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Example. Evaluate the definite integral
$$
\int_{0}^{2} \frac{4 \sqrt{2-x}-\sqrt{2+x}}{(\sqrt{x+2}+4 \sqrt{2-x})(x+2)^{2}} d x
$$ | ## Solution.
1. To make a substitution leading to an integral of a rational function, it is necessary to transform the integrand so that it contains roots of any degree, but from the same expression of the form $\frac{a x+b}{c x+d}$. Therefore, we transform the integrand, highlighting $\sqrt{\frac{2-x}{2+x}}:$
$$
\in... | \frac{\ln5}{16} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,516 |
Example. Evaluate the definite integral
$$
\int_{0}^{3 / 2} \frac{x^{2}}{\sqrt{9-x^{2}}} d x
$$ | Solution.
1. To get rid of the radical, we use the substitution
$x=3 \sin t$. Then
$$
d x=3 \cos t d t, \quad t(0)=\arcsin 0=0, \quad t\left(\frac{3}{2}\right)=\arcsin \frac{1}{2}=\frac{\pi}{6}
$$
and $\sqrt{9-x^{2}}=|3 \cos t|=3 \cos t$, since $\cos t>0$ for $t \in[0, \pi / 6]$.
2. We make a change of variable in ... | \frac{3}{4}\pi-\frac{9\sqrt{3}}{8} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,517 |
Example. Find the area of the region bounded by the graphs of the functions
$$
y=x^{2}-4 x+3, \quad y=-x^{2}+2 x+3
$$ | ## Solution.
1. Find the abscissas $a$ and $b$ of the points of intersection of the graphs. For this, solve the equation
$$
x^{2}-4 x+3=-x^{2}+2 x+3
$$
We get $a=0, \quad b=3$.
2. Investigate the sign of the function $\varphi=x^{2}-4 x+3-\left(-x^{2}+2 x+3\right)$ on the interval $[a, b]=[0,3]$. For this, assign $x... | 9 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,518 |
Example. Calculate the length of the arc of the curve
$$
y=\frac{1-e^{x}-e^{-x}}{2}, \quad 0 \leq x \leq 3
$$ | ## Solution.
1. Differentiating the equation of the curve, we get
$$
y^{\prime}=\frac{-e^{x}+e^{-x}}{2}=-\operatorname{sh} x
$$
2. We compute the differential of the arc length:
$$
d l=\sqrt{1+\left(y^{\prime}\right)^{2}} d x=\sqrt{1+\operatorname{sh}^{2} x} d x=\operatorname{ch} x d x
$$
3. We find the arc length... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,519 |
EXAMPLE. Calculate the length of the arc of a curve given parametrically
$$
\left\{\begin{array}{l}
x=\left(t^{2}-2\right) \sin t+2 t \cos t, \\
y=\left(2-t^{2}\right) \cos t+2 t \sin t,
\end{array} \quad 0 \leq t \leq \pi\right.
$$ | ## Solution.
1. Find $x_{t}^{\prime}$ and $y_{t}^{\prime}:$
$$
x_{t}^{\prime}=t^{2} \cos t, \quad y_{t}^{\prime}=t^{2} \sin t
$$
2. Calculate the differential of the arc length:
$$
d l=\sqrt{\left(x_{t}^{\prime}\right)^{2}+\left(y_{t}^{\prime}\right)^{2}} d t=\sqrt{t^{4} \cos ^{4} t+t^{4} \sin ^{4} t} d t=t^{2} d t... | \frac{\pi^{3}}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,520 |
Example. Calculate the length of the arc of the curve given by the equation in polar coordinates
$$
\varrho=6 \sin \varphi, \quad 0 \leq \varphi \leq \pi / 3
$$ | Solution.
1. Find $\varrho^{\prime}(\varphi)$:
$$
\varrho^{\prime}(\varphi)=6 \cos \varphi
$$
2. Calculate the differential of arc length:
$$
d l=\sqrt{\varrho(\varphi)^{2}+\varrho^{\prime}(\varphi)^{2}} d \varphi=\sqrt{36 \sin ^{2} \varphi+36 \cos ^{2} \varphi} d \varphi=6 d \varphi
$$
3. Find the arc length by e... | 2\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,521 |
Example. Calculate the volume of the body bounded by the surfaces
$$
\frac{x^{2}}{16}+\frac{y^{2}}{9}+\frac{z^{2}}{196}=1, \quad z=0, \quad z=7
$$ | Solution. If $S=S(z)$ is the area of the cross-section of a body by a plane perpendicular to the $O Z$ axis and intersecting it at a point with ordinate $z$, then the volume of the part of the body enclosed between the planes $z=z_{1}$ and $z=z_{2}$ is determined by the formula
$$
V=\int_{z_{1}}^{z_{2}} S(z) d z
$$
1... | 77\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,522 |
Example. Calculate the volume of the solid formed by rotating the region bounded by the graphs of the functions
$$
y=x^{3} \quad y=\sqrt{x}
$$
around the $O X$ axis. | ## Solution.
1. Define the region $D$:
a) find the abscissas $a$ and $b$ of the points of intersection of the graphs. For this, solve the equation
$$
x^{3}=\sqrt{x}
$$
We get
$$
a=0, \quad b=1
$$
b) on the interval $[0,1] \sqrt{x} \geq x^{3}$. Therefore, $u(x)=x^{3}$ and $v(x)=\sqrt{x}$.
2. Calculate the volume ... | \frac{5\pi}{14} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,523 |
EXAMPLE 1. Calculate the line integral
$$
\int_{L} \frac{z^{2}}{x^{2}+y^{2}} d l
$$
where $L-$ is the first turn of the helical line
$$
\left\{\begin{array}{l}
x=\cos t \\
y=\sin t, \quad 0 \leq t \leq 2 \pi \\
z=t
\end{array}\right.
$$ | Solution.
1. Compute: $x^{\prime}(t)=-\sin t, y^{\prime}(t)=\cos t, z^{\prime}(t)=1, d l=\sqrt{2} d t$ and $f(x, y)=z^{2} /\left(x^{2}+y^{2}\right)=t^{2}$.
2. Substitute these results into formula (1) and compute the definite integral:
$$
\int_{L} \frac{z^{2}}{x^{2}+y^{2}} d l=\int_{0}^{2 \pi} t^{2} \sqrt{2} d t=\fra... | \frac{8\sqrt{2}\pi^{3}}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,524 |
EXAMPLE 2. Calculate the line integral
$$
\int_{L}(x-y) d l
$$
where $L$ is the line segment from point $A(0,0)$ to point $B(4,3)$. | ## Solution.
1. In this case, the equation of the line is $y=3 x / 4 \quad(0 \leq x \leq 4)$ and, consequently, $y^{\prime}(x)=3 / 4$ and $d l=5 / 4 d t$.
2. Substitute these results into formula (1') and compute the definite integral:
$$
\int_{L}(x-y) d l=\int_{0}^{4}\left(x-\frac{3}{4} x\right) \frac{5}{4} d x=\fra... | \frac{5}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,525 |
Example 3. Calculate the line integral
$$
\int_{L} \operatorname{arctg} \frac{y}{x} d l
$$
where $L$ is the part of the Archimedean spiral $\varrho=\varphi(0 \leq \varphi \leq \pi / 2)$. | Solution.
1. We calculate: $\varrho^{\prime}(\varphi)=1, d l=\sqrt{\varphi^{2}+1} d \varphi$ and $f(x, y)=\varphi$, since $\operatorname{arctg}(\operatorname{tg} \varphi)=\varphi$ for $0 \leq \varphi \leq \pi / 2$.
2. Substitute these results into formula ( $1^{\prime \prime}$ ) and compute the definite integral:
$$
... | \frac{(\pi^{2}+4)^{3/2}-8}{24} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,526 |
EXAMPLE 1. Calculate the line integral
$$
\int_{L} \frac{y}{3} d x-3 x d y+x d z
$$
along the part of the curve $L$ given parametrically by
$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=1-2 \cos t-2 \sin t
\end{array} \quad 0 \leq t \leq \frac{\pi}{2}\right.
$$ | ## Solution.
1. Compute: $x^{\prime}(t)=-2 \sin t, y^{\prime}(t)=2 \cos t$ and $z^{\prime}(t)=2 \sin t-2 \cos t$.
2. Compute the line integral using formula (1):
$$
\begin{aligned}
& \int_{L} \frac{y}{3} d x-3 x d y+x d z= \\
& =\int_{0}^{\pi / 2}\left[\frac{2}{3} \sin t(-2 \sin t)-6 \cos t(2 \cos t)+2 \cos t(2 \sin ... | 2-\frac{13}{3}\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,527 |
Example 2. Calculate the line integral
$$
\int_{L}(x-y) d x+d y+z d z
$$
from point $M(2,0,4)$ to point $N(-2,0,4)(y \geq 0)$ along the curve $L$, formed by the intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $z=4$, | Solution. In the section, we obtain a circle
\[
\left\{\begin{array}{l}
x^{2}+y^{2}=4 \\
z=4
\end{array}\right.
