Multivector Functions of a Multivector Variable

We discuss a version of the fundamental theorem of calculus in several variables and some applications, of potential interest as a teaching material in undergraduate courses.

This is the sixth, concluding part of a series of papers the first five of which have been submitted to the present archive in mid 1998 and published as INR preprints in 1999. The present paper was printed as an INR preprint, too, but for nonscientific reasons was never made public in any form, electronic or hard-copy. In it I define the bivector derivative for four- and five-vector fields in the case of arbitrary Riemannian geometry; examine a more general case of five-vecto...

In our previous paper [International Journal of Theoretical Physics, 41 (2002), 1165-1190] we have shown, following the tradition of synthetic differential geometry, that div and rot are uniquely determined, so long as we require that the divergence theorem and the Stokes theorem should hold on the infinitesimal level. In this paper we will simplify the discussion considerably in terms of differential forms, leading to the natural derivation of exterior differentiation in the...

This is a draft of a textbook on differential forms. The primary target audience is sophmore level undergraduates enrolled in what would traditionally be a course in vector calculus. Later chapters will be of interest to advaced undergraduate and beginning graduate students. Applications include brief introductions to Maxwell's equations, foliations and contact structures, and DeRham cohomology.

This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text focuses on some subjects, with the intention of providing the reader with the main algebraic tools necessary for a modern course in continuum mechanics. The presentation of tensor algebra and analysis is intentionally done in Cartesian coor...

We study integrals of the form $\int_{\Omega}f\left( d\omega_1 , \ldots , d\omega_m \right), $ where $m \geq 1$ is a given integer, $1 \leq k_{i} \leq n$ are integers and $\omega_{i}$ is a $(k_{i}-1)$-form for all $1 \leq i \leq m$ and $ f:\prod_{i=1}^m \Lambda^{k_i}\left( \mathbb{R}^{n}\right) \rightarrow\mathbb{R}$ is a continuous function. We introduce the appropriate notions of convexity, namely vectorial ext. one convexity, vectorial ext. quasiconvexity and vectorial ext...

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.

In this paper, we intend to revisit Theorem 2 of [3] formulating it in a way that, weakening the hypotheses and, at the same time, highlighting the richer conclusion allowed by the proof, it can potentially be applicable to a broader range of different situations. Samples of such applications are also given.

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.

We introduce an original approach to geometric calculus in which we define derivatives and integrals on functions which depend on extended bodies in space--that is, paths, surfaces, and volumes etc. Though this theory remains to be fully completed, we present it at its current stage of development, and discuss it's connection to physical research, in particular its application to spinning particles in curved space.