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## Über dieses Buch

This is the fourth and revised edition of a well-received book that aims at bridging the gap between the engineering course of tensor algebra on the one side and the mathematical course of classical linear algebra on the other side. In accordance with the contemporary way of scientific publications, a modern absolute tensor notation is preferred throughout. The book provides a comprehensible exposition of the fundamental mathematical concepts of tensor calculus and enriches the presented material with many illustrative examples. In addition, the book also includes advanced chapters dealing with recent developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics. Hence, this monograph addresses graduate students as well as scientists working in this field. In each chapter numerous exercises are included, allowing for self-study and intense practice. Solutions to the exercises are also provided.

## Inhaltsverzeichnis

### Chapter 1. Vectors and Tensors in a Finite-Dimensional Space

We start with the definition of the vector space over the field of real numbers.

Mikhail Itskov

### Chapter 2. Vector and Tensor Analysis in Euclidean Space

In the following we consider a vector-valued function

$$\varvec{x}\left( t\right)$$

x

t

and a tensor-valued function

$$\mathbf {A}\left( t\right)$$

A

t

of a real variable

$$t$$

t

.

Mikhail Itskov

### Chapter 3. Curves and Surfaces in Three-Dimensional Euclidean Space

A curve in three-dimensional space is defined by a vector function

Mikhail Itskov

### Chapter 4. Eigenvalue Problem and Spectral Decomposition of Second-Order Tensors

So far we have considered solely real vectors and real vector spaces.

Mikhail Itskov

### Chapter 5. Fourth-Order Tensors

Fourth-order tensors play an important role in continuum mechanics where they appear as elasticity and compliance tensors.

Mikhail Itskov

### Chapter 6. Analysis of Tensor Functions

Let us consider a real scalar-valued function

$$f\left( {{\mathbf {A}}}_1,{{\mathbf {A}}}_2,\ldots ,{{\mathbf {A}}}_l\right)$$

f

A

1

,

A

2

,

,

A

l

of second-order tensors

$${{\mathbf {A}}}_k\in {\mathbf {L}}{\text {in}}^n \, \left( k=1,2,\ldots ,l\right)$$

A

k

L

in

n

k

=

1

,

2

,

,

l

.

Mikhail Itskov

### Chapter 7. Analytic Tensor Functions

In the previous chapter we discussed isotropic and anisotropic tensor functions and their general representations.

Mikhail Itskov

### Chapter 8. Applications to Continuum Mechanics

Let us consider an infinitesimal vector

$$\mathrm{{d}}\varvec{ X }$$

d

X

in the reference configuration of a material body and its counterpart

$$\mathrm{{d}}\varvec{ x }$$

d

x

in the current configuration.

Mikhail Itskov

### Chapter 9. Solutions

In this chapter, solutions of the exercises are presented.

Mikhail Itskov

### Backmatter

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