\]
Therefore, the parametric equations of the curve \( L \) are
\[
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=4
\end{array}\right.
\]
1. We compute: \( x^{\prime}(t)=-2 \sin t \), \( y^{\prime}... | 2\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,528 |
Example. Find the sum of the series
$$
\sum_{n=1}^{\infty} \frac{72}{n^{2}+5 n+4}
$$ | Solution.
1. The roots of the denominator $n=-1$ and $n=-4$ differ by an integer, i.e., $n^{2}+5 n+4=(n+1)(n+1+3)$. Therefore, the terms of the sequence of partial sums of the series $\sum_{n=1}^{\infty} a_{n}$ are easily found, as many terms in the expression $S_{n}=a_{1}+a_{2}+\ldots+a_{n}$ cancel each other out.
2.... | 26 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,529 |
Example. Investigate the convergence of the series
$$
\sum_{n=1}^{\infty} \frac{\sqrt{n^{3}+1}}{n^{2}(2+\sin n)}
$$ | Solution.
1. We have
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{n^{3}+1}}{n^{2}(2+\sin n)}=0
$$
i.e., the necessary condition for the convergence of the series is satisfied.
2. Since $-1 \leq \sin n \leq 1$ and $1 \leq 2+\sin n \leq 3$ for all $n \geq 1$ and
$$
\frac{\sqrt{n^{3}+1}}{n^{2}(2+\sin n)}>0
$$
we can ... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,530 |
Example. Investigate the convergence of the series
$$
\sum_{n=1}^{\infty} \arcsin \frac{n}{\left(n^{2}+3\right)^{5 / 2}}
$$ | SOLUTION.
1. We have
$$
\lim _{n \rightarrow \infty} \arcsin \frac{n}{\left(n^{2}+3\right)^{5 / 2}}=0
$$
2. We check that the terms of the given series are positive. Indeed,
$$
\arcsin \frac{n}{\left(n^{2}+3\right)^{5 / 2}}>0
$$
for all $n \geq 1$, since $n /\left(n^{2}+3\right)^{5 / 2} \in(0,1)$.
3. We conclude ... | Theseries\sum_{n=1}^{\infty}\arcsin\frac{n}{(n^{2}+3)^{5/2}}converges | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,531 |
Example. Investigate the convergence of the series
$$
\sum_{n=1}^{\infty} \frac{1 \cdot 4 \cdot 7 \cdot \ldots \cdot(3 n-2)}{n!} \sin \frac{1}{2^{n+1}}
$$ | ## Solution.
1. We check that the terms of the series are positive. Indeed,
$$
a_{n}=\frac{1 \cdot 4 \cdot 7 \cdot \ldots \cdot(3 n-2)}{n!} \sin \frac{1}{2^{n+1}}>0
$$
for all $n \geq 1$.
2. Since $\sin x \sim x$ as $x \rightarrow 0$, we can simplify the expression for $a_{n}$:
$$
\frac{1 \cdot 4 \cdot 7 \cdot \ld... | Theseries\sum_{n=1}^{\infty}\frac{1\cdot4\cdot7\cdot\ldots\cdot(3n-2)}{n!}\sin\frac{1}{2^{n+1}}diverges | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,532 |
Example. Investigate the convergence of the series
$$
\sum_{n=2}^{\infty} \frac{3 n}{\left(n^{2}-2\right) \ln (2 n)}
$$ | Solution.
1. Simplify the expression for $a_{n}$
$$
\frac{3 n}{\left(n^{2}-2\right) \ln (2 n)} \sim \frac{3}{n \ln n}
$$
and we will investigate the convergence of the series $\sum_{n=1}^{\infty} \frac{3}{n \ln n}$ using the integral test of Cauchy, since the function $f(x)=\frac{3}{x \ln x}$ has an obvious antideri... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,534 |
Example. Investigate the convergence of the series
$$
\sum_{n=1}^{\infty}(-1)^{n}\left(1-\cos \frac{1}{\sqrt{n}}\right)
$$ | Solution.
1. Check the necessary condition for convergence:
$$
\lim _{n \rightarrow \infty}(-1)^{n}\left(1-\cos \frac{1}{\sqrt{n}}\right)=0
$$
2. Investigate the convergence of the series of absolute values:
$$
\sum_{n=1}^{\infty}\left|(-1)^{n}\left(1-\cos \frac{1}{\sqrt{n}}\right)\right|=\sum_{n=1}^{\infty}\left(1... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,535 |
Example. Calculate the sum of the series
$$
\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{\left(1+n^{3}\right)^{2}}
$$
with accuracy $\alpha=0.001$. | Solution.
1. The given series is alternating and convergent (absolutely). The terms of the series decrease in absolute value:
$$
\frac{n+1}{\left(1+(n+1)^{3}\right)^{2}}<\frac{n}{\left(1+n^{3}\right)^{2}}, \quad n \geq 1
$$
2. To find the interval of convergence, we solve the inequality $\varrho(x)<1$.
3. Investigate... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,536 |
Example 1. Find the domain of convergence of the series
$$
\sum_{n=1}^{\infty} \frac{n^{3}}{\left(n^{2}+\sqrt{n}+1\right)^{x+1}}
$$ | ## Solution.
1. For each fixed $x$ all terms of the given series are positive:
$$
a_{n}=\frac{n^{3}}{\left(n^{2}+\sqrt{n}+1\right)^{x+1}}>0 \quad \forall n \geq 1
$$
2. We use the second (limit) comparison test. We have
$$
\frac{n^{3}}{\left(n^{2}+\sqrt{n}+1\right)^{x+1}} \sim \frac{1}{n^{2(x+1)-3}} \quad \text { a... | (1,\infty) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,537 |
Example 2. Find the domain of convergence of the series
$$
\sum_{n=1}^{\infty} \frac{4^{n}}{n^{3}\left(x^{2}-4 x+7\right)^{n}}
$$ | Solution.
1. To apply the D'Alembert's criterion, we find $\varrho(x)$ using the formula
$$
\varrho(x)=\lim _{n \rightarrow \infty} \frac{\left|f_{n+1}\right|}{\left|f_{n}\right|}=\lim _{n \rightarrow \infty} \frac{\frac{4^{n+1}}{(n+1)^{3}\left|x^{2}-4 x+7\right|^{n+1}}}{\frac{4^{n}}{n^{3}\left|x^{2}-4 x+7\right|^{n}... | (-\infty,1]\cup[3,+\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,538 |
Example 1. Find the interval of convergence of the series
$$
\sum_{n=1}^{\infty} \frac{(n+1)^{5}}{2 n+1}(x+1)^{n}
$$ | Solution.
1. In this case, $c_{n}=\frac{(n+1)^{5}}{2 n+1} \neq 0$ for all $n$. Therefore, we can use formulas (1) or (2) for the radius of convergence of the power series.
By the D'Alembert's formula
$$
R=\lim _{n \rightarrow \infty}\left|\frac{c_{n}}{c_{n+1}}\right|=\lim _{n \rightarrow \infty} \frac{(n+1)^{5}}{2 n... | (-2,0) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,539 |
Example 2. Find the interval of convergence of the series
$$
\sum_{n=1}^{\infty} \frac{1}{n+1}(x-2)^{2 n}
$$ | Solution.
1. In this case, $c_{n}=0$ for all odd $n$. Therefore, we cannot use the formulas for the radius of convergence of a power series.
We use the Cauchy criterion, for which we compute
$$
\rho(x)=\lim _{n \rightarrow \infty} \sqrt[n]{\left|f_{n}(x)\right|}
$$
where $f_{n}(x)=\frac{1}{n+1}(x-2)^{2 n}$.
We fin... | [1,3) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,540 |
Example. Find the sum of the series
$$
\sum_{n=1}^{\infty} \frac{\sin ^{n} x}{n}
$$
and specify the domain of convergence of the series to this sum. | Solution.
1. Find the interval of convergence of the series.
By the Cauchy criterion, the interval of convergence is determined by the inequality $|\sin x|<1$.
At the boundary points, when $x=\pi / 2+2 \pi k$ the series diverges, and when $x=3 \pi / 2+2 \pi k$ the series converges conditionally.
