Math Problems Vault

Problem 4376

Apply the Ostrogradsky formula to transform the following surface integral, where the smooth surface $S$ encloses the bounded volume $V$ , and $\cos \alpha$ , $\cos \beta$ , $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$ : $\displaystyle \iint_{S} x^{3}\, dy\, dz+y^{3}\, dx\, dz+z^{3}\, dx\, dy$ .

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Problem 4377

Apply the Ostrogradsky formula to transform the following surface integral, where the smooth surface $S$ encloses the bounded volume $V$ , and $\cos \alpha$ , $\cos \beta$ , $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$ : $\displaystyle \iint_{S} xy\, dx\, dy+xz\, dx\, dz+yz\, dy\, dz$ .

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Problem 4378

Apply the Ostrogradsky formula to transform the following surface integral, where a smooth surface $S$ encloses a bounded volume $V$ , and $\cos \alpha$ , $\cos \beta$ , $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$ : $\displaystyle \iint_{S} \dfrac {x\cos \alpha +y\cos \beta +z\cos \gamma }{\sqrt{x^{2}+y^{2}+z^{2}}}\, dS$ .

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Problem 4379

Apply the Ostrogradsky formula to transform the following surface integral, where a smooth surface $S$ encloses a bounded volume $V$ , and $\cos \alpha$ , $\cos \beta$ , $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$ : $\displaystyle \iint_{S} \left(\dfrac {\partial u}{\partial x}\cos \alpha +\dfrac {\partial u}{\partial y}\cos \beta +\dfrac {\partial u}{\partial z}\cos \gamma \right)\, dS$ .

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Problem 4380

Apply the Ostrogradsky formula to transform the following surface integral, where a smooth surface $S$ encloses a bounded volume $V$ , and $\cos \alpha$ , $\cos \beta$ , $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$ :

\begin{align} \iint _{S} \Bigg[\left(\dfrac {\partial R}{\partial y}-\dfrac {\partial Q}{\partial z}\right)\cos \alpha +\left(\dfrac {\partial P}{\partial z}-\dfrac {\partial R}{\partial x}\right)\cos \beta +\left(\dfrac {\partial Q}{\partial x}-\dfrac {\partial P}{\partial y}\right)\cos \gamma \Bigg]\, dS . \end{align}

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Problem 4381

Prove that if $S$ is a closed simple surface and $\overrightarrow {l}$ is any fixed direction, then

\iint _{S} \cos (\overrightarrow {n},\overrightarrow {l})\, dS = 0 ,

where $\overrightarrow {n}$ is the outward normal of the surface $S$ .

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Problem 4382

Prove that the volume enclosed by the surface $S$ equals

V = \dfrac {1}{3}\iint _{S} (x\cos \alpha +y\cos \beta +z\cos \gamma )\, dS ,

where $\cos \alpha$ , $\cos \beta$ , $\cos \gamma$ are the direction cosines of the outward normal of the surface $S$ .

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Problem 4383

Prove that the volume of the cone bounded by the smooth conical surface $F(x,y,z)=0$ and the plane $Ax+By+Cz+D=0$ equals

V = \dfrac {1}{3}SH ,

where $S$ is the area of the base of the cone lying on the given plane, and $H$ is the height of the cone.

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Problem 4384

Find the volume of the solid bounded by the surfaces $z=\pm c$ and

x = a\cos u\cos v+b\sin u\sin v ,

y = a\cos u\sin v-b\sin u\cos v ,

z = c\sin u

.

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Problem 4385

Find the volume of the body bounded by the surface

x=u\cos v , \qquad y=u\sin v , \qquad z=-u+a\cos v \qquad (u\geqslant 0)

and the planes $x=0$ , $z=0$ $(a>0)$ .

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Problem 4385.1

[NO TASK]

Problem 4386

Prove the formula

\begin{align} \dfrac {d}{dt}\left\{ \iiint _{x^{2}+y^{2}+z^{2}\leqslant t^{2}} f(x,y,z,t)\, dx\, dy\, dz\right\} \\ & = \iint _{x^{2}+y^{2}+z^{2}=t^{2}} f(x,y,z,t)\, dS + \iiint _{x^{2}+y^{2}+z^{2}\leqslant t^{2}} \dfrac {\partial f}{\partial t}\, dx\, dy\, dz \qquad (t>0) . \end{align}

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Problem 4387

Using the Ostrogradsky formula, compute the following surface integral: $\displaystyle \iint_{S} x^{2}\, dy\, dz+y^{2}\, dx\, dz+z^{2}\, dx\, dy$ , where $S$ is the outer surface of the boundary of the cube $0<x<a$ , $0<y<a$ , $0<z<a$ .

