Integrals of Field Theory and Potential Theory

If $u(x,y,z)$ is a function of a point, then the vector whose projections on the axes $Ox$, $Oy$, and $Oz$ equal the quantities $\dfrac {\partial u}{\partial x}$, $\dfrac {\partial u}{\partial y}$, $\dfrac {\partial u}{\partial z}$ is called the gradient of $u$ and denoted $\operatorname {grad}u$.

Verify that $\operatorname {grad}(x^{2}+y^{2}+z^{2})=2\overrightarrow {r}$, where $\overrightarrow {r}$ is the radius vector of the point $(x,y,z)$.

Verify that $\operatorname {grad}\dfrac {1}{\sqrt{x^{2}+y^{2}+z^{2}}}=\dfrac {\overrightarrow {r}}{r^{3}}$, where $\overrightarrow {r}$ is the radius vector of the point $(x,y,z)$, and $r=\left|\overrightarrow {r}\right|$.

Verify that $\operatorname {div}\operatorname {grad}(x^{2}+y^{2}+z^{2})=6$, where for a vector $\overrightarrow {u}$ with projections $u_{x}$, $u_{y}$, $u_{z}$, the divergence of the field is called the scalar quantity $\operatorname {div}\overrightarrow {u}=\dfrac {\partial u_{x}}{\partial x}+\dfrac {\partial u_{y}}{\partial y}+\dfrac {\partial u_{z}}{\partial z}$.

Verify that $\operatorname {div}\dfrac {\overrightarrow {r}}{r^{3}}=0$, where $\overrightarrow {r}$ is the radius vector of the point $(x,y,z)$, and $r=\left|\overrightarrow {r}\right|$.

Prove that if $u_{x}=-\dfrac {ay}{x^{2}+y^{2}}$, $u_{y}=\dfrac {ax}{x^{2}+y^{2}}$, $u_{z}=0$, then the circulation of $\overrightarrow {u}$ along a closed curve (i.e., the line integral $\int_{L} \overrightarrow {u}\cdot d\overrightarrow {r}=\int_{L} u_{x} \, dx+u_{y} \, dy+u_{z} \, dz$) equals zero if the curve does not go around the axis $Oz$. It equals $2\pi n$ if the curve goes around the axis $Oz$ $n$ times, in such a way that its projection onto the plane $xOy$ goes around the origin counterclockwise $n$ times.

Prove that for the vector $\overrightarrow {u}$ of the previous problem ($u_{x}=-\dfrac {ay}{x^{2}+y^{2}}$, $u_{y}=\dfrac {ax}{x^{2}+y^{2}}$, $u_{z}=0$), the curl $\operatorname {rot}\overrightarrow {u}=0$ at all points except those on the axis $Oz$, where for a vector $\overrightarrow {u}$ with projections $u_{x}$, $u_{y}$, $u_{z}$, the curl (rotation) of the field is called the vector with projections on the axes $Ox$, $Oy$, $Oz$ equal to the quantities $\dfrac {\partial u_{z}}{\partial y}-\dfrac {\partial u_{y}}{\partial z}$, $\dfrac {\partial u_{x}}{\partial z}-\dfrac {\partial u_{z}}{\partial x}$, $\dfrac {\partial u_{y}}{\partial x}-\dfrac {\partial u_{x}}{\partial y}$.

A medium performs a rotation, as a rigid body, about the axis $Oz$ with angular velocity $\omega$. The velocity vector $\overrightarrow {v}$ has projections on the axes $v_{x}=-\omega y$, $v_{y}=\omega x$, $v_{z}=0$. Find $\operatorname {rot}\overrightarrow {v}$.

Find the circulation of the vector $\overrightarrow {v}$ of the previous problem ($v_{x}=-\omega y$, $v_{y}=\omega x$, $v_{z}=0$, the velocity vector of a medium rotating as a rigid body about the axis $Oz$ with angular velocity $\omega$) along the circle $x^{2}+y^{2}=a^{2}$ in the positive direction.