Therefore, the give... | S(x)=-\ln(1-\sinx),\quadx\neq\pi/2+2\pik | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,541 |
Example. Find the sum of the series
$$
\sum_{n=0}^{\infty}(n+6) x^{7 n}
$$
and specify the interval of convergence of the series to this sum. | ## Solution.
1. Find the interval of convergence of the series.
By the Cauchy criterion, the interval of convergence is determined by the inequality $\left|x^{7}\right|<1$. Hence, $-1<x<1$. At the boundary points $x= \pm 1$, the series diverges because the necessary condition for convergence is not satisfied. Therefo... | \frac{6-5x^{7}}{(1-x^{7})^{2}},\quadx\in(-1,1) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,542 |
Example. Expand the function
$$
f(x)=\frac{3}{2-x-x^{2}}
$$
into a Taylor series in powers of $x\left(x_{0}=0\right)$. | ## SOLUTION.
1. To use tabular expansions, we decompose the given function into elementary fractions:
$$
\frac{3}{2-x-x^{2}}=\frac{1}{1-x}+\frac{1}{x+2} \text {. }
$$
2. Using the tabular expansion
$$
\frac{1}{1-t}=1+t+t^{2}+\ldots+t^{n}+\ldots=\sum_{n=0}^{\infty} t^{n}, \quad t \in(-1,1)
$$
$$
\begin{gathered}
\t... | \frac{3}{2-x-x^{2}}=\sum_{n=0}^{\infty}(1+\frac{(-1)^{n}}{2^{n+1}})x^{n}\quad\forallx\in(-1,1) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,543 |
Example. Calculate the integral
$$
\int_{0}^{0.1} \cos \left(100 x^{2}\right) d x
$$
with accuracy $\alpha=0.001$. | Solution.
1. We expand the integrand function into a Taylor series in powers of $x:$
$$
\begin{aligned}
\cos \left(100 x^{2}\right)=1-\frac{\left(10^{2} x^{2}\right)^{2}}{2!}+ & \frac{\left(10^{2} x^{2}\right)^{4}}{4!}-\frac{\left(10^{2} x^{2}\right)^{6}}{6!}+\ldots= \\
& =\sum_{n=0}^{\infty}(-1)^{n} \frac{\left(10^{... | 0.090\0.001 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,544 |
Example. Prove that the function $y=-\sqrt{x^{4}-x^{2}}$ satisfies the equation
$$
x y y^{\prime}-y^{2}=x^{4}
$$ | Solution. We have
$$
y^{\prime}=-\frac{x\left(2 x^{2}-1\right)}{\sqrt{x^{4}-x^{2}}}
$$
Substitute $y$ and $y^{\prime}$ into the left side of the equation and perform the necessary transformations:
$x\left(-\sqrt{x^{4}-x^{2}}\right)\left(-\frac{x\left(2 x^{2}-1\right)}{\sqrt{x^{4}-x^{2}}}\right)-\left(x^{4}-x^{2}\rig... | proof | Algebra | proof | Yes | Yes | olympiads | false | 31,545 |
EXAMPLE. Find the integral curves of the differential equation
$$
6 x d x-6 y d y=2 x^{2} y d y-3 x y^{2} d x
$$ | Solution.
1. Rewrite the original equation as
$$
3 x\left(2+y^{2}\right) d x=2 y\left(x^{2}+3\right) d y
$$
Since $x^{2}+3>0$ and $2+y^{2}>0$, we separate the variables, i.e., represent equation (2) in the form
$$
\frac{3 x}{x^{2}+3} d x=\frac{2 y}{2+y^{2}} d y
$$
2. Compute the integrals in the equation
$$
\int ... | \frac{(x^{2}+3)^{3}}{(2+y^{2})}=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,546 |
Example. Find the integral curves of the differential equation
$$
y^{\prime}=\frac{x^{2}+2 x y-5 y^{2}}{2 x^{2}-6 x y}
$$ | Solution.
1. Transform the given equation to the form
$$
y^{\prime}=\frac{1+2 \frac{y}{x}-5\left(\frac{y}{x}\right)^{2}}{2-6 \frac{y}{x}}
$$
(we divided the numerator and denominator of the right-hand side of the given equation by $x^{2}$).
2. Make the substitution $y=x z(x)$, where $z(x)$ is the new unknown functi... | 2\operatorname{arctg}\frac{y}{x}-\ln\frac{(x^{2}+y^{2})^{3}}{|x|^{5}}=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,547 |
Example. Find the solution to the Cauchy problem for the equation
$$
y^{\prime}-\frac{1}{x} y=-\frac{2}{x^{2}}
$$
with the initial condition
$$
y(1)=1 \text {. }
$$ | SOLUTION.
1st method.
1. We write the corresponding homogeneous linear equation:
$$
y^{\prime}-\frac{1}{x} y=0
$$
This is a separable equation.
2. Separating the variables and integrating, we obtain the general solution of the homogeneous equation
$$
y=C x .
$$
3. We apply the method of variation of the arbitrar... | \frac{1}{x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,548 |
Example. Find the solution to the Cauchy problem
$$
x y^{\prime}+y=x y^{2}
$$
with the initial condition
$$
y(1)=1
$$ | Solution.
By transforming the equation to the form
$$
y^{\prime}+\frac{1}{x} y=y^{2}
$$
we confirm that this is a Bernoulli equation with $\alpha=2$.
1. Using the substitution
$$
y=z^{1 /(1-\alpha)}=z^{-1}
$$
the equation (3) is transformed into a linear equation
$$
z^{\prime}-\frac{1}{x} z=-1
$$
2. We solve eq... | \frac{1}{x(1-\lnx)} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,549 |
Example. Find the integral curves of the differential equation
$$
x d x+y d y+\frac{x d y-y d x}{x^{2}+y^{2}}=0
$$ | Solution.
1. Transform the equation (3):
$$
\left(x-\frac{y}{x^{2}+y^{2}}\right) d x+\left(y+\frac{x}{x^{2}+y^{2}}\right) d y=0
$$
In this case,
$$
P(x, y)=\left(x-\frac{y}{x^{2}+y^{2}}\right), Q(x, y)=\left(y+\frac{x}{x^{2}+y^{2}}\right)
$$
These functions are continuously differentiable in the region $x^{2}+y^{2... | \frac{x^{2}+y^{2}}{2}-\operatorname{arctg}\frac{x}{y}=C | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,550 |
Example. Find the general solution of the differential equation
$$
\left(1+x^{2}\right) y^{\prime \prime}+2 x y^{\prime}=12 x^{3}
$$ | Solution.
1. Since the differential equation does not contain $y$, by setting $y^{\prime}=p(x)$, we have $y^{\prime \prime}=p^{\prime}(x)$. We obtain a first-order differential equation
$$
\left(1+x^{2}\right) p^{\prime}+2 x p=12 x^{3}
$$
## 2. The equation
$$
p^{\prime}+\frac{2 x}{1+x^{2}} p=\frac{12 x^{3}}{1+x^{2... | C_{1}\operatorname{arctg}x+x^{3}-3x+C_{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,551 |
Example. Find the general solution of the linear differential equation
$$
y^{\prime \prime}+y=x \sin x
$$ | Solution.
1. We write the corresponding homogeneous equation
$$
y^{\prime \prime}+y=0
$$
and look for its solution in the form $y=e^{\lambda x}$, where $\lambda$ is an unknown number.
Substituting $y=e^{\lambda x}, y^{\prime}=\lambda e^{\lambda x}$, and $y^{\prime \prime}=\lambda^{2} e^{\lambda x}$ into equation (6... | C_{1}\cosx+C_{2}\sinx-\frac{x^{2}}{4}\cosx+\frac{x}{4}\sinx | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,553 |
Example. Find the general solution of the linear differential equation
$$
y^{\prime \prime \prime}-100 y^{\prime}=20 e^{10 x}+100 \cos 10 x
$$ | Solution.
1. We write the corresponding homogeneous equation
$$
y^{\prime \prime \prime}-100 y^{\prime}=0
$$
and look for its solution in the form $y=e^{\lambda x}$, where $\lambda$ is an unknown number.
Substituting $y=e^{\lambda x}, y^{\prime}=\lambda e^{\lambda x}$, and $y^{\prime \prime}=\lambda^{2} e^{\lambda ... | C_{1}+C_{2}e^{10x}+C_{3}e^{-10x}+\frac{x}{10}e^{10x}-\frac{1}{20}\sin10x | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,554 |
Example. Find the solution to the Cauchy problem
$$
y^{\prime \prime}+y=\frac{1}{\cos x}
$$
with initial conditions $y(0)=1, y^{\prime}(0)=0$. | Solution.
