Math Problems Vault

Problem 36.21

(1)(2)(3)(4)

Starting from the transformation laws of the tensors $a_{jk}^{i}, a_{j}^{i}$ , $b_{j}^{i}, a_{i}, b^{ij}, a_{klm}^{ij}, \xi^{k}$ , verify the transformation law of the components of the contractions:

?

(1)

$a_{ji}^{i}$

(2)

$a_{j}^{i} b_{k}^{j}$

(3)

$a_{i} b^{ij}$

(4)

$a_{klm}^{ij} \xi^{k}$ .

Problem 36.22

(1)(2)(3)

Starting from the geometric meaning of the tensors $a_{i}, \xi^{i}$ , $a_{j}^{i}, b_{ij}$ , explain the geometric meaning of the contractions:

?

(1)

$a_{i} \xi^{i}$

(2)

$a_{j}^{i} \xi^{j}$

(3)

$b_{ij} \xi^{i} \xi^{j}$ .

Problem 36.23

(1)(2)(3)

Can one contract:

?

(1)

a vector and a covector?

(2)

a vector and a vector?

(3)

a pair of covectors?

Problem 36.24

Write the product of linear transformations in tensor notation.

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Problem 36.25

(1)(2)(3)

Tensors $a_{j}^{i}, \xi^{i}, \varkappa_{i}$ are given by the matrices: $\mathbf{A}_{232}, \mathbf{c}_{104}$ , $\mathbf{c}_{104}^{T}$ . Compute the contractions: $\mathbf{A}_{232} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{smallmatrix}\right], \mathbf{c}_{104} = \left[\begin{smallmatrix} 1 \\ -2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{104}^{T} = \left[\begin{smallmatrix} 1 & -2 & 3 \end{smallmatrix}\right]$ .

?

(1)

$a_{j}^{i} \xi^{j}$

(2)

$a_{j}^{i} \varkappa_{i}$

(3)

$a_{j}^{i} \xi^{j} \varkappa_{i}$

Problem 36.26

How many distinct tensors can be formed by contraction from a given tensor of type $(2, 2)$ ?

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Problem 36.27

(1)(2)

A tensor $a_{k}^{ij}$ is given by the matrix:

?

(1)

$\mathbf{A}_{651}$ . $\mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{655}$ . $\mathbf{A}_{655} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & -2 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 3 \\ 3 & 2 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

Find the matrices of the contractions:

(1) $a_{i}^{i j}$ ,

(2) $a_{j}^{i j}$ .

Problem 36.28

(1)(2)(3)

A tensor $a_{kl}^{ij}$ is given by the matrix:

?

(1)

$\mathbf{A}_{649}$ . $\mathbf{A}_{649} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 5 \\ 2 & 6 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 3 & 7 \\ 4 & 8 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{693}$ . $\mathbf{A}_{693} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} -1 & -2 \\ -3 & -4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 0 \\ 0 & 0 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{694}$ . $\mathbf{A}_{694} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -4 & -3 \\ -2 & -1 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} -4 & -3 \\ -2 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 6 \\ 7 & 8 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

Compute the contractions:

(1) $a_{i l}^{i j}$ ;

(2) $a_{k j}^{i j}$ ;

(3) $a_{k i}^{i j}$ ;

(4) $a_{i l}^{i j}$ ;

(5) $a_{i j}^{i j}$ ;

(6) $a_{j i}^{i j}$ .

Problem 36.29

(1)(2)

?

(1)

To each basis of the space $\mathcal{L}_{n}$ is assigned an ordered set of numbers $a_{klm}^{ij}$ (all indices range over the values from 1 to $n$ ). It is known that for an arbitrary vector $\xi^{k}$ the numbers $a_{klm}^{ij} \xi^{k}$ are the components of a tensor of type $(2, 2)$ . Prove that $a_{klm}^{ij}$ is a tensor of type $(2, 3)$ .

(2)

To each basis of the space $\mathcal{L}_{n}$ is assigned an ordered set of numbers $a_{klm}^{ij}$ (all indices range over the values from 1 to $n)$ . It is known that for an arbitrary tensor $u_{ij}^{k}$ of type $(1, 2)$ the numbers $a_{klm}^{ij} u_{ij}^{k}$ are the components of a tensor of type $(0, 2)$ . Prove that $a_{klm}^{ij}$ is a tensor of type $(2, 3)$ .