Math Problems Vault

Problem 37.1

(1)(2)

The vectors $e_{1}^{\prime }, e_{2}^{\prime }$ are given by their coordinates $(1, 0)$ and $(\cos \alpha , \sin \alpha )$ relative to some orthonormal basis $e_{1}, e_{2}$ of a two-dimensional Euclidean space. Write out the matrices:

(1) of the metric,

(2) of the contravariant metric,

(3) and of the discriminant tensors, in the bases:

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(1)

$e_{1}, e_{2}$

(2)

$e_{1}^{\prime }, e_{2}^{\prime }$ .

Problem 37.2

Prove that in an arbitrary basis of a Euclidean space the discriminant tensor has the following components: $\varepsilon_{i_{1} \ldots i_{n}} = 0$ , if among the values of the indices there are equal ones, and $\varepsilon_{i_{1} \ldots i_{n}} = (- 1)^{N\left(i_{1} \ldots i_{n}\right)} \sigma \sqrt{\operatorname {det} \Gamma }$ , if the indices are pairwise distinct. Here $\Gamma$ is the matrix of the metric tensor, $N\left(i_{1} \ldots i_{n}\right)$ is the number of inversions in the permutation $\left(i_{1}, \ldots , i_{n}\right); \sigma = 1$ for right-handed bases, $\sigma = - 1$ for left-handed bases.

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Problem 37.3

Prove that in all right-handed orthonormal bases the discriminant tensor has the following components: $\varepsilon_{i_{1} \ldots i_{n}} = 0$ , if among the indices there are equal ones, and $\varepsilon_{i_{1} \ldots i_{n}} = (- 1)^{N\left(i_{1} \ldots i_{n}\right)}$ , if $i_{1} \ldots i_{n}$ are pairwise distinct. Here $N\left(i_{1} \ldots i_{n}\right)$ is the number of inversions in the permutation $\left(i_{1}, \ldots , i_{n}\right)$ .

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Problem 37.4

What tensor is obtained if one index of the metric tensor is raised? Both indices?

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Problem 37.5

What tensor is obtained if an index of the Kronecker symbol is lowered? Raised?

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Problem 37.6

Give examples of contraction with the metric tensor that have occurred in the course of linear algebra.

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Problem 37.7

(1)(2)

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(1)

A tensor $a_{j}^{i}$ defines a linear transformation in a Euclidean space $\mathcal{E}_{n}$ . Find the tensor defining the adjoint transformation.

(2)

State the condition under which the tensor $a_{j}^{i}$ defines a self-adjoint transformation.

Problem 37.8

(1)(2)(3)

The metric tensor and the tensor $a_{ij}$ are given, respectively, by the matrices:

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(1)

$\mathbf{A}_{55}, \mathbf{A}_{9}$ . $\mathbf{A}_{55} = \left[\begin{smallmatrix} 2 & 3 \\ 3 & 5 \end{smallmatrix}\right], \mathbf{A}_{9} = \left[\begin{smallmatrix} 9 & -5 \\ -5 & 3 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{57}, \mathbf{A}_{18}$ . $\mathbf{A}_{57} = \left[\begin{smallmatrix} 2 & 5 \\ 5 & 13 \end{smallmatrix}\right], \mathbf{A}_{18} = \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{245}, \mathbf{A}_{210}$ . $\mathbf{A}_{245} = \left[\begin{smallmatrix} 21 & -10 & -4 \\ -10 & 5 & 2 \\ -4 & 2 & 1 \end{smallmatrix}\right], \mathbf{A}_{210} = \left[\begin{smallmatrix} 0 & 1 & 3 \\ 2 & 3 & 5 \\ 3 & 5 & 7 \end{smallmatrix}\right]$ .

Find the matrices of the tensors:

(1) $a_{\cdot j}^{i}$ ;

(2) $a_{i \cdot }^{\cdot j}$ ;

(3) $a^{i j}$ .

Problem 37.9

(1)(2)

Is the following statement true: if the matrix of the tensor $a_{ij}$ is symmetric, then the matrices of the tensors are also symmetric:

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(1)

$a^{\cdot j}_{i \cdot }$ ;

(2)

$a^{ij}$ .

Problem 37.10

(1)(2)

The metric tensor and the tensor $a_{j}^{i}$ are given, respectively, by the matrices:

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(1)

$\mathbf{A}_{58}, \mathbf{A}_{40}$ . $\mathbf{A}_{58} = \left[\begin{smallmatrix} 1 & -2 \\ -2 & 5 \end{smallmatrix}\right], \mathbf{A}_{40} = \left[\begin{smallmatrix} 4 & -2 \\ -3 & 1 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{207}, \mathbf{A}_{232}$ . $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right], \mathbf{A}_{232} = \left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{smallmatrix}\right]$ .

