Vectors and Tensors in a Finite-Dimensional Space

Abstract

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.

Exercises

1.1.

Prove that if $\varvec{x}\in \mathbb {V}$ is a vector and $\alpha \in \mathbb {R}$ is a scalar, then the following identities hold.

(a) $-\varvec{ 0 }=\varvec{ 0 }$,   (b) $\alpha \varvec{ 0 }=\varvec{ 0 }$,   (c) $0\varvec{x}=\varvec{ 0 }$,   (d) $-\varvec{x}= \left( -1\right) \varvec{x}$,   (e) if $\alpha \varvec{x}=\varvec{ 0 }$, then either $\alpha = 0$ or $\varvec{x} =\varvec{ 0 }$ or both.

1.2.

Prove that $\varvec{x}_i \ne \varvec{ 0 } \, (i =1,2,\ldots , n)$ for linearly independent vectors $\varvec{x}_1,\varvec{x}_2,$ $\ldots ,\varvec{x}_n$. In other words, linearly independent vectors are all non-zero.

1.3.

Prove that any non-empty subset of linearly independent vectors $\varvec{x}_1, \varvec{x}_2,$ $\ldots ,\varvec{x}_n$ is also linearly independent.

1.4.

Write out in full the following expressions for n = 3: (a) $\delta _j^i a^j$,   (b) $\delta _{ij} x^i x^j$,   (c) $\delta _i^i$,   (d) $\dfrac{\partial f_i}{\partial x^j}dx^j$.

1.5.

Prove that

$$\begin{aligned} \varvec{ 0 } \cdot \varvec{x} = 0, \; \forall \varvec{x} \in \mathbb {E}^n. \end{aligned}$$

(1.173)

1.6.

Prove that a set of mutually orthogonal non-zero vectors is always linearly independent.

1.7.

Prove the so-called parallelogram law: $\left\| \varvec{x} + \varvec{y} \right\| ^2 = \left\| \varvec{x} \right\| ^2 + 2\varvec{x} \cdot \varvec{y} + \left\| \varvec{y} \right\| ^2$.

1.8.

Let $\mathcal {G}=\left\{ \varvec{g}_1,\varvec{g}_2,\ldots ,\varvec{g}_n \right\}$ be a basis in $\mathbb {E}^n$ and $\varvec{a}\in \mathbb {E}^n$ be a vector. Prove that $\varvec{a} \cdot \varvec{g}_i = 0 \,\left( i =1,2,\ldots , n\right)$ if and only if $\varvec{a} = \varvec{ 0 }$.

1.9.

Prove that $\varvec{a} = \varvec{b}$ if and only if $\varvec{a} \cdot \varvec{x} = \varvec{b}\cdot \varvec{x}, \ \forall \varvec{x} \in \mathbb {E}^n$.

1.10.

(a) Construct an orthonormal set of vectors orthogonalizing and normalizing (with the aid of the procedure described in Sect. 1.4) the following linearly independent vectors:

$$ \varvec{g}_1 = \left\{ {\begin{array}{ccc} 1 \\ 1 \\ 0 \\ \end{array} } \right\} , \quad \varvec{g}_2 = \left\{ {\begin{array}{ccc} 2 \\ 1 \\ -2 \\ \end{array} } \right\} , \quad \varvec{g}_3 = \left\{ {\begin{array}{ccc} 4 \\ 2 \\ 1 \\ \end{array} } \right\} , $$

where the components are given with respect to an orthonormal basis.

(b) Construct a basis in $\mathbb {E}^3$ dual to the given above utilizing relations (1.16)$_2$, (1.18) and (1.19).

(c) As an alternative, construct a basis in $\mathbb {E}^3$ dual to the given above by means of (1.21)$_1$, (1.24) and (1.25)$_2$.

(d) Calculate again the vectors $\varvec{g}^i$ dual to $\varvec{g}_i\,\left( i=1,2,3\right)$ by using relations (1.33) and (1.35). Compare the result with the solution of problem (b).

1.11.

Verify that the vectors (1.33) are linearly independent.

1.12.

Prove identities (1.41) and (1.42) by means of (1.18), (1.19) and (1.24), respectively.

1.13

Prove relations (1.40) and (1.44) by using (1.39) and (1.43), respectively.

1.14.

Verify the following identities involving the permutation symbol (1.36) for n = 3: (a) $\delta ^{ij} e_{ijk} = 0$, (b) $e^{ikm} e_{jkm} = 2\delta _j^i$, (c) $e^{ijk} e_{ijk} = 6$, (d) $e^{ijm} e_{klm} = \delta _k^i \delta _l^j - \delta _l^i \delta _k^j$.

1.15.

