Tensor calculus/Tensor calculus

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Problem: In \( \mathbb{R}^{3} \) given a basis \( \mathcal{E}=\left\{e_{1}, e_{2}, e_{3}\right\} \) three linear forms by coordinate notation in \( \mathcal{E} \), forming a basis \( \mathcal{T}=\left\{f_{1}(x), f_{2}(x), f_{3}(x)\right\} \) of a dual space: \( e_{1}=(1 ; 3 ; 2), e_{2}=(1 ; 0 ; 1), e_{3}=(2 ;-1 ; 1) ; \) \( f_{1}(x)=2 x_{1}-x_{2}+2 x_{3} \), \( f_{2}(x)=x_{1}+x_{2}+x_{3}, f_{3}(x)=3 x_{1}+x_{3} \). 1) Find a basis in \( \mathbb{R}^{3} \), reciprocal with basis \( \mathcal{T} \). 2) Write in basis \( \varepsilon \) the basis \( e^{\prime}=\left\{e_{1}^{\prime}, e_{2}^{\prime}, e_{3}^{\prime}\right\} \), reciprocal with the standard basis in \( \mathbb{R}^{3} \). 3) Find a basis reciprocal with \( \varepsilon \).

1.11.1 Tensor calculus

7.8 $

Problem: Given a symmetric tensor: \[ A_{i j}=\left(\begin{array}{ccc} 1 & 0 & 2 \\ 0 & -1 & 1 \\ 2 & 1 & 3 \end{array}\right) \text {. } \] Find: 1) Eigenvalues and eigenvectors of the tensor, 2) orts of the coordinate system associated with the principal axes of the tensor, 3) the rotation matrix to the principal axes of the tensor, 4) tensor invariants, 5) equation and type of the characteristic surface of the tensor and depict it.

1.11.2 Tensor calculus

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Problem: \( a_{i k} u^{i} u^{k} \) - scalar for any choice of contravariant vector \( u^{i} \). Show that \( a_{(i k)} \) is a tensor.

1.11.3 Tensor calculus

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Problem: Find the tensor of inertia for the following homogeneous solid bodies, assuming that the center of rotation coincides with the center of inertia, and the mass of the bodies is equal to \( m \) : 1) Rectangular plate with sides \( a, b \), 2) Ball of radius \( R \).

1.11.4 Tensor calculus

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Problem: Tensor type (1,2) in the basis \( \mathcal{E}=\left\{e_{1}, e_{2}\right\} \) of the space \( V_{2} \) is given by the matrix \( A= \) \( \left(\begin{array}{ll|ll}-4 & 2 & 3 & 4 \\ -5 & 3 & 5 & 7\end{array}\right) \), where \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}-e_{2} \\ e_{2}^{\prime}=-e_{1}+2 e_{2} \end{array} .\right. \] Find tensor matrix in basis \( \mathcal{E}^{\prime}=\left\{e_{1}^{\prime}, e_{2}^{\prime}\right\} \).

1.11.5 Tensor calculus

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Problem: Given the vectors \[ x_{1}=(3 ; 1), x_{2}=(5 ; 0), x_{3}=(1 ;-1) \text {. } \] Find the components of the tensor \[ x=x_{1} \otimes x_{2}+x_{2} \otimes x_{3} \text {. } \]

1.11.6 Tensor calculus

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Problem: Given the tensor: \[ A=\left(\begin{array}{ll|ll} 1 & 1 & 1 & -1 \\ 1 & 1 & 1 & -1 \end{array}\right)=a_{i j k} . \] Find components of tensors: 1) \( a_{(i j) k} \), 2) \( \left.a_{i(j k)}, 3\right) \) \( a_{(i|j| k)} \),4) \( a_{(i j k)} \).

1.11.7 Tensor calculus

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Problem: Let \( a=\operatorname{det}\left(a_{i k}\right) \) for tensor \( a_{i k} \). Prove that \[ a=\frac{1}{6} \varepsilon_{i j k} \cdot \varepsilon^{m n p} \cdot a_{i m} \cdot a_{j n} \cdot a_{k p}, \] where \( \varepsilon_{i j k} \) is Levi - Civita symbol.

1.11.8 Tensor calculus

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Problem: Given the covariant metric tensor \( \left(g_{i j}\right)=\left(\begin{array}{ll}2 & 3 \\ 3 & 5\end{array}\right) \) and tensor \( \left(a_{\cdot j k}^{i}\right)=\left(\begin{array}{ll|ll}3 & 4 & 2 & 5 \\ 5 & 7 & 1 & 3\end{array}\right) \). Find matrices of the tensors: 1) \( a_{i j k} \), 2) \( a_{\cdot \cdot k}^{i j} \), 3) \( a_{. j .}^{i \cdot k} \), 4) \( a^{i j k} \).

1.11.9 Tensor calculus

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Problem: Given the permittivity tensor: \( \varepsilon_{i j}= \) \( \left(\begin{array}{ccc}3 & 2 & 0 \\ 2 & 4 & -2 \\ 0 & -2 & 5\end{array}\right) \), the material is placed in a uniform field with intensity \( E=E_{0}\{-2 ; 2 ; 1\} \). Find: 1) dielectric susceptibility tensor \( \alpha_{i j} \) of dielectric 2) polarization vector \( P \), 3) electric induction vector \( D \), 4) pairwise angles between vectors \( E, P, D \).

1.11.10 Tensor calculus

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Problem: Given the tensors: \[ C=\left(C_{i j}\right)=\left(\begin{array}{cc} 3 & 0 \\ -1 & 5 \end{array}\right), \quad A=\left(a^{i}\right)=(3 ; 1) . \] Calculate convolutions \( a_{j}=C_{i j} a^{i}, \quad b_{i}=C_{i j} \cdot a^{j} \), and scalar \( u=C_{i j} \cdot a^{i} \cdot a^{j} \).

1.11.12 Tensor calculus

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Problem: Given the tensors: \[ \left(C^{i j}\right)=C=\left(\begin{array}{cc} 3 & 0 \\ -1 & 5 \end{array}\right), \quad d_{i j}=D=\left(\begin{array}{cc} 5 & -6 \\ 3 & 1 \end{array}\right) . \] Calculate convolutions \( a_{k}^{j}=C^{i j} d_{i k}, \quad b_{k}^{i}=C^{i j} d_{k j} \), and scalar \( u=C^{i j} d_{i j} \).

1.11.11 Tensor calculus

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