V. V. Fernández
State University of Campinas
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Featured researches published by V. V. Fernández.
Advances in Applied Clifford Algebras | 2001
V. V. Fernández; A. M. Moya; Waldyr Alves Rodrigues
LetV be ann-dimensional real vector space. In this paper we introduce the concept ofeuclidean Clifford algebraCℓ (V, GE) for a given euclidean structure onV , i.e., a pair (V, GE) where GE is an euclidean metric forV (also called an euclidean scalar product). Our construction ofCℓ(V, GE) has been designed to produce a powerful computational tool. We start introducing the concept ofmultivectors overV. These objects are elements of a linear space over the real field, denoted by ΛV. We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of twocontraction operators on ΛV; and the concept of euclideaninterior algebras. Equipped with these notions an euclidean Clifford product is easily introduced. We worked out with considerable details several important identities and useful formulas, to help the reader to develope a skill on the subject, preparing himself for the reading of the following papers in this series.
Advances in Applied Clifford Algebras | 2001
A. M. Moya; V. V. Fernández; Waldyr Alves Rodrigues
In this paper we introduce the concept of metric Clifford algebraCℓ(V; g) for ann-dimensional real vector spaceV endowed with a metric extensor g whose signature is (p; q), withp+q=n. The metric Clifford product onCℓ (V; g) appears as a well-defined deformation (induced by g) of an euclidean Clifford product onCℓ (V). Associated with the metric extensorg; there is a gauge metric extensorh which codifies all the geometric information just contained ing: The precise form of suchh is here determined. Moreover, we present and give a proof of the so-calledgolden formula, which is important in many applications that naturally appear in our studies of multivector functions, and differential geometry and theoretical physics.
Advances in Applied Clifford Algebras | 2001
A. M. Moya; V. V. Fernández; Waldyr Alves Rodrigues
This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar to the usual theory of vector functions of a real variable, has some subtle issues which make its presentation worhtwhile.We refer in particular to the derivative rules involving exterior and Clifford products, and also to the rule for derivation of a composition of an ordinary scalar function with a multivector function of a real variable.
Advances in Applied Clifford Algebras | 2001
A. M. Moya; V. V. Fernández; Waldyr A. Rodrigues
In this paper we develop with considerable details a theory of multivector functions of ap-vector variable. The concepts of limit, continuity and differentiability are rigorously studied. Several important types of derivatives for these multivector functions are introduced, as e.g., theA-directional derivative (whereA is ap-vector) and the generalized concepts of curl, divergence and gradient. The derivation rules for different types of products of multivector functions and for compositon of multivector functions are proved.
Archive | 2000
V. V. Fernández; A. M. Moya; Waldyr A. Rodrigues
We present a general theory of covariant derivative operators (linear connections) on a Minkowski manifold (represented as an affine space (M, M*) using the powerful multiform calculus. When a gauge metric extensor G (generated by a gauge distortion extensor h) is introduced in the Minkowski manifold, we get a theory that permits the introduction of general Riemann-Cartan-Weyl geometries. The concept of gauge covariant derivatives is introduced as the key notion necessary to generate linear connections that are compatible with G, thus, permitting the construction of Riemann-Cartan geometries. Many results of genuine mathematical interest are obtained. Moreover, such results are fundamental for building a consistent formulation of a theory of the gravitational field in flat spacetime. Some important examples of applications of our theory are worked in details.
arXiv: Mathematical Physics | 2001
A. M. Moya; V. V. Fernández; Waldyr Alves RodriguesJr.
We present the Lagrangian formalism for multiform fields on Minkowski spacetime based on the multiform and extensor calculus. The formulation gives a unified mathematical description for the relativistic field theories including the gravitational field. We work out examples including the Dirac—Hestenes field on the gravitational background.
Advances in Applied Clifford Algebras | 2001
V. V. Fernández; A. M. Moya; Waldyr Alves Rodrigues
In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms oftensors andextensors. A nice property of metric extensors is that they have inverses which are also themselves metric extensors. This property is not shared by metric tensors because tensors donot have inverses. We relate the definition of determinant of a metric extensor with the classical determinant of the corresponding matrix associated to the metric tensor in a given vector basis. Previous identifications of these concepts are equivocated. The use of metric extensor permits sophisticated calculations without the introduction of matrix representations.
International Journal of Geometric Methods in Modern Physics | 2007
V. V. Fernández; R. da Rocha; A. M. Moya; Waldyr Alves Rodrigues
Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection ∇) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of a triple (M, ∇, g) is investigated for each particular open set U ⊂ M through the introduction of a geometric structure on U, i.e. a triple (U, γ, g), where γ is a general connection field on U and g is a metric extensor field associated to g. The relation between geometrical structures related to gauge extensor fields is clarified. These geometries may be said to be deformations one of each other. Moreover, we study the important case of a class of deformed Levi–Civita geometrical structures and prove key theorems about them that are important in the formulation of geometric theories of the gravitational field.
Archive | 2005
V. V. Fernández; A. M. Moya; Waldyr Alves Rodrigues; R. da Rocha
arXiv: Differential Geometry | 2005
V. V. Fernández; Waldyr Alves Rodrigues; A. M. Moya; Roldao da Rocha