Find the circulation of the same vector $\overrightarrow {v}$ ($v_{x}=-\omega y$, $v_{y}=\omega x$, $v_{z}=0$, the velocity vector of a medium rotating as a rigid body about the axis $Oz$ with angular velocity $\omega$) along the circle $(x-2)^{2}+y^{2}=2$.

The formula of Ostrogradsky-Gauss relates the triple integral over a volume $V$ to the double integral over the surface $\sigma$ bounding it:

where $\overrightarrow {n}$ is the unit outward normal vector to $\sigma$, and $P$, $Q$, $R$ are the projections of the field vector $\overrightarrow {u}$ on the axes $Ox$, $Oy$, $Oz$; in vector form the same formula reads $\iiint_{V}\operatorname {div}\overrightarrow {u} \, dv=\iint_{\sigma }\overrightarrow {u}\cdot \overrightarrow {n} \, d\sigma =\iint_{\sigma }u_{n} \, d\sigma$.

Prove the equality $\iint_{S}\cos (\overrightarrow {n},x) \, d\sigma =0$, where $S$ is a closed surface and $\overrightarrow {n}$ is the outward normal to it.

Find the value of the integral $\displaystyle \iint \limits_{S}(x\cos \alpha +y\cos \beta +z\cos \gamma ) \, ds$, where $\alpha$, $\beta$, $\gamma$ are the angles of the outward normal to the surface $S$, and the integral is taken over the entire surface $S$ of some body of volume $v$.

Find the integral $\displaystyle \iint \limits_{S}(x^{3}\cos \alpha +y^{3}\cos \beta +z^{3}\cos \gamma ) \, ds$, taken over the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$. Here $\alpha$, $\beta$, $\gamma$ are the angles of the outward normal with the axes.

Find the integral $\displaystyle \iint \limits_{S}\big((z^{n}-y^{n})\cos \alpha +(x^{n}-z^{n})\cos \beta +(y^{n}-x^{n})\cos \gamma \big) \, ds$, taken over the upper half of the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$; $\alpha$, $\beta$, $\gamma$ are the angles of the normal outward from the sphere.

Applying the formula of Ostrogradsky-Gauss

to the vector $u\operatorname {grad}v$, obtain Green’s formula:

Here $\omega$ is a volume; $\sigma$ is its surface; $\dfrac {\partial v}{\partial n}=\dfrac {\partial v}{\partial x}\cos \alpha +\dfrac {\partial v}{\partial y}\cos \beta +\dfrac {\partial v}{\partial z}\cos \gamma$ is the derivative along the inward normal, and $\Delta v=\dfrac {\partial^{2}v}{\partial x^{2}}+\dfrac {\partial^{2}v}{\partial y^{2}}+\dfrac {\partial^{2}v}{\partial z^{2}}$.

Starting from the previous result (Green's formula, problem 3772: $\iiint_{\omega }\big(u\Delta v+\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}+\frac{\partial u}{\partial z}\frac{\partial v}{\partial z}\big) \, d\omega = -\iint_{\sigma }u\frac{\partial u}{\partial n} \, d\sigma$), prove Green’s second formula:

A function $u$ for which $\dfrac {\partial^{2}v}{\partial x^{2}}+\dfrac {\partial^{2}v}{\partial y^{2}}+\dfrac {\partial^{2}v}{\partial z^{2}}=0$ is called harmonic. Prove that for harmonic functions the formula

holds.

Prove a formula similar to the formula of Ostrogradsky-Gauss

Here $\sigma$ is an area, and $S$ is the line bounding it; $\alpha$ and $\beta$ are the angles of the outward normal with the axes.

