generalised vector products in three-dimensional g…
We study the general rational trigonometry of a tetrahedron, based on quadrances, spreads and solid spreads, using vector products associated to an arbitrary symmetric bilinear form over a general field, not of characteristic two. This…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
Vector calculus in three-dimensional space is ubiquitous in applications of mathematics in physics and engineering. Its two-dimensional version is, however, quite rare. Here we try to provide a pedagogical account of the subject. It is…
Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…
A vector space over a field $\mathbb{F}$ is a set $V$ together with two binary operations, called vector addition and scalar multiplication. It is standard practice to think of a Euclidean space $\mathbb{R}^n$ as an $n$-dimensional real…
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…
This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space $\R{E} ^3 $ we consider a surface $\varPhi %\colon \vect{x} = \vect{x}(u^1,u^2) $ with position vector field $\vect{x}$, which is…
The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main…
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…
Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.
The idea of standardised co-ordinates in three-dimensional affine space is defined, by way of the Standard tetrahedron. By performing an affine map on a general tetrahedron, we may replace the study of a general tetrahedron over a specific…
Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…
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…
This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields.…
It is shown that scalar product of two vectors can be introduced in any geometry (metric space) independently of possibility of the linear space introduction. In general, linear properties of scalar product are restricted. Domain of…
One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this…
We consider a generalized angle in complex normed vector spaces. Its definition corresponds to the definition of the well known Euclidean angle in real inner product spaces. Not surprisingly it yields complex values as `angles'. This…
We develop the basic formulae of hyperspherical trigonometry in multidimensional Euclidean space, using multidimensional vector products, and their conversion to identities for elliptic functions. We show that the basic addition formulae…
We suggest a generalization of vector calculus for the case of non-integer dimensional space. The first and second orders operations such as gradient, divergence, the scalar and vector Laplace operators for non-integer dimensional space are…