1. We write the corresponding homogeneous equation:
$$
y^{\prime \prime}+y=0
$$
We find the fundamental system of solutions $y_{1}=\cos x$ and $y_{2}=\sin x$ and the general solution of the homogeneous equation
$$
y=C_{1} \cos x+C_{2} \sin x
$$
2. We apply Lagrange's method (method of variation of arbitr... | \cosx(\ln\cosx+1)+x\sinx | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,555 |
Example. Change the order of integration
$$
I=\int_{0}^{1} d y \int_{0}^{\sqrt{y}} f(x, y) d x+\int_{1}^{\sqrt{2}} d y \int_{0}^{\sqrt{2-y^{2}}} f(x, y) d x
$$ | Solution.
1. The region of integration consists of two regions $D_{1}$ and $D_{2}$. Define them using inequalities
$$
\begin{gathered}
D_{1}=\left\{\begin{array}{cc}
(x, y): \begin{array}{c}
0 \leq y \leq 1 \\
0 \leq x \leq \sqrt{y}
\end{array}
\end{array}\right\} \\
D_{2}=\left\{\begin{array}{cc}
(x, y): & 1 \leq y ... | \int_{0}^{1}\int_{x^{2}}^{\sqrt{2-x^{2}}}f(x,y) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,556 |
Example. Evaluate the double integral
$$
\iint_{D}\left(54 x^{2} y^{2}+150 x^{4} y^{4}\right) d x d y
$$
where the region $D$ is bounded by the lines $x=1, y=x^{3}$ and $y=-\sqrt{x}$. | Solution.
1. Let's define the region $D$ by inequalities. It is obvious that $-\sqrt{x} \leq x^{3}$. Therefore, $-\sqrt{x} \leq y \leq x^{3}$. Since $x$ appears under the square root, $x \geq 0$. For $x$, the possible inequalities are $0 \leq x \leq 1$ or $1 \leq x$. In the second case, the region is unbounded, which ... | 11 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,557 |
Example. Evaluate the double integral
$$
\iint_{D} \frac{x}{y^{5}} d x d y
$$
where the region $D$ is defined by the inequalities
$$
1 \leq \frac{x^{2}}{16}+y^{2} \leq 3, \quad y \geq \frac{x}{4}, \quad x \geq 0
$$ | SOLUTION.
1. The region $D$ is defined by inequalities in the Cartesian coordinate system:
$$
D=\left\{(x, y): \begin{array}{c}
1 \leq \frac{x^{2}}{16}+y^{2} \leq 3 \\
\\
y \geq \frac{x}{4}, \quad x \geq 0
\end{array}\right\}
$$
2. Since the region $D$ is bounded by ellipses and lines passing through the origin, it ... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,559 |
Example 1. Find the volume of the body bounded by the surfaces
$$
x=17 \sqrt{2 y}, \quad x=2 \sqrt{2 y}, \quad z=1 / 2-y, \quad z=0
$$ | ## Solution.
1. By formula (1) with $f_{2}=1 / 2-y$ and $f_{1}=0$, the desired volume is
$$
V=\iint_{D}\left(\frac{1}{2}-y\right) d x d y
$$
where $D$ is the projection of the body onto the $X O Y$ plane.
2. To find $D$, we define the body using inequalities and eliminate $z$ from them. In this case, the body is de... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,560 |
Example 2. Find the volume of the body bounded by the surfaces
$$
x^{2}+y^{2}+2 x=0, \quad z=\frac{25}{4}-y^{2}, \quad z=0
$$ | Solution.
1. By formula (1) with $f_{2}=25 / 4-y^{2}$ and $f_{1}=0$, the desired volume is
$$
V=\iint_{D}\left(\frac{25}{4}-y^{2}-0\right) d x d y
$$
where $D$ is the projection of the body onto the $X O Y$ plane.
2. To find $D$, we define the body using inequalities and eliminate $z$ from them. In this case, the b... | 6\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,561 |
Example. Find the area of the region $D$ bounded by the lines
$$
x^{2}+y^{2}=12, \quad x \sqrt{6}=y^{2} \quad(x \geq 0)
$$ | Solution.
1. Let's define the region $D$ by inequalities. The region cannot be outside the circle, as it would then be unbounded. The region cannot be to the left of the parabola, as in this case, its points could have negative abscissas, which is excluded by the condition $x \geq 0$. Therefore,
$$
D=\left\{\begin{ar... | 3\pi+2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,562 |
Example. Find the area of the figure bounded by the given lines
$$
y^{2}-4 y+x^{2}=0, \quad y^{2}-8 y+x^{2}=0, \quad y=\frac{x}{\sqrt{3}}, \quad x=0
$$ | ## Solution.
1. Since the region $D$ is bounded by circles and lines passing through the origin, it is easier to solve the given problem by transitioning to polar coordinates
$$
\left\{\begin{array}{l}
x=\rho \cos \varphi \\
y=\rho \sin \varphi
\end{array}\right.
$$
In this case, the region $D$ will transform into t... | 4\pi+3\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 31,563 |
Example 1. Find the mass of the plate $D$ with surface density $\mu=16 x+9 y^{2} / 2$, bounded by the curves
$$
x=\frac{1}{4}, \quad y=0, \quad y^{2}=16 x \quad(y \geq 0)
$$ | Solution.
1. The mass of the plate $D$ with surface density $\mu=16 x+9 y^{2} / 2$ is determined by the formula
$$
m=\iint_{D}\left(16 x+\frac{9 y^{2}}{2}\right) d x d y
$$
2. We compute the obtained double integral in Cartesian coordinates:
a) define the region $D$ by a system of inequalities:
$$
\left\{\begin{ar... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,564 |
Example 2. Find the mass of the plate $D$ with surface density $\mu=x^{2} /\left(x^{2}+y^{2}\right)$, bounded by the curves
$$
y^{2}-4 y+x^{2}=0, \quad y^{2}-8 y+x^{2}=0, \quad y=\frac{x}{\sqrt{3}}, \quad x=0
$$ | ## Solution.
1. The mass of the plate $D$ with surface density $\mu=x^{2} /\left(x^{2}+y^{2}\right)$ is determined by the formula
$$
m=\iint_{D} \frac{x^{2}}{x^{2}+y^{2}} d x d y
$$
2. We compute the given double integral:
a) since the region $D$ is bounded by circles and lines passing through the origin, it is easi... | \pi+\frac{3\sqrt{3}}{8} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,565 |
Example 3. Find the mass of the plate $D$ with surface density $\mu=x / y^{5}$, bounded by the curves
$$
\frac{x^{2}}{16}+y^{2}=1, \quad \frac{x^{2}}{16}+y^{2}=3, \quad y=\frac{x}{4}, \quad x=0 \quad\left(y \geq \frac{x}{4}, x \geq 0\right)
$$ | Solution.
1. The mass of the plate $D$ with surface density $\mu=x / y^{5}$ is determined by the formula
$$
m=\iint_{D} \frac{x}{y^{5}} d x d y
$$
2. We calculate the obtained double integral:
a) define the region $D$ by inequalities in Cartesian coordinates
$$
D=\left\{(x, y): \begin{array}{c}
1 \leq \frac{x^{2}}... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,566 |
Example. Calculate the triple integral
$$
\iint_{\Omega} \int^{2} \operatorname{sh}(x y) d x d y d z
$$
where $\Omega$ is bounded by the planes
$$
x=2, \quad y=\frac{x}{2}, \quad y=0, \quad z=0, \quad z=1
$$ | Solution.
1. Let's define the region $\Omega$ by inequalities. Clearly, $0 \leq z \leq 1$. For $y$, the possible inequalities are $0 \leq y \leq x / 2$ or $x / 2 \leq y \leq 0$. If $0 \leq y \leq x / 2$, then $x \geq 0$ and for $x$ we have $0 \leq x \leq 2$. If $x / 2 \leq y \leq 0$, then $x \leq 0$ and the region doe... | \operatorname{sh}2-2 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,567 |
Example. Calculate the triple integral
$$
\iiint_{\Omega} \frac{x^{2}}{x^{2}+y^{2}} d x d y d z
$$
where the region $\Omega$ is bounded by the surfaces
$$
z=\frac{9}{2} \sqrt{x^{2}+y^{2}}, \quad z=\frac{11}{2}-x^{2}-y^{2}
$$ | Solution.
1. Since $\Omega$ is a body of revolution around the $O Z$ axis, it is convenient to switch to cylindrical coordinates
$$
\left\{\begin{array}{l}
x=\varrho \cos \varphi \\
y=\varrho \sin \varphi \\
z=z
\end{array}\right.
$$
In this case, $(\varrho, \varphi, z) \in \Omega^{\prime}$, and the desired integral... | \pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,568 |
Example. Compute the triple integral
$$
\iint_{\Omega} \int \frac{x^{2}}{x^{2}+y^{2}} d x d y d z
$$
where the region $\Omega$ is bounded by the surfaces
$$
z=\sqrt{36-x^{2}-y^{2}}, \quad z=\sqrt{\frac{x^{2}+y^{2}}{3}}
$$ | Solution.
