This web page is intended to be a compilation of the books written by Javier de Montoliu Siscar, Doctor of Industrial Engineering,
published in PDF,
referring to vector spaces, tensor algebra, tensor calculus and applications of the tensor algebra in mathematics and engineering.
For any comments on any of the books, please contact us by email:
jmontoliu@mautar.com
- Vector spaces
In this essay, an attempt is made to summarize the main general
properties of vector spaces, as well as the relationships that
can be established between them when they are structured on the same field.
Specifically, linear and multilinear applications are discussed,
both in their general expression and in the form of products between vectors.
Among the products, we will consider especially the tensor product
that defines the tensors and those scalar products that define the
dual spaces. For this purpose and for greater ease, we will not
dispense with the consideration of complex conjugate vector spaces.
The most important vector spaces are succinctly outlined,
and we will be particularly interested in those of n dimensions
with finite n, although some characteristics of infinite n are mentioned.
Finally, the set of affine tensors to a Euclidean or properly
Euclidean space of finite dimension n is studied, for which
an algebra structure is established whose application to
properly Euclidean spaces is the object of development in another work.
The purpose of this essay is not only to present in an elementary
way the general properties of vector spaces, but also to facilitate
the understanding of the algebra established in the last chapter,
which we consider of interest, since it not only allows us to represent
tensorially any linear or multilinear operator, but also to find the
result of its application on a given tensor or vector by means of a
simple algebraic operation, with an intrinsic character.
- Tensor Algebra and Analysis
The aim of this essay is to facilitate calculations in Linear Algebra,
especially those referring to the different subjects of physical and
geometric sciences in their relationship with a specific affine space,
always properly Euclidean, based on considering three-dimensionality
as a particular case of n-dimensionality with n finite.
Its possible applications to relativistic and quantum physics
are not developed for the moment.
This Algebra and Tensor Calculus is especially useful because it does not
only operate with tensor magnitudes as such (which include vectors and scalars),
but also allows linear or multilinear operators to be considered as such in the calculus,
thus facilitating the formulation of the images they determine.
Tensorically expressible operators include derivatives, directional
or partial derivatives, as well as integral quantities.
The symbolism chosen for the tensors is intrinsic, and the
use and description of their characteristic components and
of the adopted vector bases has been limited to the cases
in which it has been necessary or convenient for a definition or a demonstration.
As for the expression of its characteristic components
in figures, this is done exceptionally as an example or particular case.
The algebra used is defined in the second chapter of the text,
and in the rest of the text, the algebra is applied to a detailed partial study of various tensors and their relationships.
In the first part, special attention is devoted to
second order tensors and their relation to square matrices.
In the second part we emphasize the applications of algebra
to the tensor expression of differentiable or integrable quantities
as well as spatial and differential derivatives and the intrinsic
expression of the Stokes and Ostrogradski formulas.
- Linear varieties
In this essay we will study in an elementary way the use of
curvilinear coordinate systems, and in particular their application
to Riemannian spaces. For this we will follow in general lines
the order of the text of "Elements of Tensor Algebra" by Lichnerowicz.
Our object is not to go deeper into these topics nor to make
demonstrations, but only to try to see if the intrinsic method
of algebra and tensor analysis that we have applied in previous
writings can be useful for the study of a Riemannian space.
The study is divided into two parts.
In the first part, a Euclidean space is considered in general
(even if it is not properly Euclidean), through the adoption
of a curvilinear coordinate system. This first part is an
introduction to the second part, which is devoted to Riemannian spaces.
- Quadrics
This essay aims to develop the knowledge of quadrics
and their properties using preferably a method of algebra and
tensor analysis based on intrinsic properties of tensor
quantities that has been exposed and developed by the author
in his Algebra and Tensor Analysis.
We can consider the present study as a continuation
of the author's previous essay on linear varieties.
Although no single element used by the method is
unknown per se, as a whole it appears to be unknown, and
although the intrinsic procedures are appreciated today, in
practice they do not crystallize as generally accepted
methods.
The calculation method used here is presented in a
specific practical application, the study of quadrics, and we
submit it to the consideration of the reader so that he can
judge for himself about its effectiveness and possible general
interest.
- Curvilinear Coordinates Riemannian Spaces
In this essay we will study in an elementary way the
use of curvilinear coordinate systems, and especially their
application to Riemannian spaces. For this we will follow in
general lines the order of the text of "Elements of Tensor
Algebra" by Lichnerowicz, although nowadays it has been
superseded.
Our object is not to go deeper into these topics nor
to make demonstrations, but only to try to see if the
intrinsic method of algebra and tensor analysis that we have
applied in previous writings can be useful for the study of a
Riemannian space.
The study is divided into two parts.
In the first part, a Euclidean space is considered
in general (even if it is not properly Euclidean), through the
adoption of a curvilinear coordinate system. This first part
is an introduction to the second part, which is devoted to
Riemannian spaces.
- Elasticity
The purpose of this essay was to verify the
usefulness of the tensor calculus method developed by the
author in "Algebra and Tensor Calculus" in ordinary calculus
and specifically in the study of elastic mechanics.
This method is a natural extension of ordinary
scalar algebra, which intrinsically considers vectors and
tensors of any order and simplifies most of the usual
operations.
With its application to the theory of elasticity,
which naturally leads to the conclusions we all know, we
believe it has proved its usefulness.
Regardless of this, this work has highlighted some
circumstances that have an impact on the subject and that are
often omitted. We mainly point out the following:
a) The velocity of the material points is a point
vector quantity, but this is not always the case with its
components relative rotational velocity and relative
deformation velocity. Consequently, the same is true for the
displacements.
b) The deformation tensor, which is normally used
for small deformations and is defined with the displacements,
is now considered the differential of a point magnitude tensor
and is defined by the velocity field.
c) A displacement vector is considered as point
magnitude, as well as another point magnitude vector that we
call original position vector.
d) The two forms of material waves are not exactly
proper to the movement of material points. One of the forms,
the condensation waves, lies in the relative expansion
coefficient, and the other form, the distortion waves, in the
rotational velocity, and they act independently of each other.
- Electrostatics and direct current
This text is essentially a transcription of
electrostatics and direct currents by Dr. Jose Maria Codina
Vidal, professor emeritus of electrical engineering at the
faculty of physical sciences of Barcelona.
The purpose of this paper is to express these themes
of the book, adapting them to the language I have used in my
work on algebra and tensor calculus.
Independently of the transcription, some topics that
do not appear in the original book are also developed.