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Problem: What is the probability that the sum of \( n \) independent random variables, uniformly distributed in [0,1], will not exceed 1.

15.7.3 Limit theorems

3 $

Problem: Find the gradient of the scalar field \( u(x, y, z)= \) \( z e^{x^{2}+y^{2}+z^{2}} \) at an arbitrary point and at the point \( M_{0}(0,0,0) \). For the resulting vector field \( \bar{a}= \) \( \Delta u(x, y, z) \) find \( \operatorname{div} \bar{a} \) and \( \operatorname{rot} \bar{a} \) at the point \( M_{0} \).

1.12.1 Vector analysis

3.75 $

Problem: Show that the gradient field of the scalar field \( \varphi=6 x^{3} y-2 x y^{4}+z^{4} x^{2} y \) is irrotational.

1.12.3 Vector analysis

0 $

Problem: Find the vector lines of the field \( \vec{a}=(z-y) \vec{\imath}+ \) \( +(z+1) \vec{\jmath}+(y+1) \vec{k} \).

1.12.4 Vector analysis

1.25 $

Problem: Calculate the gradient of the scalar field \( \varphi(x, y, z) \), find and draw level surfaces \[ \varphi=0 ; \pm 1 ; \pm 3, \quad \text { where } \quad \varphi(x, y, z)=\frac{12 z}{x^{2}+y^{2}} . \]

1.12.5 Vector analysis

2.5 $

Problem: Given a vector field \( \vec{a}=(2 x \vec{\imath}+3 y \vec{\jmath}+z \vec{k}) \cos |\vec{r}| \). Calculate scalar and vector products: 1. \( (\nabla \cdot \vec{a}), 2 \). \( [\nabla \times \vec{a}] \).

1.12.6 Vector analysis

3.75 $

Problem: Calculate the gradient of the scalar field \( \varphi(x, y, z) \), find and draw level surfaces \[ \varphi=0 ; \pm 1 ; \pm 3, \quad \text { where } \quad \varphi(x, y, z)=\frac{3 y^{2}+x}{z^{2}} . \]

1.12.7 Vector analysis

2.5 $

Problem: Find the vector lines of gradient field of scalar function: \[ \varphi=y+y z-\frac{1}{2} x^{2}-z . \]

1.12.8 Vector analysis

2.5 $

Problem: For an arbitrary scalar field \( u \) prove that \( \operatorname{div}(\operatorname{grad} u)=\Delta u \).

1.12.9 Vector analysis

1.25 $

Problem: For any scalar field \( u \) calculate: a) \( \operatorname{div}(u \operatorname{grad} u) \), b) \( \operatorname{rot}(u \operatorname{grad} u) \).

1.12.10 Vector analysis

3 $

Problem: Are the vectors \( \overrightarrow{c_{1}} \) and \( \overrightarrow{c_{2}} \), constructed from the vectors \( \vec{a} \) and \( \vec{b} \) collinear? \[ \begin{array}{l} \vec{a}=\{-2 ; 7 ;-1\}, \quad \vec{b}=\{-3 ; 5 ; 2\}, \\ \overrightarrow{c_{1}}=2 \vec{a}+3 \vec{b}, \quad \overrightarrow{c_{2}}=3 \vec{a}+2 \vec{b} . \end{array} \]

1.1.10 Vector Algebra

1 $

Problem: Find the cosine of the angle between the vectors \( \overrightarrow{A B} \) and \( \overrightarrow{A C} \). \[ A(6 ; 2 ;-3), \quad B(6 ; 3 ;-2), \quad C(7 ; 3 ;-3) \text {. } \]

1.1.11 Vector Algebra

0.75 $

Problem: Calculate the area of the parallelogram formed by the vectors \( \vec{a} \) and \( \vec{b} \). \[ \begin{array}{l} \vec{a}=7 \vec{p}+\vec{q}, \quad \vec{b}=\vec{p}-3 \vec{g}, \quad|\vec{p}|=3, \\ |\vec{q}|=1, \quad\left(\widehat{\vec{p}, \vec{q})}=\frac{3 \pi}{4} .\right. \end{array} \]

1.1.12 Vector Algebra

2 $

Problem: Find the decomposition of the vector \( \vec{v} \) into the vectors \( \vec{p}, \vec{q} \) and \( \vec{r} \). \[ \begin{array}{l} \vec{v}=\{8 ; 9 ; 4\}, \quad \vec{p}=\{1 ; 0 ; 1\}, \\ \vec{q}=\{0 ;-2 ; 1\}, \quad \vec{r}=\{1 ; 3 ; 0\} . \end{array} \]

1.1.13 Vector Algebra

1 $

Problem: Are the vectors \( \overrightarrow{c_{1}} \) and \( \overrightarrow{c_{2}} \), formed by the vectors \( \vec{a} \) and \( \vec{b} \) collinear? \[ \begin{array}{ll} \vec{a}=\{-1 ; 2 ;-1\}, & \vec{b}=\{2 ;-7 ; 1\}, \\ \overrightarrow{c_{1}}=6 \vec{a}-2 \vec{b}, & \overrightarrow{c_{2}}=\vec{b}-3 \vec{a} . \end{array} \]

1.1.14 Vector Algebra

0 $

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