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A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of extensor fields is present using algebraic and analytical tools developed in previous papers. Several important formulas are derived.

A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.

Using the theory of extensors developed in a previous paper we present a theory of the parallelism structure on arbitrary smooth manifold. Two kinds of Cartan connection operators are introduced and both appear in intrinsic versions (i.e., frame independent) of the first and second Cartan structure equations. Also, the concept of deformed parallelism structures and relative parallelism structures which play important role in the understanding of geometrical theories of the gravitational field are investigated.

In this paper we study in details the properties of the duality product of multivectors and multiforms (used in the definition of the hyperbolic Clifford algebra of multivefors) and introduce the theory of the k multivector and l multiform variables multivector (or multiform) extensors over V studying their properties with considerable detail.

We give in this paper which is the third in a series of four a theory of covariant derivatives of representatives of multivector and extensor fields on an arbitrary open set U of M, based on the geometric and extensor calculus on an arbitrary smooth manifold M. This is done by introducing the notion of a connection extensor field gamma defining a parallelism structure on U, which represents in a well defined way the action on U of the restriction there of some given connection del defined 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 connection operator fields are identified, and both appear in the intrinsic Cartan's structure equations satisfied by the Cartan's torsion and curvature extensor fields. We introduce moreover a metrical extensor g in U corresponding to the restriction there of given metric tensor \slg defined on M and also introduce the concept a geometric structure (U,gamma,g) for U and study metric compatibility of covariant derivatives induced by the connection extensor gamma. This permits the presentation of the concept of gauge (deformed) derivatives which satisfy noticeable properties useful in differential geometry and geometrical theories of the gravitational field. Several derivatives operators in metric and geometrical structures, like ordinary and covariant Hodge coderivatives and some duality identities are exhibit., Comment: This paper is an improved version of material contained in math.DG/0501560, math.DG/0501561, math.DG/0502001

This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the theory of its deformations leading to metric geometric algebras Cl(V,G) and some special types of extensors. Those tools permit obtaining, the remarkable golden formula relating calculations in Cl(V,G) with easier ones in Cl(V,G_{E}) (e.g., a noticeable relation between the Hodge star operators associated to G and G_{E}). Several useful examples are worked in details fo the purpose of transmitting the "tricks of the trade"., Comment: This paper (to appear in Int. J. Geom. Meth. Mod. Phys. 4 (6) 2007) is an improved version of material appearing in math.DG/0501556, math.DG/0501557, math.DG/0501558

This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel presentation of several topics of the differential geometry of (smooth) manifolds of arbitrary global topology. A key tool for the development of our program is the mastering of the euclidean geometrical algebra of multivectors that is detailed in the present paper.

This paper, sixth in a series of eight, uses the geometric calculus on manifolds developed in previous papers of the series to introduce through the concept of a metric extensor field g a metric structure for a smooth manifold M. The associated Christoffel operators, a notable decomposition of that object and the associated Levi-Civita connection field are given. The paper introduces also the concept of a geometrical structure for a manifold M as a triple (M,g,gamma), where gamma is a connection extensor field defining a parallelism structure for M . Next, the theory of metric compatible covariant derivatives is given and a relationship between the connection extensor fields and covariant derivatives of two deformed (metric compatible) geometrical structures (M,g,gamma) and (M,eta,gamma') is determined.

This paper, the third in a series of eight introduces some of the basic concepts of the theory of extensors needed for our formulation of the differential geometry of smooth manifolds . Key notions such as the extension and generalization operators of a given linear operator (a (1,1)-extensor) acting on a real vector space V are introduced and studied in details. Also, we introduce the notion of the determinant of a (1,1)-extensor and the concepts of standard and metric Hodge (star) operators, disclosing a non trivial and useful relation between them., Comment: revised version

The objective of the present paper (the second in a series of four) is to give a theory of multivector and extensor fields on a smooth manifold M of arbitrary topology based on the powerful geometric algebra of multivectors and extensors. Our approach does not suffer the problems of earlier attempts which are restricted to vector manifolds. It is based on the existence of canonical algebraic structures over the so-called canonical space associated to a local chart (U_{o},phi_{o}) of the maximal atlas of M. The key concepts of a-directional ordinary derivatives of multivector and extensor fields are defined and their properties studied. Also, we introduce the Lie algebra of smooth vector fields and the Hestenes derivatives whose properties are studied in details., Comment: revised version

We present an introduction to the mathematical theory of the Lagrangian formalism for multiform fields on Minkowski spacetime based on the multiform and extensor calculus. Our formulation gives a unified mathematical description for the main relativistic field theories including the gravitational field (which however will be discussed in a separate paper). We worked out several examples (including tricks of the trade), from simple to very sophisticated ones (like, e.g., the Dirac-Hestenes field on the more general gravitational background) which show the power and beauty of the formalism.

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.

In this paper we introduce the concept of \emph{multivector functionals.} We study some possible kinds of derivative operators that can act in interesting ways on these objects such as, e.g., the $A$-directional derivative and the generalized concepts of curl, divergence and gradient. The derivation rules are rigorously proved. Since the subject of this paper has not been developed in previous literature, we work out in details several examples of derivation of multivector functionals., Comment: Some references changed

In this paper we develop with considerable details a theory of multivector functions of a $p$-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., the $A$% -directional derivative (where $A$ is a $p$-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., Comment: Some references changed

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 the geometric information just contained in $g.$ The precise form of such $h$ is here determined. Moreover, we present and give a proof of the so-called \emph{golden formula,} which is important in many applications that naturally appear in ours studies of multivector functions, and differential geometry and theoretical physics.

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 of \emph{tensors} and \emph{extensors}. 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 do \emph{not} 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.

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 $V.$ These objects are elements of a linear space over the real field, denoted by $\bigwedge V.$ We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of two \emph{contraction operators} on $\bigwedge V,$ and the concept of euclidean \emph{interior} 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., Comment: Latex accent in author(s) was introduced Latex commands in abstract were corrected