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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

7.72 $

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

9 $

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

0.77 $

§ Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{l} x=x^{2}+y^{2} \\ z=x y, \quad z=0 \end{array}\right. \]

9.5.9 Volume of a solid

1.8 $

(3) Problem: Calculate the volume of the solid \( G \), bounded by the given surfaces: \[ \begin{array}{c} G=\left\{(x, y, z): x^{2}+y^{2}-4 x-4 y+z \leq 14\right. \\ \left.z \geq x^{2}+y^{2}\right\} \end{array} \]

9.5.10 Volume of a solid

1.8 $

Problem: Find the particular solution of the differential equation: \[ \frac{d^{2} u}{d x^{2}}-5 u=0, \quad u(0)=1, \quad u(1)=7 . \] clip22net.com

8.1.2.26 Second order differential equations

1.29 $

Problem: Solve the Cauchy problem: \[ y^{\prime \prime}=x \ln x, y(1)=0, y^{\prime}(1)=0 \text {. } \]

8.1.2.30 Second order differential equations

1.54 $

Problem: A sandwich, as usual, has fallen down with the buttered side on the horizontal heated surface, having the initial speed of translational motion \( V_{0} \). Assuming that the force of resistance to the motion of the sandwich along the surface was proportional to the momentum, i.e. \( F_{\text {res. }}=-k m V \), where \( k \) is the coefficient of proportionality, find the space passed by the sandwich until the butter is completely melted. The sandwich has left a fixed trace of melted butter of constant thickness and width on the surface, and the speed of the slide to the indicated moment of the time has decreased twice. The initial mass of the butter on the sandwich is equal to the mass of a slice of bread.

8.1.4.31 Geometric and physical applications

3.34 $

Problem: Find the general solution of the first order differential equation: \[ y^{\prime} \sin ^{2} 5 x-y^{2}+5=0 . \]

8.1.1.32 First order differential equations

0 $

Problem: Find the general solution of the second order differential equation with constant coefficients: \[ y^{\prime \prime}+4 y^{\prime}=5 x^{2}-1 \text {. } \] clip2net.com

8.1.2.34 Second order differential equations

1.29 $

Problem: Find the general solution of the differential equation: \[ \frac{d y}{\sin 7 x}+\cos ^{2} 5 y d x=0 \text {. } \]

8.1.1.35 First order differential equations

0 $

Problem: Find the general solution of the differential equation: \[ y^{\prime}=\frac{y}{x}+\cos ^{2} \frac{y}{x} \text {. } \]

8.1.1.36 First order differential equations

1.54 $

Problem: Find the general solution of the second order differential equation with constant coefficients: \[ y^{\prime \prime}+9 y=5 e^{3 x} \text {. } \] clip2net.com

8.1.2.37 Second order differential equations

0 $

Problem: Find the particular solution of the differential equation: \[ y^{\prime} x \cos y+x \sin y=\sin y, y(1)=\frac{\pi}{2} . \]

8.1.1.38 First order differential equations

1.29 $

Problem: Find the particular solution of the differential equation: \[ \frac{y d y+z d z}{\sqrt{y^{2}+z^{2}}}+\frac{y d z-z d y}{y^{2}}=0, \quad z(1)=0 . \]

8.1.1.39 First order differential equations

2.57 $

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