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Problem 4388

Using the Ostrogradsky formula, compute the following surface integral: $\displaystyle \iint_{S} x^{3}\, dy\, dz+y^{3}\, dx\, dz+z^{3}\, dx\, dy$ , where $S$ is the outer surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ .

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Problem 4389

Using the Ostrogradsky formula, compute the following surface integral: $\displaystyle \iint_{S} (x-y+z)\, dy\, dz+(y-z+x)\, dx\, dz+(z-x+y)\, dx\, dy$ , where $S$ is the outer surface of the surface $\left|x-y+z\right|+\left|y-z+x\right|+\left|z-x+y\right|=1$ .

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Problem 4390

Compute

\iint _{S} (x^{2}\cos \alpha +y^{2}\cos \beta +z^{2}\cos \gamma )\, dS ,

where $S$ is the part of the conical surface $x^{2}+y^{2}=z^{2}$ $(0\leqslant z\leqslant h)$ , and $\cos \alpha$ , $\cos \beta$ , $\cos \gamma$ are the direction cosines of the outward normal of this surface.

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Problem 4391

Prove the formula

\iiint _{V} \dfrac {d\xi \, d\eta \, d\zeta }{r} = \dfrac {1}{2}\iint _{S} \cos (\overrightarrow {r},\overrightarrow {n})\, dS ,

where $S$ is the closed surface enclosing the volume $V$ , $\overrightarrow {n}$ is the outward normal at the moving point $(\xi ,\eta ,\zeta )$ on the closed surface $S$ , and

r = \sqrt{(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}} ,

with $\overrightarrow {r}$ the radius vector from the point $(x,y,z)$ to the point $(\xi ,\eta ,\zeta )$ .

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Problem 4392

Compute the Gauss integral

I(x,y,z) = \iint _{S} \dfrac {\cos (\overrightarrow {r},\overrightarrow {n})}{r^{2}}\, dS ,

where $S$ is a simple closed smooth surface containing the volume $V$ , $\overrightarrow {n}$ is the outward normal to the surface $S$ at the point $(\xi ,\eta ,\zeta )$ , $\overrightarrow {r}$ is the radius vector connecting the point $(x,y,z)$ and the point $(\xi ,\eta ,\zeta )$ ,

r = \sqrt{(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}} .

Consider two cases: (a) when the surface $S$ does not enclose the point $(x,y,z)$ ; (b) when the surface $S$ encloses the point $(x,y,z)$ .

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Problem 4393

(a)(b)

Prove that if

\Delta u = \dfrac {\partial ^{2}u}{\partial x^{2}}+\dfrac {\partial ^{2}u}{\partial y^{2}}+\dfrac {\partial ^{2}u}{\partial z^{2}}

and $S$ is a smooth surface enclosing a bounded volume $V$ , then the following formulas hold:

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(a)

$\displaystyle \iint_{S} \dfrac {\partial u}{\partial n}\, dS = \iiint_{V} \Delta u\, dx\, dy\, dz$ ;

(b)

$\displaystyle \iint_{S} u\dfrac {\partial u}{\partial n}\, dS$

\begin{align} = \iiint _{V} \left[\left(\dfrac {\partial u}{\partial x}\right)^{2}+\left(\dfrac {\partial u}{\partial y}\right)^{2}+\left(\dfrac {\partial u}{\partial z}\right)^{2}\right]\, dx\, dy\, dz +\iiint _{V} u\Delta u\, dx\, dy\, dz , \end{align}

where $u$ and its partial derivatives up to second order are continuous functions in the domain $V+S$ , and $\dfrac {\partial u}{\partial n}$ is the derivative along the outward normal of the surface $S$ .

