Physical applications of line, surface integrals and field theory

Find the work done by gravity (the $Oz$ axis pointing vertically upward) as a point of mass $m$ moves from position $(x_{1},y_{1},z_{1})$ to position $(x_{2},y_{2},z_{2})$.

An elastic force is directed toward the origin of coordinates, its magnitude proportional to the distance of the point from the origin. Suppose the point traces out, counterclockwise, one quarter of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$; find the work done by the elastic force.

Find the work done by the gravitational force $F=\dfrac {k}{r^{2}}$ (where $r=\sqrt{x^{2}+y^{2}+z^{2}}$) acting on a unit mass as it moves from the point $M_{1}(x_{1},y_{1},z_{1})$ to the point $M_{2}(x_{2},y_{2},z_{2})$.

Let $u=u(x,y)$ and $v=v(x,y)$ be the components of the velocity of a liquid. Find the amount of liquid flowing per unit time through the domain $S$ bounded by the contour $C$ (i.e., the difference between the amount of liquid flowing out and the amount flowing in). If the liquid is incompressible and there are no sources or sinks within the domain $S$, what equation must the functions $u$ and $v$ satisfy?

According to the Biot--Savart law, the current $i$ flowing through the line element $ds$ produces a magnetic field at the point $M(x,y,z)$ in space, with intensity

where $\overrightarrow {r}$ is the vector connecting the element $d\overrightarrow {s}$ to the point $M$, and $k$ is a constant of proportionality. For the case of a closed wire $C$, find the projections $H_{x}$, $H_{y}$, $H_{z}$ of the intensity of the magnetic field $\overrightarrow {H}$ at the point $M$.

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).

A body rotates counterclockwise about the $Oz$ axis with a certain angular velocity $\overrightarrow {\omega }$. Find the divergence of the velocity vector $\overrightarrow {v}$ and the acceleration vector $\overrightarrow {w}$ at the point $M(x,y,z)$ in space, and at a given moment in time.

A body rotates with constant angular velocity $\omega$ about the axis $\overrightarrow {l}\left\{ \cos \alpha ,\cos \beta ,\cos \gamma \right\}$. Find the curl of the velocity vector $\overrightarrow {v}$ at the point $M(x,y,z)$ of space at a given instant of time.

The quantity of heat entering a temperature field $u$ per unit time through a surface element $dS$ equals

where $k$ is the coefficient of internal thermal conductivity and $\overrightarrow {n}$ is the unit normal vector of the surface $S$. Find the quantity of heat accumulated per unit time by a body $V$. Study the rate of temperature rise so as to derive the equation satisfied by the temperature of the body (the heat conduction equation).

An incompressible fluid in motion occupies a volume $V$. Assuming that there are no sources or sinks in the region $V$, derive the continuity equation:

where $\rho =\rho (x,y,z,t)$ is the density of the fluid, $\overrightarrow {v}$ is the velocity vector, and $t$ is time.

The steady incompressible flow of a plane is described by the velocity vector

Find: (1) the amount $Q$ of fluid flowing out through a closed contour $C$ enclosing a region $S$ (the fluid discharge); (2) the circulation $\Gamma$ of the velocity vector along the contour $C$. If the flow field has no sources, no sinks, and no vorticity, what equations must the functions $u$ and $v$ satisfy?

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.

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.

A wire which forms a closed plane curve $C$ carries a unit electric current and exerts a force $F$ on a unit magnetic pole in the plane of, and interior to, $C$. Given $A$, the area enclosed by $C$, prove that $F$ is a minimum when $C$ is a circle and the magnetic pole is at its center.