Find the matrices of the tensors:

(1) $a_{i j}^{\cdot }$

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

Problem 37.11

(1)(2)(3)

The metric tensor and the tensor $a_{jk}^{i}$ are given, respectively, by the matrices:

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(1)

$\mathbf{A}_{55}, \mathbf{A}_{650}$ . $\mathbf{A}_{55} = \left[\begin{smallmatrix} 2 & 3 \\ 3 & 5 \end{smallmatrix}\right], \mathbf{A}_{650} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{55}, \mathbf{A}_{651}$ . $\mathbf{A}_{55} = \left[\begin{smallmatrix} 2 & 3 \\ 3 & 5 \end{smallmatrix}\right], \mathbf{A}_{651} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 3 & 4 \\ 2 & 5 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 7 \\ 1 & 3 \end{smallmatrix}\right] \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{207}, \mathbf{A}_{722}$ . $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right], \mathbf{A}_{722} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 2 & -2 \\ -2 & 0 & 2 \\ 2 & -2 & 0 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

Find the matrix of the tensor:

(1) $a_{i j k}$ ;

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

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

(4) $a^{i j k}$ .

Problem 37.12

(1)(2)

The metric tensor and the tensor $a_{kl}^{ij}$ are given, respectively, by the matrices:

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(1)

$\mathbf{A}_{57}, \mathbf{A}_{697}$ . $\mathbf{A}_{57} = \left[\begin{smallmatrix} 2 & 5 \\ 5 & 13 \end{smallmatrix}\right], \mathbf{A}_{697} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 1 \\ 0 & 1 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} 1 & 1 \\ 2 & 2 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 0 & 1 \\ 0 & 2 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{19}, \mathbf{A}_{694}$ . $\mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right], \mathbf{A}_{694} = \left[\begin{smallmatrix} \left[\begin{smallmatrix} 1 & 2 \\ 3 & 4 \end{smallmatrix}\right] & \left[\begin{smallmatrix} -4 & -3 \\ -2 & -1 \end{smallmatrix}\right] \\ \left[\begin{smallmatrix} -4 & -3 \\ -2 & -1 \end{smallmatrix}\right] & \left[\begin{smallmatrix} 5 & 6 \\ 7 & 8 \end{smallmatrix}\right]\end{smallmatrix}\right]$ .

Find the matrices of the tensors:

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

(2) $a^{i j k l}$ .

Problem 37.13

(1)(2)(3)

Simplify the expressions:

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(1)

$\left(a_{ij} g^{jk} + \delta_{i}^{j} a_{lj} g^{lk}\right) g_{ks}$

(2)

$\delta_{j}^{i} \delta_{k}^{j} g^{kl} a_{lj}$

(3)

$a_{ij} g^{jk} g_{kl} g^{ls}$ .

Problem 37.14

It is known that $a_{k}^{ij} = g^{il} g^{jm} a_{lmk}$ . Express $a_{lmk}$ in terms of $a_{k}^{ij}$ .

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Problem 37.15

Let $\varphi$ be a linear transformation of a Euclidean space, and $\varphi^{ * }$ the adjoint transformation. The index of the tensor corresponding to the product of transformations $\varphi \varphi^{ * }$ is lowered. Show that the resulting tensor has type $(0, 2)$ and is symmetric.

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Problem 37.16

In a two-dimensional Euclidean space $\mathcal{E}_{2}$ , to a vector $\xi^{k}$ is assigned the vector $g^{ij} \varepsilon_{jk} \xi^{k}$ . Prove that this defines a linear transformation of the space $\mathcal{E}_{2}$ , and determine its geometric meaning.

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Problem 37.17

In a three-dimensional Euclidean space, to a pair of vectors $\xi^{i}, \eta^{j}$ is assigned the vector $\zeta^{k} = g^{kl} \varepsilon_{lij} \xi^{i} \eta^{j}$ . Prove that the vector $\zeta^{k}$ is the cross product of the vectors $\xi^{i}$ and $\eta^{j}$ .

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Problem 37.18

In a four-dimensional Euclidean space, to the vectors $x, y, z$ with components $\xi^{i}, \eta^{j}, \zeta^{k}$ is assigned the linear function $f$ with coefficients $\varkappa_{l} = \varepsilon_{ijkl} \xi^{i} \eta^{j} \zeta^{k}$ . Prove that $\mathrm{f}(x) = \mathrm{f}(y) = \mathrm{f}(z) = 0$ .

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Problem 37.19

(1)(2)

In a four-dimensional Euclidean space, to the vectors $x, y, z$ with components $\xi^{i}, \eta^{j}, \zeta^{k}$ is assigned the vector $u$ with components $g^{ml} \varepsilon_{lijk} \xi^{i} \eta^{j} \zeta^{k}$ .

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(1)

Prove that the vector $u$ is orthogonal to the vectors $x, y, z$ .

(2)

Prove that the vector $u$ corresponding to the triple $x, y, z$ differs by the factor $-1$ from the vector $u$ corresponding to the triple $y, x, z$ .