Prove a formula similar to Green's formula ($\iiint_{\omega }\big(u\Delta v+\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}+\frac{\partial u}{\partial z}\frac{\partial v}{\partial z}\big) \, d\omega = -\iint_{\sigma }u\frac{\partial u}{\partial n} \, d\sigma$, problem 3772):

Here $\sigma$ is an area, $S$ is its contour, $n$ is the inward normal, $\Delta v=\dfrac {\partial^{2}v}{\partial x^{2}}+\dfrac {\partial^{2}v}{\partial y^{2}}$.

Prove the equality:

where $\sigma$ is an area, $S$ is its contour, $n$ is the inward normal, $\Delta v=\dfrac {\partial^{2}v}{\partial x^{2}}+\dfrac {\partial^{2}v}{\partial y^{2}}$ and similarly for $\Delta u$ (notation of problem 3776).

Prove that for harmonic functions of two variables (i.e., functions $u$ for which $\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}=0$) the equality

holds.

For the same functions as in the previous problem (harmonic functions of two variables), prove the formula

Prove that for harmonic functions of three variables

Using Green's second formula ($\iiint_{\omega }(u\Delta v-v\Delta u) \, d\omega = -\iint_{\sigma }\big(u\frac{\partial v}{\partial n}-v\frac{\partial u}{\partial n}\big) \, d\sigma$, see 3773), applied to the region between the surface $\sigma$ and a sphere of radius $\rho \to 0$ centered at the point $M(\xi ,\eta ,\zeta )$ lying inside the surface $\sigma$, prove for a harmonic function of three variables the equality

where $r$ is the distance from the point $M$ to a variable point of the surface, and $\overrightarrow {n}$ is the outward normal vector.

Prove that

according to whether the point from which the radius vectors $\overrightarrow {r}$ are drawn to the points of the surface lies inside the surface or outside it. The surface is assumed to be simply connected, closed, and without singular points.

Prove that for harmonic functions of two variables the formula

holds, where the derivatives are taken along the inward normal, and the point $M(x,y)$ is inside the contour $S$.

Prove the equality

where $\varphi$ is the angle at which the line $AB$ is seen from the point $(\xi ,\eta )$, $r=\sqrt{(x-\xi )^{2}+(y-\eta )^{2}}$, and $\overrightarrow {n}$ is the normal to the line $AB$.

Prove that for harmonic functions of three variables the formula

holds, where $\sigma$ is the surface of a sphere of radius $r$ centered at the point $(x,y,z)$. (Gauss.)

Prove for harmonic functions of two variables the formula

where $S$ is the circle of radius $r$ centered at the point $(x,y)$.

Prove that for a function $u(x,y,z)$ harmonic inside a sphere of volume $\omega$, the formula

holds, where $u_{c}$ is the value of $u$ at the center of the sphere; and for a function $u(x,y)$ harmonic inside a disk of area $\sigma$, prove the formula

where $u_{c}$ is the value of $u$ at the center of the disk.

Using the Taylor series expansion, prove the equality

where

and so on, and the integration is carried out over the circle of radius $r$ centered at the point $(x,y)$.

Prove the equality

where

and the integration is taken over the surface of a sphere of radius $r$ centered at the point $(x,y,z)$.

Prove Maxwell's formula

Here $\sigma$ is the surface of the volume $\omega$, in which the function $u$ is continuous together with its second derivatives.

Prove Riemann's equality

Here $S$ is the contour of the area $\sigma$,

and the expressions for $M$ and $N$ are to be found.

If the projections of the velocity of a fluid at time $t$ at the point $(x,y,z)$ equal the functions $u(x,y,z,t)$, $v(x,y,z,t)$, and $w(x,y,z,t)$, then the amount of fluid flowing through the surface $\sigma$ during the time from $t_{1}$ to $t_{2}$ equals the integral

where $\alpha$, $\beta$, $\gamma$ are the angles of the normal to the surface $\sigma$ with the axes. Deduce from this that for an incompressible fluid the condition

must hold.

