Tensors Unravelled
C. Pozrikidis
Chester & Bennington
420 pages including covers
Download the first version
tun_26.01.pdf
Download the accompaning Matlab programs
as a tar file:
TUNLIB_26.01.tar
Please forgive typographical, conceptual, and other errors.
The book should still be readable, mostly accurate,
and should fulfil its intended role.
Further revisions will correct these errors.
Why now?
I am in poor health and uncertain as to
when I will kick the bucket.
Remember:
It's not the size of the dog in the fight,
it's the size of fight in the dog.
What is this?
A vector is a physical entity
endowed with magnitude and orientation;
examples are the position, the velocity, and the acceleration.
A vector is typically described by its components in a chosen frame
of reference, Cartesian or non-Cartesian, rectilinear or curvilinear.
A tensor is also a physical entity
described by a higher number of
components in a chosen frame of reference.
In computational practice,
the components of a tensor are typically stored in a
two- or higher-dimensional array.
Vectors and tensors are distinguished by our ability to
deduce their components in a certain frame
of reference defined by a base from those in any
other frame of reference defined by another base by simple
geometrical transformations.
To indicate this ability, we say that the physical entity
represented by a tensor is objective or frame invariant.
All physical entities should be frame invariant;
if they were not, observation and computation
would be subjective,
that is, the results would depend on the position and orientation
of an observer or measuring instrument.
A zeroth-order tensor is a scalar whose value is frame-independent;
examples are the temperature of a star,
the angle between two vectors,
and the distance between two cities.
A first-order tensor is a vector
that has the same magnitude and points in the same direction
independent of the location of an observer.
The stress tensor is a second-order tensor
encapsulating the tractions exerted
on three small mutually perpendicular faces in a solid or fluid.
Higher-order tensors and their components in an
arbitrary frame of reference can be defined.
Examples
are the alternating three-index tensor
and the Riemann--Christoffel four-index curvature tensor.
My goal in this book is to present a concise
and accessible introduction
to vectors and tensors in Cartesian
or non-Cartesian, rectilinear or curvilinear coordinates
in a way that couples theory and numerical computation.
The notion of uniadic, dyadic, and multiadic bases is emphasized,
differential operations on vector and tensor fields
inside volumes and over surfaces are derived
in terms of the Christoffel symbols and the curvature tensor,
and applications in fluid mechanics, membrane theory,
and theory of shells are discussed.
Original derivations and novel approaches are presented,
including the construction of covariant coordinate fields
and the derivation of Green's function
of the convection--diffusion equation.
In Chapter 1, the concept of vectors endowed with magnitude
and orientation is introduced
and the description of a vector
in terms of its components in a specified base is discussed.
Transformation rules for vector components
naturally leads us
to the notion of vectors as first-order tensors
as opposed to mere one-dimensional numerical arrays.
Dyadic bases and the concept of tensors
are introduced in a similar way,
the description of tensors
in terms of their components in a specified base is discussed,
and transformation rules
are established.
In Chapter 2,
vectors and tensor representations in
dual biorthogonal bases
are discussed,
the apparatus of covariant and contravariant
bases and associated components is explained,
and relevant transformation rules
for vectors and tensors are established.
The discussion serves as a natural introduction
to the subject ot curvilinear coordinates
where biorthonormal bases are constructed with reference
to contravariant and covariant coordinates
defined by families of curved lines in space.
This natural introduction serves to emphasize
that a tensor is a tensor is a tensor:
the concept of covariant and contravariant tensors
is not appropriate.
In Chapter 3,
basic notions
and fundamental concepts underlying
the structure, construction, and properties
of curvilinear coordinates in two dimensions
are discussed.
In particular, the dual bases discussed in Chapter 2
are reintroduced with reference to contravariant and covariant
coordinates and associated base vectors.
Following this introduction,
finite-difference methods for solving the Laplace and Poisson equations
on structured grids
are developed and implemented to demonstrate
the practical usefulness of the
theoretical apparatus.
In Chapter 4,
a comprehensive
discussion of tensors in non-Cartesian coordinates is presented
from the viewpoint of applied mathematics,
physics, and engineering.
The Christoffel symbols are defined in terms
of derivatives of covariant base vectors
with respect to contravariant coordinates,
and the notion of covariant derivatives
of vector and tensor components is discussed.
A covariant derivative is a derivative
of a vector or tensor component with respect to
a contravariant coordinate.
In Chapter 5,
vector and tensor calculus
on non-Cartesian coordinates discussed,
and expressions for the divergence,
the curl, the gradient, the laplacian,
and other differential operations
are derived using an expedient method that circumvents
a great deal of manipulations.
Having introduced the necessary framework,
applications
in mathematical physics
are discussed.
While stating the contravariant or covariant components
of the governing equations
in arbitrary curvilinear coordinates
is straightforward,
subtleties arise in the case of moving coordinates.
The notion of convected coordinates is introduced
and expressions for Green's functions of the
convection--diffusion equation are derived.
In Chapter 6,
the apparatus of curvilinear coordinates
is specialized to surfaces embedded in space
with the introduction of the curvature tensor.
Surface calculus is discussed
and the Gauss surface divergence theorem is established
with applications
to force and bending moment
equilibria of membranes and thin shells.
A suite of
Matlab codes
encapsulated in a library named TUNLIB accompany the text.
The codes confirm theoretical derivations
presented in the book
and encode numerical methods
for computing solutions
of selected partial differential equations.
This book is suitable for self study and
as a text in an upper-level undergraduate
or graduate level core or elective course.
The theoretical discussion and computational developments
assume an upper-level undergraduate
or entry-level graduate level knowledge
of applied mathematics
on readily accessible topics.
Random acts of kindness
If you enjoyed reading this book,
please consider performing
a random act of kindness,
saying a prayer for the author,
saying another prayer for the pretender,
or donating what you can afford
to the St. Jude Children's Research Hospital
in Memphis, Tennessee.
Please be kind to one another,
you never know what kind of trauma lies
behind the smile.
Fly high in Heaven Robin and Chester.