# Tensors Unravelled

#### C. Pozrikidis

Forthcoming in 2020

Publisher

### Software

This book is accompanied by the Matlab/Octave library TUNLIB_20.04
The main directories (folders) correspond to the book chapters

### From the Preface

A vector is a physical entity endowed with magnitude and orientation; examples are the position, the velocity, and the acceleration vector. A vector is typically described by its components in a chosen frame of reference: Cartesian or non-Cartesian, rectilinear or curvilinear. A tensor is 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-index or higher-dimensional array.

Vectors and tensors are distinguished by our ability to deduce their components in a certain frame of reference from those in any other frame of reference using simple geometrical transformation rules. To indicate this ability, we say that the physical entity represented by a tensor is frame invariant. All physical entities should be frame invariant, otherwise observation and computation will be subjective, that is, they will depend on the position and orientation of an observer or measuring instrumentation.

A zeroth-order tensor is a scalar whose value is frame-independent; examples are the temperature or color of a star 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 material. Higher-order tensors and their components in an arbitrary frame of reference can be defined. An example is the four-index Riemann--Christoffel curvature tensor.

My purpose in this book is to present a concise yet accessible introduction to vectors and tensors in Cartesian or non-Cartesian, rectilinear or curvilinear coordinates. 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. Some original material is included.

Tensors Unravelled is suitable for self-study and as a main or supplemental text in an upper-level undergraduate or graduate level core or elective course in mathematics, physics, and engineering.

### Why this book?

Tensors unravelled is distinguished by the following unique features:

• An introduction to tensors is presented preceding the notion of curvilinear coordinates. This is an essential and unique rather than superficial distinction.
• The concept of uniadic, dyadic, and multiadic bases is emphasized with reference to one-, two-, and higher-index tensor arrays.
• Non-Cartesian coordinates are discussed from the viewpoint of applied mathematics and engineering; comprehensive expressions from differential calculus are derived using a novel methodology.
• Applications in several fields of science and engineering are discussed, including fluid mechanics, transport, membrane mechanics, and mechanics of shells.
• Convected coordinates are discussed and expressions for Green's function of the convection--diffusion equation in curvilinear coordinates are derived.
• The apparatus of surface curvilinear coordinates is discussed in terms of the curvature tensor with applications.
• Theory and computation are coupled by way of computer codes that illustrate and implement theoretical predictions.
• A suite of computer codes that confirm theoretical derivations and encode numerical methods for computing solutions of selected differential equations accompany the text.

### Contents

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 Cartesian 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. In Chapter 2, dyadic bases and the notion of tensors are introduced in a similar way, the description of a tensor in terms of its components in a specified base is discussed, and transformation rules for tensor components are established.

Vectors and tensor descriptions in biorthogonal frames are discussed in Chapter 3 where the apparatus of covariant and contravariant bases and associated vector components is explained, and relevant transformation rules are established. The discussion serves as a prelude to curvilinear coordinates where the biorthogonal bases are constructed with reference to contravariant and covariant coordinates defined by two families of curved lines.

Basic notions and fundamental concepts underlying the construction of curvilinear coordinates in two dimensions are discussed in Chapter 4. Following an abbreviated theoretical discussion, finite-difference methods for solving the Laplace and Poisson equations are developed to demonstrate the practical usage of the theoretical apparatus.

Non-Cartesian coordinates are discussed in detail in Chapter 5 from the viewpoint of applied mathematics and engineering. Following this discussion, differential calculus in non-Cartesian coordinates is introduced in Chapter 6 and expressions for partial derivatives, directional derivatives, the divergence, the curl, the gradient, and other differential operations are derived.

Applications of calculus in non-Cartesian coordinates in mathematical physics are discussed in Chapter 7. While stating the contravariant or covariant component of equations in arbitrary curvilinear coordinates is straightforward, subtleties are encountered in the case of moving coordinates with regard to the meaning of the inherent time derivative. The notion of convected coordinates is introduced and Green's functions of the convection--diffusion equation are derived.

In Chapter 8, the apparatus of curvilinear coordinates is specialized to surfaces with the introduction of the curvature tensor. Surface calculus is discussed in Chapter 9, and the Gauss surface divergence theorem is established with applications to the equilibrium of membranes and thin shells.

Forthcoming.