Study the variation of the function $P(x)$, where $P(x)=\mu \int_{a}^{b}\left|y-x\right| \, dy$, as $x$ varies over the interval $(-\infty ,+\infty )$.

Find the second derivative of

where the function $\mu (y)$ is continuous.

Using Poisson's integral

find the logarithmic potential of a homogeneous disk, i.e., the integral $\mu \iint_{\sigma } \ln r \, d\sigma$, where $r$ is the distance between a point $M$ and a point on the area of a disk $\sigma$ of radius $R$.

The logarithmic potential of a homogeneous segment $(-1,1)$ is called the integral $P=\int_{-1}^{+1}\ln r \, d\xi$, where $r$ is the distance between the points $(x,y)$ and $(\xi ,0)$. Find the equation of the curve on which $P(x,y)=P(1,0)$.

The logarithmic potential of an area $\sigma$ at the point $(x,y)$ is called the integral

Find the logarithmic potential of the disk $x^{2}+y^{2}=a^{2}$, if the density $\mu$ at the point $(x,y)$ is a given function of the radius vector from the origin: $\mu =f(r)$, where $r=\sqrt{x^{2}+y^{2}}$.

Find at the point $M(x,y,z)$ the Newtonian potential of a homogeneous shell of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ with surface density $\mu$ equal to some constant, i.e., find the integral $\iint_{\sigma }\mu \dfrac {d\sigma }{r}$, where $\sigma$ is the surface of the sphere, and $r$ is the distance from a variable point $(\xi ,\eta ,\zeta )$ on this surface to the given point $M$.

Find at the point $M(x,y,z)$ the Newtonian potential of a homogeneous sphere, i.e., the integral

where $\omega$ is a sphere of radius $a$ centered at the origin, $\mu$ is the constant density, $(\xi ,\eta ,\zeta )$ is a point inside the sphere, $r=\sqrt{(x-\xi )^{2}+(y-\eta )^{2}+(z-\zeta )^{2}}$ is the distance from the point $(\xi ,\eta ,\zeta )$ to the point $(x,y,z)$.

The same question (find at the point $M(x,y,z)$ the Newtonian potential of a sphere, i.e., the integral $P=\iiint_{\omega } \mu \, d\xi \, d\eta \, d\zeta / r$, where $\omega$ is a sphere of radius $a$ centered at the origin, $r$ is the distance from the point $(\xi ,\eta ,\zeta )$ inside the sphere to the point $(x,y,z)$) for a sphere whose density $\mu$ equals a given function $f(\rho )$, where $\rho^{2}=\xi^{2}+\eta^{2}+\zeta^{2}$.

At the point $(0,0,z)$ find the Newtonian potential of the upper half of the surface of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ with constant density $\mu$.

Find the Newtonian potential of a homogeneous spherical shell between two concentric spheres of radii $a$ and $b$, where $a<b$.

Find at the apex of a cone the Newtonian potential of the solid cone with base radius $r$ and height $h$.

Show that among bodies of constant density and given mass, the one that attracts a material point most strongly (by Newton's law) is bounded by the surface $R=a\sqrt{\cos \theta }$ (the material point is taken as the origin, the direction of the attractive force as the axis $Oz$; $R$, $\theta$ are spherical coordinates). Find this force and compare it with the force of attraction by a sphere (of the same mass) on a material point lying on its surface.

Find at the point $M(x,y)$ the Newtonian potential of a non-homogeneous segment $(-a,a)$, i.e., the integral $\displaystyle \int_{-a}^{a}\frac{\mu (\xi ) \, d\xi }{\sqrt{(\xi -x)^{2}+y^{2}}}$, if $\mu (\xi )=\dfrac {\xi }{a}$.

Find the Newtonian potential of a homogeneous circular plate $x^{2}+y^{2}=a^{2}$, $z=0$, at the point $(0,0,z)$.

