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## Über dieses Buch

This is the fifth and revised edition of a well-received textbook that aims at bridging the gap between the engineering course of tensor algebra on the one hand and the mathematical course of classical linear algebra on the other hand. In accordance with the contemporary way of scientific publication, 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. As such, this new edition also discusses such modern topics of solid mechanics as electro- and magnetoelasticity. 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 textbook addresses graduate students as well as scientists working in this field and in particular dealing with multi-physical problems. 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 $$\mathbb {R}$$. A vector space is a set $$\mathbb {V}$$ of elements called vectors satisfying the following axioms.
Mikhail Itskov

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

In the following we consider a vector-valued function $$\varvec{x}\left( t\right)$$ and a tensor-valued function $$\mathbf {A}\left( t\right)$$ of a real variable t. Henceforth, we assume that these functions are continuous such that
\begin{aligned} \lim \limits _{t \rightarrow t_0} \left[ \varvec{x}\left( t \right) - \varvec{x}\left( t_0 \right) \right] = \varvec{ 0 }, \quad \lim \limits _{t \rightarrow t_0} \left[ \mathbf {A}\left( t \right) - \mathbf {A}\left( {t_0 } \right) \right] = \mathbf {0} \end{aligned}
for all $$t_0$$ within the definition domain. The functions $$\varvec{x}\left( t\right)$$ and $$\mathbf {A}\left( t\right)$$ are called differentiable if the following limits
\begin{aligned} \frac{\mathrm{d}\varvec{x}}{\mathrm{d}t} = \lim \limits _{s \rightarrow 0} \frac{\varvec{x}\left( t + s\right) - \varvec{x}\left( t \right) }{s}, \quad \frac{\mathrm{d}\mathbf {A}}{\mathrm{d}t} = \lim \limits _{s \rightarrow 0} \frac{\mathbf {A}\left( t + s \right) - \mathbf {A}\left( t \right) }{s} \end{aligned}
exist and are finite. They are referred to as the derivatives of the vector- and tensor-valued functions $$\varvec{x}\left( t\right)$$ and $$\mathbf {A}\left( t\right)$$, respectively.
Mikhail Itskov

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

A curve in three-dimensional space is defined by a vector function
$$\varvec{r}=\varvec{r}\left( t\right) , \quad \varvec{r}\in \mathbb {E}^3,$$
where the real variable t belongs to some interval: $$t_1\le t \le t_2$$. Henceforth, we assume that the function $$\varvec{r}\left( t\right)$$ is sufficiently differentiable and
\begin{aligned} \frac{\mathrm{d}\varvec{r}}{\mathrm{d}t}\ne \varvec{ 0 } \end{aligned}
over the whole definition domain. Specifying an arbitrary coordinate system (2.​16) as
\begin{aligned} \theta ^i=\theta ^i\left( \varvec{r}\right) , \quad i=1,2,3, \end{aligned}
the curve (3.1) can alternatively be defined by
\begin{aligned} \theta ^i=\theta ^i\left( t\right) , \quad i=1,2,3. \end{aligned}
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. For the purposes of this chapter an introduction of complex vectors is, however, necessary. Indeed, in the following we will see that the existence of a solution of an eigenvalue problem even for real second-order tensors can be guaranteed only within a complex vector space. In order to define the complex vector space let us consider ordered pairs $$\left\langle \varvec{x},\varvec{y}\right\rangle$$ of real vectors $$\varvec{x}$$ and $$\varvec{y} \in \mathbb {E}^n$$. The sum of two such pairs is defined by [18]
$$\left\langle \varvec{x}_1,\varvec{y}_1\right\rangle + \left\langle \varvec{x}_2,\varvec{y}_2\right\rangle =\left\langle \varvec{x}_1+\varvec{x}_2,\varvec{y}_1 + \varvec{y}_2\right\rangle .$$
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. In this section we define fourth-order tensors and learn some basic operations with them. To this end, we consider a set $$\varvec{\mathcal {L}}\text {in}^n$$ of all linear mappings of one second-order tensor into another one within $$\mathbf {L}\text {in}^n$$. Such mappings are denoted by a colon as
$$\mathbf {Y}=\varvec{\mathcal {A}} : \mathbf {X}, \quad \varvec{\mathcal {A}}\in \varvec{\mathcal {L}}\text {in}^n, \; \mathbf {Y}\in \mathbf {L}\text {in}^n, \; \forall \mathbf {X}\in \mathbf {L}\text {in}^n.$$
Mikhail Itskov