Problem 4394

Prove the second Green's formula in space

\iiint _{V} \begin{vmatrix} \Delta u & \Delta v \\ u & v \end{vmatrix}\, dx\, dy\, dz = \iint _{S} \begin{vmatrix} \dfrac {\partial u}{\partial n} & \dfrac {\partial v}{\partial n} \\ u & v \end{vmatrix}\, dS ,

where the volume $V$ is enclosed by the surface $S$ , $\overrightarrow {n}$ is the outward normal direction of the surface $S$ , and the functions $u=u(x,y,z)$ , $v=v(x,y,z)$ are twice differentiable in the domain $V+S$ .

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Problem 4395

(a)(b)

The function $u=u(x,y,z)$ has continuous derivatives up to second order in some domain; if

\Delta u = \dfrac {\partial ^{2}u}{\partial x^{2}}+\dfrac {\partial ^{2}u}{\partial y^{2}}+\dfrac {\partial ^{2}u}{\partial z^{2}} = 0 ,

then $u(x,y,z)$ is called a harmonic function in this domain.

Prove that if $u$ is a harmonic function in the bounded closed domain $V$ enclosed by a smooth surface $S$ , then the following formulas hold:

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(a)

$\displaystyle \iint_{S} \dfrac {\partial u}{\partial n}\, dS = 0$ ;

(b)

$\displaystyle \iiint_{V} \left[\left(\dfrac {\partial u}{\partial x}\right)^{2}+\left(\dfrac {\partial u}{\partial y}\right)^{2}+\left(\dfrac {\partial u}{\partial z}\right)^{2}\right]\, dx\, dy\, dz$

= \iint _{S} u\dfrac {\partial u}{\partial n}\, dS ,

where $\overrightarrow {n}$ is the outward normal of the surface $S$ .

Using formula (2), prove that a harmonic function in the domain $V$ is uniquely determined by its values on the boundary $S$ .

Problem 4396

Prove that if the function $u=u(x,y,z)$ is a harmonic function in the bounded closed domain $V$ enclosed by a smooth surface $S$ , then

u(x,y,z) = \dfrac {1}{4\pi }\iint _{S} \left[u\dfrac {\cos (\overrightarrow {r},\overrightarrow {n})}{r^{2}}+\dfrac {1}{r}\dfrac {\partial u}{\partial n}\right]\, dS ,

where $\overrightarrow {r}$ is the radius vector drawn from the interior point $(x,y,z)$ of the domain $V$ to the moving point $(\xi ,\eta ,\zeta )$ on the surface $S$ , and

r = \sqrt{(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}} ,

with $\overrightarrow {n}$ the outward normal vector of the surface $S$ at the point $(\xi ,\eta ,\zeta )$ .

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Problem 4397

Prove that if $u=u(x,y,z)$ is a harmonic function inside the ball $S$ of radius $R$ centered at the point $(x_{0},y_{0},z_{0})$ , then

u(x_{0},y_{0},z_{0}) = \dfrac {1}{4\pi R^{2}}\iint _{S} u(x,y,z)\, dS

(the mean value theorem).

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Problem 4398

Prove that a function $u=u(x,y,z)$ that is continuous on a bounded closed domain $V$ and harmonic in its interior, if it is not constant, cannot attain its maximum and minimum values at interior points of the domain (the maximum principle).

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Problem 4399

A body $V$ is completely submerged in a liquid. Starting from Pascal's law, prove that the buoyant force of the liquid equals the weight of the liquid displaced by the body, directed vertically upward (Archimedes' principle).

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Problem 4400

Let $S_{t}$ be the moving sphere $(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}=t^{2}$ , and let the function $f(\xi ,\eta ,\zeta )$ be continuous. Prove that the function

u(x,y,z,t) = \dfrac {1}{4\pi }\iint _{S_{t}} \dfrac {f(\xi ,\eta ,\zeta )}{t}\, dS_{t}

satisfies the wave equation

\dfrac {\partial ^{2}u}{\partial x^{2}}+\dfrac {\partial ^{2}u}{\partial y^{2}}+\dfrac {\partial ^{2}u}{\partial z^{2}} = \dfrac {\partial ^{2}u}{\partial t^{2}}

together with the initial conditions $u\big|_{t=0}=0$ , $\dfrac {\partial u}{\partial t}\bigg|_{t=0}=f(x,y,z)$ .

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