1. Since $\Omega$ is a region bounded by the upper hemisphere and the upper half-cone, it is convenient to switch to spherical coordinates
$$
\left\{\begin{array}{l}
x=\varrho \cos \varphi \sin \theta \\
y=\varrho \sin \varphi \sin \theta \\
z=\varrho \cos \theta
\end{array}\right.
$$
In this case, $(\varr... | 36\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,569 |
Example 1. Find the volume of the body $\Omega$, bounded by the surfaces
$$
x=17 \sqrt{2 y}, \quad x=2 \sqrt{2 y}, \quad z=\frac{1}{2}-y, \quad z=0
$$ | Solution.
1. Define the region $\Omega$ by inequalities. Since $17 \sqrt{2 y} \geq 2 \sqrt{2 y}$, for $x$ we have the inequalities $2 \sqrt{2 y} \leq x \leq 17 \sqrt{2 y}$. Since $y$ appears under the square root, $y \geq 0$. For $z$, the possible inequalities are $0 \leq z \leq 1 / 2-y$ or $1 / 2-y \leq z \leq 0$. In... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,570 |
Example 2. Find the volume of the body $\Omega$, bounded by the surfaces
$$
z=\frac{9}{2} \sqrt{x^{2}+y^{2}}, \quad z=\frac{11}{2}-x^{2}-y^{2}
$$ | Solution.
1. Since $\Omega$ is a body of revolution around the $O Z$ axis, it is convenient to use cylindrical coordinates
$$
\left\{\begin{array}{l}
x=\varrho \cos \varphi \\
y=\varrho \sin \varphi \\
z=z
\end{array}\right.
$$
In this case, $(\varrho, \varphi, z) \in \Omega^{\prime}$, and the desired volume is dete... | 2\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,571 |
Example 3. Find the volume of the body $\Omega$, bounded by the surfaces
$$
z=\sqrt{36-x^{2}-y^{2}}, \quad z=\sqrt{\frac{x^{2}+y^{2}}{3}}
$$ | Solution.
1. Since $\Omega$ is the region bounded by the upper hemisphere and the upper half-cone, it is convenient to switch to spherical coordinates
$$
\left\{\begin{array}{l}
x=\varrho \cos \varphi \sin \theta \\
y=\varrho \sin \varphi \sin \theta \\
z=\varrho \cos \theta
\end{array}\right.
$$
In this case, $(\va... | 72\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,572 |
Example 1. Find the mass of the body $\Omega$ with density $\mu=2 x$, bounded by the surfaces
$$
x=2 \sqrt{2 y}, \quad x=\sqrt{2 y}, \quad z=1-y, \quad z=0
$$ | Solution.
1. The mass of the body $\Omega$ with density $\mu=2 x$ is determined by the formula
$$
m=\iiint_{\Omega} 2 x d x d y d z
$$
2. Let's define the region $\Omega$ using inequalities. Since $2 \sqrt{2 y} \geq \sqrt{2 y}$, for $x$ we have the inequalities $\sqrt{2 y} \leq x \leq 2 \sqrt{2 y}$. Since $y$ appear... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,573 |
Example 2. Find the mass of the body $\Omega$ with density $\mu=z$, bounded by the surfaces
$$
x^{2}+y^{2}=4, \quad z=0, \quad z=\frac{x^{2}+y^{2}}{2}
$$ | ## Solution.
1. The mass of the body $\Omega$ with density $\mu=z$ is determined by the formula
$$
m=\iiint_{\Omega} z d x d y d z
$$
Since $\Omega$ is a body of revolution around the $O Z$ axis, it is convenient to switch to cylindrical coordinates
$$
\left\{\begin{array}{l}
x=\varrho \cos \varphi \\
y=\varrho \si... | \frac{8\pi}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,574 |
Example 3. Find the mass of the body $\Omega$ with density $\mu=20 z$, bounded by the surfaces
$$
z=\sqrt{1-x^{2}-y^{2}}, \quad z=\sqrt{\frac{x^{2}+y^{2}}{4}}
$$ | Solution.
1. The mass of the body $\Omega$ with density $\mu=20 z$ is determined by the formula
$$
m=\iiint_{\Omega} 20 z d x d y d z
$$
Since $\Omega$ is the region bounded by the upper hemisphere and the upper half-cone, it is convenient to switch to spherical coordinates:
$$
\left\{\begin{array}{l}
x=\varrho \co... | 4\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,575 |
Example. Calculate the surface integral
$$
\iint_{\Sigma}(-x+3 y+4 z) d \sigma
$$
where $\Sigma-$ is the part of the plane
$$
x+2 y+3 z=1
$$
located in the first octant (i.e., $x \geq 0, y \geq 0, z \geq 0$ ). | SOLUTION.
1. The unit normal vectors $\vec{n}_{0}=\{\cos \alpha, \cos \beta, \cos \gamma\}$ to the surface defined by the equation $F(x, y, z)=0$ are given by the formula
$$
\vec{n}_{0}= \pm \frac{\operatorname{grad} F}{|\operatorname{grad} F|}
$$
In this case, $F(x, y, z)=x+2 y+3 z-1$. Therefore,
$$
\vec{n}_{0}= \... | \frac{\sqrt{14}}{18} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,576 |
Example. Compute the surface integral
$$
\iint_{\Sigma}\left(x^{2}+y^{2}\right) d \sigma
$$
where $\Sigma$ is the part of the surface $x^{2}+y^{2}=1$, cut off by the planes $z=0, \quad z=2$. | SOLUTION.
1. We introduce curvilinear coordinates on the given surface (cylinder)
$$
\left\{\begin{array}{l}
x=\varrho \cos \varphi \\
y=\varrho \sin \varphi \\
z=z
\end{array}\right.
$$
In these coordinates, the surface is defined by the conditions
$$
\Sigma=\left\{\begin{array}{ll}
\varrho=1 \\
(\varrho, \varphi,... | 4\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,577 |
Example. Compute the surface integral
$$
\iint_{\Sigma}\left(x^{2}+y^{2}\right) d \sigma
$$
where $\Sigma-$ is the upper hemisphere
$$
x^{2}+y^{2}+z^{2}=9, \quad z \geq 0
$$ | SOLUTION.
1. We introduce curvilinear coordinates on the given surface (sphere)
$$
\left\{\begin{array}{l}
x=\varrho \cos \varphi \sin \theta \\
y=\varrho \sin \varphi \sin \theta \\
z=\varrho \cos \theta
\end{array}\right.
$$
In these coordinates, the surface is defined by the conditions
$$
\Sigma=\left\{\begin{ar... | 108\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,578 |
Example. Find the vector lines of the vector field
$$
\vec{a}=9 z \vec{j}-4 y \vec{k}
$$ | Solution.
1. Since the first coordinate of the field $P(x, y, z)=0$, then $d x=0$ and, consequently, $x=C$. Therefore, we write the differential equation of vector lines as:
$$
\frac{d y}{9 z}=-\frac{d z}{4 y} \quad \text { when } \quad x=C
$$
2. Solving the differential equation, we get
$$
\left\{\begin{array}{l}
... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,579 |
Example. Find the flux of the vector field
$$
\vec{a}=-x \vec{i}+2 y \vec{j}+z \vec{k}
$$
through the part of the plane
$$
x+2 y+3 z=1
$$
located in the first octant (the normal forms an acute angle with the $O Z$ axis). | Solution.
1. The field of unit normals to the surface defined by the equation $F(x, y, z)=0$ is given by the formula
$$
\vec{n}_{0}= \pm \frac{\operatorname{grad} F}{|\operatorname{grad} F|}
$$
In this case, $F(x, y, z)=x+2 y+3 z-1$ and, therefore,
$$
\vec{n}_{0}= \pm \frac{\{1,2,3\}}{\sqrt{14}}
$$
Considering tha... | \frac{1}{18} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,580 |
Example. Find the flux of the vector field
$$
\vec{a}=x \vec{i}+y \vec{j}+z \vec{k}
$$
through the part of the surface
$$
x^{2}+y^{2}=1
$$
cut by the planes $z=0$ and $z=2$. (The normal is outward to the closed surface formed by these surfaces). | Solution.