### Chapter 6. Analysis of Tensor Functions

Let us consider a real scalar-valued function $$f\left( \mathbf {A}_1,\mathbf {A}_2,\ldots ,\mathbf {A}_l\right)$$ of second-order tensors $$\mathbf {A}_k\in \mathbf {L}\text {in}^n \, \left( k=1,2,\ldots , l\right)$$. The function f is said to be isotropic if
\begin{aligned}&f\left( \mathbf {Q}\mathbf {A}_1\mathbf {Q}^\text {T},\mathbf {Q}\mathbf {A}_2\mathbf {Q}^\text {T},\ldots , \mathbf {Q}\mathbf {A}_l\mathbf {Q}^\text {T}\right) \nonumber \\&\qquad \qquad \qquad \qquad \qquad =f\left( \mathbf {A}_1,\mathbf {A}_2,\ldots ,\mathbf {A}_l\right) , \quad \forall \mathbf {Q}\in \mathbf {O}\text {rth}^n. \end{aligned}
Mikhail Itskov

### Chapter 7. Analytic Tensor Functions

In the previous chapter we discussed isotropic and anisotropic tensor functions and their general representations. Of particular interest in continuum mechanics are isotropic tensor-valued functions of one arbitrary (not necessarily symmetric) tensor. For example, the exponential function of the velocity gradient or other non-symmetric strain rates is very suitable for the formulation of evolution equations in large strain anisotropic plasticity. In this section we focus on a special class of isotropic tensor-valued functions referred here to as analytic tensor functions. In order to specify this class of functions we first deal with the general question how an isotropic tensor-valued function can be defined.
Mikhail Itskov

### Chapter 8. Applications to Continuum Mechanics

Let us consider an infinitesimal vector $$\text {d}\varvec{{X}}$$ in the reference configuration of a material body and its counterpart $$\text {d}\varvec{{x}}$$ in the current configuration. By virtue of the representation for the deformation gradient (2.​67) we get
\begin{aligned} \text {d}\varvec{{x}} = \frac{\partial \varvec{x}}{\partial X ^j} \text {d}X ^j = \left( \frac{\partial \varvec{x}}{\partial X ^j} \otimes \varvec{e}^j \right) \left( \text {d}X ^k \varvec{e}_k\right) =\mathbf {F}\text {d}\varvec{{X}}. \end{aligned}
Thus, the deformation gradient $$\mathbf {F}$$ maps every infinitesimal vector from the reference configuration to the current one by $$\text {d}\varvec{{x}} = \mathbf {F}\text {d}\varvec{{X}}$$.
Mikhail Itskov

### Chapter 9. Solutions

(a) (A.4), (A.3):
$$\varvec{ 0 } = \varvec{ 0 } + \left( -\varvec{ 0 }\right) = -\varvec{ 0 }.$$
(b) (A.1)–(A.4), (B.3):
\begin{aligned} \alpha \varvec{ 0 } = \varvec{ 0 } + \alpha \varvec{ 0 }= & {} \alpha \varvec{x} + \left( - \alpha \varvec{x}\right) + \alpha \varvec{ 0 } \nonumber \\= & {} \alpha \left( \varvec{ 0 } + \varvec{x}\right) + \left( - \alpha \varvec{x}\right) = \alpha \varvec{x} + \left( - \alpha \varvec{x}\right) = \varvec{ 0 }. \nonumber \end{aligned}
Mikhail Itskov

### Backmatter

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