Prove the equality

Here $r_{1}^{2}=x^{2}+y^{2}+z^{2}$, $r$ is the distance from the point $(x,y,z)$ to an arbitrary point $(\xi ,\eta ,\zeta )$ inside the volume $\omega$.

Prove the identity

where $\sigma$ is the surface of the body of volume $\omega$; $\overrightarrow {n}$ is the outward normal to $\sigma$; $\overrightarrow {r}$ is the vector from the point $(x,y,z)$ to the point $(\xi ,\eta ,\zeta )$; $r$ is its length.

Under the same conditions as in the previous problem ($\sigma$ is the surface of the body of volume $\omega$; $\overrightarrow {n}$ is the outward normal to $\sigma$; $\overrightarrow {r}$ is the vector from the point $(x,y,z)$ to the point $(\xi ,\eta ,\zeta )$; $r$ is its length), prove the identity

Write down analogous identities (see problem 3809) for an area $\sigma$ and its contour $S$.

Prove the equality $\displaystyle \int_{S} \frac{\cos (\overrightarrow {r},\overrightarrow {n})}{r} \, ds = 0$, if the point $(x,y)$ lies outside the region bounded by the contour $S$, and $=2\pi$, if the contour $S$ makes one complete circuit around the point $(x,y)$.

The projections on the axes $Ox$, $Oy$, $Oz$ of the force with which a point of mass $m$ is attracted, by Newton's law, by a body of density $\mu$, are expressed by the formulas:

where $P=\iiint_{\omega } \mu \dfrac {d\omega }{r}$, $r=\sqrt{(\xi -x)^{2}+(\eta -y)^{2}+(\zeta -z)^{2}}$, $f$ is the gravitational constant. Using this, compute the force of attraction by a sphere of radius $a$ with density $\mu$ on a point of mass $m$ located on its surface.

By the Biot-Savart law, each element $d\overrightarrow {s}$ of a current of strength $I$ creates at the point $M(x,y,z)$ a magnetic field whose intensity can be expressed by the vector product $-\dfrac {kI}{r^{2}}d\overrightarrow {s}\times \overrightarrow {r}$, where $\overrightarrow {r}$ is the vector going from $d\overrightarrow {s}$ to $M$. Prove that the field intensity $\overrightarrow {H}$ created by a closed current flowing along the contour $S$ is expressed by the vector integral $\overrightarrow {H}=-kI\int_{S} \dfrac {d\overrightarrow {s}\times \overrightarrow {r}}{r^{2}}$; here $k$ is the proportionality coefficient.

Using the previous result (problem 3813: $\overrightarrow {H}=-kI\int_{S} \frac{d\overrightarrow {s}\times \overrightarrow {r}}{r^{2}}$, the intensity of the magnetic field of a closed current by the Biot-Savart law) and Stokes's formula

(the circulation of a vector around a closed contour $L$ equals the flux of the curl of the vector through the surface $\sigma$ bounded by the contour), prove the equality $\overrightarrow {H}=-kI\operatorname {grad}W$, where $W$ is the solid angle at which the area $\sigma$ with contour $S$ is seen from the point $M$, equal to the integral $\iint_{\sigma } \dfrac {\cos (\overrightarrow {r},\overrightarrow {n})}{r^{2}} \, d\sigma$.

Consider the integral $\displaystyle \int_{s_{1}} d\overrightarrow {s}_{1}\int_{s_{2}} \frac{d\overrightarrow {s}_{2}\times \overrightarrow {r}}{r^{3}}$, taken over two closed contours $s_{1}$ and $s_{2}$, where $\overrightarrow {r}$ is the vector joining variable points on the contours. Regarding $\dfrac {d\overrightarrow {s}_{1}\times d\overrightarrow {s}_{2}\cdot \overrightarrow {e}_{r}}{r^{2}}$ as a solid angle, prove that this double integral equals $0$ if one contour is not linked with the other, and equals $\pm 4\pi$ if it is linked (once).