1. The field of unit normals to the surface defined by the equation $F(x, y, z)=0$ is given by the formula
$$
\vec{n}_{0}= \pm \frac{\operatorname{grad} F}{|\operatorname{grad} F|}
$$
In this case, $F(x, y, z)=x^{2}+y^{2}-1$ and, therefore,
$$
\vec{n}_{0}= \pm \frac{\{x, y, 0\}}{\sqrt{x^{2}+y^{2}}}
$$
Co... | 4\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,581 |
Example. Find the flux of the vector field
$$
\vec{a}=x \vec{i}+(y+z) \vec{j}+(z-y) \vec{k}
$$
through the part of the surface
$$
x^{2}+y^{2}+z^{2}=9
$$
cut by the plane $z=0 \quad(z \geq 0)$ (the normal is external to the closed surface formed by these surfaces). | Solution.
1. The external normal at each point of the sphere $x^{2}+y^{2}+z^{2}=9$ coincides with the radius vector, i.e.,
$$
\vec{n}_{0}=\frac{\{x, y, z\}}{\sqrt{x^{2}+y^{2}+z^{2}}}
$$
2. We find the scalar product
$$
\left(\vec{a}, \overrightarrow{n_{0}}\right)=\frac{x^{2}+y(y+z)+z(z-y)}{\sqrt{x^{2}+y^{2}+z^{2}}}... | 54\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,582 |
Example. Find the flux of the vector field
$$
\vec{a}=\left(y^{2}+z^{2}\right) \vec{i}+\left(x y+y^{2}\right) \vec{j}+(x z+z) \vec{k}
$$
through the closed surface $\Sigma$, which is the complete surface of the cylinder
$$
x^{2}+y^{2}=1, \quad z=0, \quad z=1
$$
(normal outward). | ## Solution.
1. We compute the divergence of the vector field:
$$
\operatorname{div} \vec{a}=\frac{\partial\left(y^{2}+z^{2}\right)}{\partial x}+\frac{\partial\left(x y+y^{2}\right)}{\partial y}+\frac{\partial(x z+z)}{\partial z}=2 x+2 y+1
$$
2. We define the region $\Omega$ using inequalities.
The surface $\Sigma$... | 2\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,583 |
Example. Find the work of the force
\[
\vec{F}=(x-y) \vec{i}+\vec{j}
\]
when moving along the curve \( L \)
\[
x^{2}+y^{2}=4 \quad(y \geq 0)
\]
from point \( M(2,0) \) to point \( N(-2,0) \). | Solution.
1. The work $A$ of a force field is equal to the line integral of the second kind along the curve $L$:
$$
A=\int_{L}(\vec{F}, d \vec{r})=\int_{L}(x-y) d x+d y
$$
2. We compute the line integral. For this:
a) since $L$ is the upper semicircle, its parametric equations are written as
$$
\left\{\begin{array... | 2\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,584 |
Example. Find the circulation of the vector field
$$
\vec{a}=\frac{y}{3} \vec{i} + 3 - 3 x \vec{j} + x \vec{k}
$$
along the closed contour $\Gamma$
$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=1-2 \cos t-2 \sin t
\end{array} t \in[0,2 \pi]\right.
$$ | Solution.
1. By definition, the circulation of a vector field is equal to the second kind of curvilinear integral along the curve $\Gamma$:
$$
A=\oint_{\Gamma}(\vec{a}, d \vec{r})=\oint_{\Gamma} \frac{y}{3} d x-3 x d y+x d z
$$
2. We compute the curvilinear integral by reducing it to a definite integral:
$$
A=\oint... | -\frac{52\pi}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,585 |
Example. Find the modulus of the circulation of the vector field
$$
\vec{a}=y \vec{i}-x z \vec{j}+x y \vec{k}
$$
along the closed contour
$$
\Gamma=\left\{(x, y, z): \begin{array}{l}
x^{2}+y^{2}+z^{2}=9 \\
x^{2}+y^{2}=9
\end{array}\right\}
$$ | SOLUTION.
1. In this case, it is obvious that $\Gamma$ is the circle $x^{2}+y^{2}=9$ lying in the plane $z=0$. We choose the direction of traversal of the contour $\Gamma$ counterclockwise when viewed from the end of the vector $\vec{k}$.
2. We choose the surface $\Sigma$ stretched over the contour $\Gamma$.
Naturall... | 9\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,586 |
Example 1. Construct the hodograph of the vector $\mathbf{r}=\boldsymbol{t}+\boldsymbol{t} \mathbf{j}+$ $t^{2} \mathbf{k}$. | Solution. 1. This construction can be carried out point by point, by making a table:
Fig. 3
2. We can also proceed as follows. Denoting by $x, y, z$ the coordinates of the vector $\mathbf{... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,587 |
Example 1. Show that the vector $\alpha(t)=t i+\sin t \mathbf{j}$ is an infinitesimal vector as $t \rightarrow 0$. | ## Solution. We have
$$
|\alpha(t)|=|t i+\sin t j| \leqslant|t|+|\sin t| \leqslant 2|t|
$$
from which it is clear that if for any $\varepsilon>0$ we take $\delta=\frac{\varepsilon}{2}$, then for $|t-0|<\delta$, the inequality $|\alpha(t)|<\varepsilon$ holds. This shows that for any $\varepsilon>0$ there exists $\delt... | proof | Calculus | proof | Yes | Yes | olympiads | false | 31,588 |
Example 1. Find $\frac{d \mathbf{r}}{d t}$, if $\mathbf{r}=a \cos t \mathbf{i}+b \sin t \mathbf{j}$ (the point moves along an ellipse). | Solution. According to the formula (1)
$$
\frac{d \mathbf{r}}{d t}=-a \sin t \mathbf{i}+b \cos t \mathbf{j}
$$
By analogy with the differential of a scalar function, the differential of a vector function $\mathbf{r}=\mathbf{r}(t)$ is a vector $d \mathbf{r}$, defined by the equation
$$
d \mathbf{r}=\frac{d \mathbf{r}... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,589 |
Example 1. Find the indefinite integral of the vector function $\mathbf{a}(t)=$ $\mathbf{i} \cos t+\mathbf{j} e^{-t}+\mathbf{k}$ | Solution. According to formula (1)
$$
\int \mathrm{a}(t) d t=1 \int \cos t d t+\mathrm{J} \int e^{-t} d t+\mathrm{k} \int d t=1 \sin t-\mathrm{j} e^{-t}+\mathrm{k} t+\mathrm{c}
$$
where $\mathbf{c}-$ is an arbitrary constant vector.
## Problems for Independent Solution
Find the integrals of the following vector-fun... | {i}\sin-{j}e^{-}+{k}+{} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,590 |
Example 2. Calculate $\int_{0}^{\pi / 2} \mathbf{a}(t) d t$, where $\mathbf{a}(t)=\mathbf{i} \cos t-\mathbf{j} \sin ^{2} t$. | Solution. By formula (3)
$\int_{0}^{\pi / 2} \mathbf{a}(t) d t=\mathbf{i} \int_{0}^{\pi / 2} \cos t d t-\mathbf{j} \int_{0}^{\pi / 2} \sin ^{2} t d t=\left.\mathbf{i} \sin t\right|_{0} ^{\pi / 2}-\left.\mathbf{j}\left(\frac{t}{2}-\frac{\sin 2 t}{4}\right)\right|_{0} ^{\pi / 2}=\mathbf{i}-\frac{\pi}{4} \mathbf{j} . \qu... | {i}-\frac{\pi}{4}{j} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,591 |
Example 3. An electric current of strength $I$ flows from bottom to top along an infinite wire coinciding with the $O z$ axis. Find the vector $\mathbf{H}$ of the magnetic field strength created by this current at an arbitrary point $M(x, y, z)$ in space (Fig. 6). | Solution. Consider a sufficiently small element $P P_{1}=d \zeta$ of the $O z$ axis. According to the Biot-Savart law, the magnetic field intensity $d \mathbf{H}$ created at point $M$ by the current flowing through the wire element $d \zeta$ is in the direction of the vector product $\left[d \zeta, \mathbf{r}_{1}\right... | {H}=\frac{2I}{\rho^{2}}(-y{i}+x{j}),\quad\text{or}\quad{H}=\frac{2}{\rho^{2}}[{I},{r}] | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,592 |
Example 1. Calculate the curvature of the helical line
$$
\mathbf{r}=a \cos t \mathbf{i}+a \sin t \mathbf{j}+h t \mathbf{k}
$$ | Solution. Since
$$
\begin{aligned}
\frac{d \mathbf{r}}{d t} & =-a \sin t \mathbf{i} + a \cos t \mathbf{j} + h \mathbf{k} \\
\frac{d^{2} \mathbf{r}}{d t^{2}} & =-a \cos t \mathbf{i} - a \sin t \mathbf{j}
\end{aligned}
$$
then the vector product
$$
\left[\frac{d \mathbf{r}}{d t}, \frac{d^{2} \mathbf{r}}{d t^{2}}\right... | \frac{}{^{2}+^{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,594 |
Example 1. Find the torsion of the helical line
$$
r=a \cos t i + a \sin t j + h t k
$$ | Solution. We find the derivatives of the given vector
$$
\begin{aligned}
& \frac{d \mathbf{r}}{d t}=-a \sin t \mathbf{i}+a \cos t \mathbf{j}+h \mathbf{k} \\
& \frac{d^{2} \mathbf{r}}{d t^{2}}=-a \cos t \mathbf{i}-a \sin t \mathbf{j} \\
& \frac{d^{3} \mathbf{r}}{d t^{3}}=a \sin t \mathbf{i}-a \cos t \mathbf{j}
\end{ali... | \frac{}{^{2}+^{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,595 |
Example 2. Find the level surfaces of the scalar field
$$
u=x^{2}+y^{2}-z^{2}
$$ | Solution. Level surfaces are defined by the equation
$$
x^{2}+y^{2}-z^{2}=C, \text { where } C=\text { const. }
$$
When $C=0$, we obtain a circular cone. For any $C>0$, we obtain one-sheet hyperboloids of rotation with the axis coinciding with the $Oz$ axis. When $C<0$, we obtain a two-sheet hyperboloid of rotation. | x^{2}+y^{2}-z^{2}=C | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,596 |
Example 5. Find the level lines of the scalar field
$$
u=x^{2}-y^{2}
$$ | Solution. The level lines of the field are defined by the equations
$$
x^{2}-y^{2}=C, \quad C=\text { const }
$$
For $C=0$ we obtain a pair of straight lines
$$
y=x, \quad y=-x
$$
For $C \neq 0$ we obtain a family of hyperbolas (Fig. 11).