Prove the following identities

$$\begin{aligned} \left( \varvec{a}\times \varvec{b}\right) \times \varvec{c} =\left( \varvec{a}\cdot \varvec{c}\right) \varvec{b}-\left( \varvec{b}\cdot \varvec{c}\right) \varvec{a}, \end{aligned}$$

(1.174)

$$\begin{aligned} \widehat{\varvec{a}\times \varvec{b}} = \varvec{b}\otimes \varvec{a} - \varvec{a}\otimes \varvec{b},\quad \forall \varvec{a},\varvec{b},\varvec{c} \in \mathbb {E}^3. \end{aligned}$$

(1.175)

1.16.

Prove relations (1.64) using (1.45).

1.17.

Prove that $\mathbf {A}\varvec{ 0 } = \varvec{ 0 }\mathbf {A} = \varvec{ 0 }, \, \forall \mathbf {A} \in \mathbf {L}\text {in}^n$.

1.18.

Prove that $0\mathbf {A} = \mathbf {0}, \, \forall \mathbf {A} \in \mathbf {L}\text {in}^n$.

1.19.

Prove formula (1.58), where the negative tensor $-\mathbf {A}$ is defined by (1.53).

1.20.

Prove that not every second order tensor in $\mathbf {L}\text {in}^n$, where $n>1$, can be represented as a tensor product of two vectors $\varvec{a},\varvec{b}\in \mathbb {E}^n$ as $\varvec{a}\otimes \varvec{b}$.

1.21.

Prove relation (1.88).

1.22.

Prove (1.91) using (1.90) and (1.15).

1.23.

Evaluate the tensor $\mathbf {W} = \hat{\varvec{w}} = \varvec{w} \times$, where $\varvec{w} = w^i \varvec{g} _i$.

1.24.

Evaluate components of the tensor describing a rotation about the axis $\varvec{e}_3$ by the angle $\alpha$.

1.25.

Represent a tensor rotating by the angle $\omega =\pi /4$ about an axis specified by a vector $\varvec{d}=-\varvec{e}_2+\varvec{e}_3$. Calculate the vector obtained from a vector $\varvec{a}=\varvec{e}_1-2\varvec{e}_2$ by this rotation, where $\varvec{e}_i\, (i=1,2,3)$ represent an orthonormal basis in $\mathbb {E}^3$.

1.26.

Express components of the moment of inertia tensor $\mathbf {J}=\text {J}^{ij}\varvec{e}_i\otimes \varvec{e}_j$ (1.80), where $\varvec{r}=x^i\varvec{e}_i$ is represented with respect to the orthonormal basis $\varvec{e}_i\, (i=1,2,3)$ in $\mathbb {E}^3$.

1.27.

Let $\mathbf {A} = \text {A}^{ij} \varvec{g}_i \otimes \varvec{g}_j$ , where

$$ \left[ \text {A}^{ij} \right] = \left[ \begin{array}{ccc} 0 &{} {- 1} &{} 0 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \\ \end{array} \right] $$

and the vectors $\varvec{g}_i\, (i=1,2,3)$ are given in Exercise 1.10. Evaluate the components $\text {A}_{ij}$, $\text {A}^i_{\cdot j}$ and $\text {A}_{i \cdot }^{\; j}$.

1.28.

Prove identities (1.108) and (1.110).

1.29.

Let $\mathbf {A} = \text {A}_{\cdot j}^i \varvec{g}_i \otimes \varvec{g}^j$, $\mathbf {B} = \text {B}_{\cdot j}^i \varvec{g}_i \otimes \varvec{g}^j$, $\mathbf {C} = \text {C}_{\cdot j}^i \varvec{g}_i \otimes \varvec{g}^j$ and $\mathbf {D} = \text {D}_{\cdot j}^i \varvec{g}_i \otimes \varvec{g}^j$, where

$$ \left[ \text {A}_{\cdot j}^i\right] = \left[ \begin{array}{ccc} 0 &{} 2 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array} \right] , \; \left[ \text {B}_{\cdot j}^i\right] = \left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ \end{array} \right] , \; \left[ \text {C}_{\cdot j}^i\right] = \left[ \begin{array}{ccc} 1 &{} 2 &{} 3 \\ 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ \end{array} \right] , $$

$$ \left[ \text {D}_{\cdot j}^i\right] = \left[ \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 1/2 &{} 0 \\ 0 &{} 0 &{} 10 \\ \end{array} \right] . $$

Find commutative pairs of tensors.

1.30.

Let A and B be two commutative tensors. Write out in full $\left( \mathbf {A} + \mathbf {B}\right) ^k$, where $k = 2,3,\ldots$

1.31.

Prove that

$$\begin{aligned} \exp \left( \mathbf {A} + \mathbf {B} \right) = \exp \left( \mathbf {A}\right) \exp \left( \mathbf {B}\right) , \end{aligned}$$

(1.176)

where A and B commute.

1.32.

Evaluate $\exp \left( \mathbf {0}\right)$ and $\exp \left( \mathbf {I}\right)$.

1.33.

Prove that $\exp \left( -\mathbf {A}\right) \exp \left( \mathbf {A}\right) =\exp \left( \mathbf {A}\right) \exp \left( \mathbf {-A}\right) =\mathbf {I}$.

1.34.

