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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

3.81 $

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

3.05 $

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

1.78 $

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

3.81 $

Problem: Find the coordinates of the vector \( X \) in the basis \( e^{\prime}=\left\{e_{1}^{\prime} ; e_{2}^{\prime} ; e_{3}^{\prime}\right\} \), if \( X=(1 ;-4 ; 8) \) in the basis \( e=\left\{e_{1} ; e_{2} ; e_{3}\right\} \) and \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}+e_{2}-3 e_{3} \\ e_{2}^{\prime}=\frac{3}{4} e_{1}-e_{2} \\ e_{3}^{\prime}=-e_{1}+e_{2}+e_{3} \end{array}\right. \]

1.1.6 Vector Algebra

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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

1.27 $

Problem: Find the coordinates of the vector \( X \) in the basis \( e^{\prime}=\left\{e_{1}^{\prime} ; e_{2}^{\prime} ; e_{3}^{\prime}\right\} \), if in the basis \( e=\left\{e_{1} ; e_{2} ; e_{3}\right\} \), \( X=(2 ; 6 ;-3) \), and the conversion from \( e \) to \( e^{\prime} \) is: \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}+e_{2}-2 e_{3} \\ e_{2}^{\prime}=\frac{2}{3} e_{1}-e_{2} \\ e_{3}^{\prime}=-e_{1}+e_{2}+e_{3} \end{array}\right. \]

1.1.7 Vector Algebra

1.02 $

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

1.27 $

Problem: A function of three variables is given: \( f\left(x_{1}, x_{2}, x_{3}\right) \). a) Find the stationary point of function \( f\left(x_{1}, x_{2}, x_{3}\right) \), calculate the value of the function and the gradient at it. (Indicate the method for solving the system of equations). b) Determine the extremums of function \( f\left(x_{1}, x_{2}, x_{3}\right) \), if there are any. \[ \begin{array}{l} f\left(x_{1}, x_{2}, x_{3}\right)=3 x_{1}^{2}+8 x_{2}^{2}+13 x_{3}^{2}-4 x_{1} x_{2}- \\ -8 x_{1} x_{3}-6 x_{2} x_{3}-55 x_{1}-18 x_{2}-21 x_{3} . \end{array} \]

2.11.8 Function extrema

3.05 $

Problem: Function \( f\left(x_{1}, x_{2}, x_{3}\right) \) and its constraints are given. a) Compose the Lagrange function; b) Determine the stationary point of the Lagrange function; c) Determine the extremums of function \( f\left(x_{1}, x_{2}, x_{2}\right) \), if there are any. \[ \begin{array}{l} f\left(x_{1}, x_{2}, x_{3}\right)=15 x_{1}^{2}+7 x_{2}^{2}+10 x_{3}^{2}+20 x_{1} x_{2}+ \\ +x_{1} x_{3}+3 x_{2} x_{3}+10 x_{1}-25 x_{2}-20 x_{3}, \\ 11 x_{1}+20 x_{2}+10 x_{3}=300, \\ 21 x_{1}+60 x_{2}+x_{3}=270 . \end{array} \]

2.11.9 Function extrema

3.05 $

Problem: Find the leading term in the asymptotic expansion of the function: \[ F(\lambda)=\int_{-1}^{\infty} e^{-\lambda \sqrt{\frac{x^{2}}{1+x}}} d x \]

2.12.18 Asymptotic analysis

5.59 $

Problem: Investigate the integral over the parameter for uniform convergence: \[ \int_{0}^{1} \sin \left(x^{a}(\ln x)^{2}\right) d x, \quad a \leq a_{0}<1 . \]

9.2.27 Integrals depending on a parameter

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