## Problems for Independent Solution
Find the level lines of the following ... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,597 |
Example 1. Find the derivative of the scalar field
$$
u=x y z
$$
at the point $M_{0}(1,-1,1)$ in the direction from point $M_{0}$ to point $M_{1}(2,3,1)$. | Solution. We find the direction cosines of the vector $\overrightarrow{M_{0} M_{1}}=\{1,4,0\}$, the length of which is $\left|\overrightarrow{M_{0} M}\right|=\sqrt{17}$. We have
$$
\cos \alpha=\frac{1}{\sqrt{17}}, \quad \cos \beta=\frac{4}{\sqrt{17}}, \quad \cos \gamma=0
$$
The values of the partial derivatives of th... | \frac{3}{2\sqrt{5}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,598 |
Example 3. Find the derivative of the scalar field $u=x z^{2}+2 y z$ at the point $M_{0}(1,0,2)$ along the circle
$$
\left\{\begin{array}{l}
x=1+\cos t \\
y=\sin t-1 \\
z=2
\end{array}\right.
$$ | Solution. The vector equation of the circle has the form
$$
\mathbf{r}(t)=(1+\cos t) \mathbf{i}+(\sin t-1) \mathbf{j}+2 \mathbf{k} .
$$
We find the vector $T$, tangent to it at any point $M$. We have
$$
\left.T=\frac{d r}{d t}=-\sin t \mathbf{i}+\cos t \mathbf{j}\right]
$$
The given point $M_{0}(1,0,2)$ lies in the... | -4 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,599 |
Example 7. Find the angle $\Theta$ between the gradients of the functions
$$
u=\sqrt{x^{2}+y^{2}} \text { and } v=x+y+2 \sqrt{x y}
$$
at the point $M_{0}(1, 1)$. | Solution. We find the gradients of the given functions at the point $M_{1}(1,1)$. We have
$$
\begin{aligned}
& \left.\operatorname{grad} u\right|_{M_{1}}=\left.\frac{x i+y j}{\sqrt{x^{2}+y^{2}}}\right|_{M_{0}} \pm \frac{1}{\sqrt{2}} I+\frac{1}{\sqrt{2}} j \\
& \left.\operatorname{grad} v\right|_{M_{1}}=\left.\left[\le... | \Theta=0 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,602 |
Example 8. Find the directional derivative along the radius vector $\mathbf{r}$ for the function $u=\sin r$, where $r=|\mathbf{r}|$. | Solution. According to formula (2), the derivative of the function with respect to the direction of the radius vector $\mathbf{r}$ is
$$
\frac{\partial u}{\partial r}=\left(\operatorname{grad} \sin r, r^{0}\right)
$$
We find the gradient of this function:
$$
\begin{aligned}
\text { grad } \sin r & =\frac{\partial(\s... | \cosr | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,603 |
Example 10. Find the direction of the greatest increase of the scalar field $u=x y+y z+x z$ at the point $M_{0}(1,1,1)$ and the magnitude of this greatest increase at this point. | Solution. The direction of the greatest change in the field is indicated by the vector grad $\chi(M)$. Let's find it:
$$
\operatorname{grad} u(M)=(y+z) \dagger+(x+c) j+(y+x) \mathbf{k}
$$
Thus, $\operatorname{grad} u(M)=2(\mathbf{I}+\mathbf{J}+\mathbf{k})$. This vector determines the direction of the greatest increas... | 2\sqrt{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,605 |
Example 1. Find the vector lines of the vector field
$$
\mathbf{a}=[\mathbf{c}, \mathbf{r}]
$$
where $\mathbf{c}$ - a constant vector. | Solution. We have
$$
\mathbf{c}=c_{1} \mathbf{1}+c_{2} \mathbf{j}+c_{3} \mathbf{k}, \quad \mathbf{r}=x \mathbf{1}+y \mathbf{j}+z \mathbf{k}
$$
so that
$$
\mathrm{x}=[\mathbf{c}, \mathrm{r}]=\left|\begin{array}{ccc}
1 & i & \mathbf{k} \\
c_{1} & c_{2} & c_{3} \\
x & y & z
\end{array}\right|=\left(c_{2} z-c_{3} y\righ... | {\begin{aligned}x^{2}+y^{2}+z^{2}&=A_{1}\\c_{1}x+c_{2}y+c_{3}z&=A_{2}\end{aligned}.} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 31,606 |
Example 3. Find the flux of the vector field
$$
\mathbf{a}=\frac{\mathbf{r}}{|\mathbf{r}|^{3}}
$$
through a sphere of radius $\boldsymbol{R}$ centered at the origin. | Solution. Since the normal p to the sphere is collinear with the radius-vector $\mathbf{r}$, we can take $\boldsymbol{\text { m }} \mathbf{n}^{\text {n }}=\mathbf{r}^{n}=\frac{\mathbf{r}}{|\mathbf{r}|}$. Therefore,
$$
\left(\mathbf{m}, \mathbf{n}^{\mathbf{n}}\right)=\left(\frac{\mathbf{r}}{|\boldsymbol{r}|^{1}}, \frac... | 4\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,609 |
Example 4. Find the flux of the vector field $\mathbf{a} = (x-2z)\mathbf{i} + (x+3y+z)\mathbf{j} + (5x+y)\mathbf{k}$ through the upper side of the triangle $ABC$ with vertices at points $A(1,0,0)$, $B(0,1,0)$, $C(0,0,1)$. | Solution. The equation of the plane in which triangle $ABC$ lies is $z+y+z=1$, from which $z=1-x-y$. Triangle $ABC$ is projected one-to-one onto the $xOy$ plane into the region $D_{xy}$, which is the triangle $OAB$ (Fig. 18).