Prove that $\exp \left( k{\mathbf {A}}\right) = \left[ {\exp \left( \mathbf {A} \right) }\right] ^k$ for all integer k.

1.35.

Prove that $\exp {\left( \mathbf {A}+\mathbf {B}\right) }=\exp {\left( \mathbf {A}\right) }+\exp {\left( \mathbf {B}\right) }-\mathbf {I}$ if $\mathbf {A}\mathbf {B}=\mathbf {B}\mathbf {A}=\mathbf {0}$.

1.36.

Prove that $\exp {\left( \mathbf {QA}\mathbf {Q}^\text {T}\right) }=\mathbf {Q}\exp {\left( \mathbf {A}\right) }\mathbf {Q}^\text {T}, \; \forall \mathbf {Q}\in \mathbf {O}\text {rth}^n$.

1.37.

Compute the exponential of the tensors $\mathbf {D} = \text {D}_{\cdot j}^i \varvec{g}_i \otimes \varvec{g}^j$, $\mathbf {E} = \text {E}_{\cdot j}^i \varvec{g}_i \otimes \varvec{g}^j$ and $\mathbf {F} = \text {F}_{\cdot j}^i \varvec{g}_i \otimes \varvec{g}^j$, where

$$ \left[ \text {D}_{\cdot j}^i \right] = \left[ \begin{array}{*{20}c} 2 &{} 0 &{} 0 \\ 0 &{} 3 &{} 0 \\ 0 &{} 0 &{} 1 \\ \end{array}\right] , \; \left[ \text {E}_{\cdot j}^i \right] = \left[ \begin{array}{*{20}c} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array} \right] , \; \left[ \text {F}_{\cdot j}^i \right] = \left[ \begin{array}{*{20}c} 0 &{} 2 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ \end{array} \right] . $$

1.38.

Prove that $\left( \mathbf {ABCD}\right) ^\text {T} = \mathbf {D}^\text {T}\mathbf {C}^\text {T}\mathbf {B}^\text {T}\mathbf {A}^\text {T}$.

1.39.

Verify that $\left( \mathbf {A}^k\right) ^\text {T} = \left( \mathbf {A}^\text {T}\right) ^k$, where $k = 1,2,3,\ldots$

1.40.

Evaluate the components $\text {B}^{ij}$, $\text {B}_{ij}$, $\text {B}^i_{\cdot j}$ and $\text {B}_{i \cdot }^{\; j}$ of the tensor $\mathbf {B}=\mathbf {A}^\text {T}$, where A is defined in Exercise 1.27.

1.41.

Prove relation (1.136).

1.42.

Verify that $\left( \mathbf {A}^{-1}\right) ^k = \left( \mathbf {A}^k\right) ^{-1}=\mathbf {A}^{-k}$, where $k = 1,2,3,\ldots$

1.43.

Prove identity (1.146) using (1.90) and (1.145).

1.44.

Prove by means of (1.145)–(1.147) the properties of the scalar product (D.1–D.3).

1.45.

Verify that $\left[ \left( \varvec{a}\otimes \varvec{b}\right) \left( \varvec{c}\otimes \varvec{d}\right) \right] \text { : }\mathbf {I} = \left( \varvec{a} \cdot \varvec{d}\right) \left( \varvec{b} \cdot \varvec{c}\right)$.

1.46.

Express $\text {tr}\mathbf {A}$ in terms of the components $\text {A}_{\cdot j}^i$, $\text {A}_{ij}$, $\text {A}^{ij}$.

1.47.

Let $\mathbf {W} = \text {W}^{ij} \varvec{g}_i \otimes \varvec{g}_j$ , where

$$ \left[ \text {W}^{ij} \right] = \left[ \begin{array}{rrr} 0 &{} -1 &{} -3 \\ 1 &{} 0 &{} 1 \\ 3 &{} -1 &{} 0 \\ \end{array} \right] $$

and the vectors $\varvec{g}_i\, (i=1,2,3)$ are given in Exercise 1.10. Calculate the axial vector of W.

1.48.

Prove relations (1.166).

1.49.

Calculate , where $\mathbf {A}$ is given in Exercise 1.27.

1.50.

Prove that $\mathbf {M}\text { : }\mathbf {W} = 0$, where M is a symmetric tensor and W a skew-symmetric tensor.

1.51.

Evaluate $\text {tr}\mathbf {W}^k$, where W is a skew-symmetric tensor and $k =1,3,5,\ldots$

1.52.

Verify that $\text {sym}\left( \text {skew} \mathbf {A}\right) = \text {skew}\left( \text {sym} \mathbf {A}\right) = \mathbf {0}, \; \forall \mathbf {A} \in \mathbf {L}\text {in}^n$.

1.53.

Prove that $\text {sph}\left( \text {dev}\mathbf {A}\right) =\text {dev}\left( \text {sph}\mathbf {A}\right) =\mathbf {0},\; \forall \mathbf {A}\in \mathbf {L}\text {in}^n$.