By the condition, the normal $\boldsymbol{n}^{0}$ to the plane in which triangle $ABC$ lies f... | \frac{5}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,610 |
Example 5. Find the flux of the vector field $\mathbf{a} = \boldsymbol{y}^{2} \mathbf{j} + z \mathbf{k}$ through the part of the surface $z = x^{2} + y^{2}$, cut off by the plane $z = 2$. The normal is taken outward with respect to the region bounded by the paraboloid.
 is projected one-to-one onto the plane $x O y$ as a circle $D_{x y}$ (Fig. 19). We find the unit normal $\boldsymbol{n}^{0}$ to the surface $S$:
$$
n^{\prime \prime}=\frac{\operatorname{grad}\left(z-x^{2}-y^{2}\right)}{\left|\operatorname{grad}\left(z-x^{2}-y^{2... | -2\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,611 |
Example 6. Find the flux of the vector field $\mathbf{a} = \mathbf{i} - \mathbf{j} + x y z \mathbf{k}$ through the circle $S$, obtained by the intersection of the sphere $x^{2} + y^{2} + z^{2} \leqslant R^{2}$ with the plane $y = x$. Take the side of the circle facing the positive part of the $O x$ axis. | Solution. Since the plane $y=x$ is perpendicular to the coordinate plane $x O y$, the circle $S$ lying on this plane projects onto the plane $x O y$ as a segment $A_{1} A_{1}$, and thus the uniqueness of the projection is violated. On the other coordinate planes, the circle $S$ projects uniquely. Projecting the circle,... | \sqrt{2}R^{2}\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,612 |
Example 8. Calculate the flux of the vector field $\mathbf{a}=x \mathbf{i}+y \mathbf{j}+\sqrt{x^{2}+y^{2}-1} \mathbf{k}$ through the outer side of the one-sheet hyperboloid $z=\sqrt{x^{2}+y^{2}-1}$, bounded by the planes $z=0, z=\sqrt{3}$ | Solution. The given surface is projected unambiguously onto the $x O y$ plane in the region $D_{x y}$, bounded by a circle
We find the outer normal $\mathbf{n}$:
$z= \pm \operatorname{grad}... | 2\sqrt{3}\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,613 |
Example 9. Calculate the flux of the vector field $\mathbf{a} = y \mathbf{i} + z \mathbf{j} + x \mathbf{k}$ through the closed surface bounded by the cylinder $x^{2} + y^{2} = R^{2}$ and the planes $z = x, z = 0 (z \geqslant 0)$. | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,614 | |
Example 13. Find the flux of the vector field $\mathbf{v}=(x-2 y+1) \mathbf{i}+(2 x+ y-3 z) \mathbf{j}+(3 y+z) \mathbf{k}$ through the part of the sphere $x^{2}+y^{2}+z^{2}$, located in the first octant, into the region where $x^{2}+y^{2}+z^{2}>1$. | Solution. In this case, we have
$$
\begin{gathered}
R=1, \quad \varphi_{1}=0, \quad \varphi_{2}=\frac{\pi}{2} \\
\theta_{1}=0, \quad \theta_{2}=\frac{\pi}{2}, \quad n^{n}=x 1+y J+z k, \quad\left(\mathrm{a}, n^{0}\right)=x^{2}+y^{2}+z^{2}+x
\end{gathered}
$$
We proceed on the sphere $x^{2}+y^{2}+z^{2}=1$ with coordina... | \frac{3}{4}\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,615 |
Example 1. Calculate the flux of the vector field $\mathbf{a}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}$ through the closed surface $x^{2}+y^{2}+z^{2}=R^{2}, z=0(z>0)$. | Solution. By formula (I)
$$
\Pi=\iiint_{V}(2 x+2 y+2 z) d v
$$
It is convenient to calculate the integral (2) in spherical coordinates $r, \boldsymbol{\theta}, \varphi$. We have
$$
x=r \sin \theta \cos \varphi, \quad y=r \sin \theta \sin \varphi, \quad z=r \cos \theta
$$
and the volume element
$$
d v=r^{2} \sin \t... | \frac{\piR^{4}}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,616 |
Example 2. Compute the flux of the vector field a $=4 x i-y j+z \mathbf{k}$ through the surface of the torus. | Solution. Using the Ostrogradsky-Gauss theorem, we find that the sought flux $\Pi$ is equal to
$$
\begin{aligned}
\Pi & =\oiint_{s}\left(\mathrm{a}, \mathrm{n}^{0}\right) d \sigma= \\
& =\iiint_{V}\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right) d v=4 V
\end{aligne... | \pi^{2}(R_{2}-R_{1})^{2}(R_{2}+R_{1}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,617 |
Example 1. Using the invariant definition, calculate the divergence of the vector $a=x \mathbf{i}$ at the point $O(0,0,0)$, choosing as the surface $\sigma$ surrounding the point $O$, a sphere $\sigma_{\varepsilon}$ of radius $\varepsilon$ centered at this point. | Solution. By the definition of divergence at the given point, we have
$$
\operatorname{div} a(0)=\lim _{\left(\sigma_{k}\right) \rightarrow 0} \frac{\int\left(a, n^{0}\right) d \sigma}{v_{\varepsilon}}
$$
where $v_{\varepsilon}$ is the volume of the ball bounded by the sphere $\sigma_{\varepsilon}$, or
$$
\operatorn... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,618 |
Example 3. Calculate div $(u \mathbf{a})$, where $u(M)$ is a scalar function, $\mathbf{z}(M)=P(x, y, z) \mathbf{i}+Q(x, y, z) \mathbf{j}+R(x, y, z) \mathbf{k}$ is a vector function. | Solution. Using formula (3), we find
$v(u z)=\frac{\partial(u P)}{\partial x}+\frac{\partial(u Q)}{\partial y}+\frac{\partial(u R)}{\partial z}=u \frac{\partial P}{\partial x}+P \frac{\partial u}{\partial x}+u \frac{\partial Q}{\partial y}+Q \frac{\partial u}{\partial y}+$
$$
+u \frac{\partial R}{\partial x}+R \frac{... | \operatorname{div}(u{z})=u\cdot\operatorname{div}{}+({},\mathrm{grad}u) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,620 |
Example 4. Find the divergence of the vector
$$
\mathbf{a}=\varphi(r) \mathbf{r}^{0}=\frac{\varphi(r)}{r} \mathbf{r}
$$
where $r=|r|$ - the distance from the origin to the variable point $M(x, y, z)$. | Solution. Using formula (5), we get
$$
\mathrm{dlva}=\frac{\varphi(r)}{r} \operatorname{div} \mathrm{r}+\left(r, \operatorname{grad} \frac{\varphi(r)}{r}\right)
$$
Further,
$$
\text { div } r=3, \quad \text { grad } \frac{\varphi(r)}{r}=\left(\frac{\varphi(r)}{r}\right)^{\prime} \operatorname{grad} r=\frac{r \varphi... | \operatorname{div}2\frac{\varphi(r)}{r}+\varphi^{\}(r) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,621 |
Example 6. Calculate the circulation of the vector field $\mathbf{a} = y e^{x y} \mathbf{i} + x e^{x y} \mathbf{j} + x y z \mathbf{k}$ along the curve $L$, obtained by the intersection of the cone $x^{2} + y^{2} = (z-1)^{2}$ with the coordinate planes (Fig. 31). | Solution. Line $\boldsymbol{L}$ consists of two segments BC and CA, located on the coordinate planes $y O z$ and $z O z$ respectively, and the arc AB of the circle $x^{2}+y^{2}=\mathrm{L}_{1} z=0$. Therefore, the circulation of the given vector field will be
$\mu=\oint_{L}(\mathrm{a}, d r)=\int_{B C}(\mathrm{a}, d r)+... | -\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,624 |
Example 1. Find the rotor of the vector $\mathbf{v}=(x+z) \mathbf{i}+(y+z) \mathbf{j}+\left(x^{2}+z\right) \mathbf{k}$. | Solution. Using formula (2), we have
$$
\operatorname{rota}=\left|\begin{array}{ccc}
1 & j & k \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
x+z & y+z & x^{2}+z
\end{array}\right|
$$
expanding the determinant by the elements of the first row and understanding the operat... | rot{}=-{i}-(2x-1){j} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,625 |
Example 1. Calculate the circulation of the vector field $a=y i+z^{2} j-$ $z \mathbf{k}$ along the contour $L:\left\{\begin{array}{r}x^{2}+y^{2}=4, \\ z=3,\end{array}\right.$ 1) directly, 2) using Stokes' theorem. | Solution. 1) The contour $L-$ is a circle with a radius $=2$, lying in the plane $z=3$ (Fig. 32). We choose the orientation on it as indicated in the figure. Parametric equations of the line $\mathbf{L}$:
$$
\left\{\begin{array}{l}
x=2 \cos t \\
y=2 \sin t \\
z=3 \quad(0 \leqslant t<2 \pi)
\end{array}\right.
$$
BK 41... | -4\pi | Calculus | math-word-problem | Yes | Yes | olympiads | false | 31,627 |
Example 1. Show that for the vector field $\mathbf{a} = x y^{2} z \mathbf{i} + z^{2} y z \mathbf{j} + \frac{1}{2} x^{2} y^{2} \mathbf{k}$, the line integral $\int (\mathbf{a}, d \mathbf{r})$ does not depend on the form of the integration path $L$. | Solution. The coordinates of the vector field $\mathbf{a}$ are everywhere continuous functions. Since the domain of definition $G$ of the vector $\mathbf{a}$ is a simply connected region, in this region we have
$$
\mathrm{rot} \mathbf{a}=\left|\begin{array}{ccc}
1 & 1 & k \\
\frac{\partial}{\partial x} & \frac{\partia... | proof | Calculus | proof | Yes | Yes | olympiads | false | 31,628 |
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