Divergence and curl (rot) of vector fields

Find $\displaystyle \lim_{d(S)\to 0} \dfrac {1}{S}\oint_{C} (\overrightarrow {F},\overrightarrow {n})\, ds$, where $S$ is the area enclosed by the contour $C$ containing the point $(x_{0},y_{0})$, $d(S)$ is the diameter of the region $S$, $\overrightarrow {n}$ is the unit vector along the outward normal of the contour $C$, and $\overrightarrow {F}(x,y)$ is a vector that is continuously differentiable on $S+C$.

Prove the formula

where

Prove by direct computation that the divergence of the vector $\overrightarrow {a}$ is independent of the choice of Cartesian coordinate system.

Prove

where $S$ is a closed surface enclosing the point $M$ and bounding a volume $V$, $\overrightarrow {n}$ is the outward normal of the surface $S$, and $d(S)$ is the diameter of the surface $S$.

Find:

Prove:

Prove that for twice differentiable fields $u$ and $v$

Verify the indicated equalities in coordinate form, and also write them and verify them using the symbol $\nabla$ and the rules for operating with it ($\alpha , \beta$ numbers, $u, \mathbf{a}, \mathbf{b}$ differentiable scalar and vector fields):

Setting $\mathbf{r} = x \mathbf{i}+y \mathbf{j}+z \mathbf{k}, r = \left|\mathbf{r}\right|$, find $\operatorname {div} \mathbf{a}$, if:

Express $\operatorname {div} \operatorname {grad} u$ in coordinate form.

Find:

Find $(\mathbf{r} = x \mathbf{i}+y \mathbf{j}+z \mathbf{k}), r = \left|\mathbf{r}\right|)$ :

Solve the equation $(\mathbf{r} = x \mathbf{i}+y \mathbf{j}+z \mathbf{k}, r = \left|\mathbf{r}\right|)$ :

Find the divergence of the gravitational field of several point masses.

A medium rotates as a rigid body about an axis with constant angular velocity $\omega$. Find, at a fixed instant of time, the divergence of the field of linear velocities $\mathbf{v}$ and the field of accelerations $\mathbf{w}$ of the points of the medium.

Prove that div a does not depend on the choice of Cartesian coordinate system.

Find $\operatorname {div}$ a of the plane field a in polar coordinates.

Find div a of a three-dimensional field:

Find $\left(r = \sqrt{x^{2}+y^{2}+z^{2}}\right)$ :

Verify the indicated equalities in coordinate form, and also write and verify them using the symbol $\nabla$ and the rules for operating with it ($\alpha , \beta$ numbers, $u, \mathbf{a}, \mathbf{b}$ differentiable scalar and vector fields, c a constant vector):

Find $(\mathbf{r} = x \mathbf{i}+y \mathbf{j}+z \mathbf{k}, r = \left|\mathbf{r}\right|$, $\mathbf{a}$ and $\mathbf{b}$ constant vectors, $u(r)$ a differentiable field):

Calculate $\operatorname {rot} \mathbf{a}$ at the point $M_{0}$, if:

For any vector $\mathbf{p}$ the vectors $[\mathbf{p}, \mathbf{a}]$ and a are perpendicular (if they are nonzero). Is this true for the vectors $[\nabla , \mathbf{a}]$ and $\mathbf{a}$?

Find the angle between $\operatorname {rot} \mathbf{a}\left(M_{1}\right)$ and $\operatorname {rot} \mathbf{a}\left(M_{2}\right)$, if:

Find:

Verify in coordinate form:

Verify the equality $\operatorname {rot} \operatorname {rot} \mathbf{a} = \operatorname {grad} \operatorname {div} \mathbf{a}-\Delta \mathbf{a}$ in coordinate form, and also write and derive it using the symbol $\nabla$ and the rules for operating with it.

Find $\operatorname {rot} \operatorname {grad}(1 / r)$.

Derive the formulas:

Show that:

Find the components of $\operatorname {rot} \mathbf{a}$ of the plane field a in polar coordinates.

Find the components of $\operatorname {rot} \mathbf{a}$ of a three-dimensional field a:

Find $\operatorname {rot}(u(r) \mathbf{a}(r)), r = \sqrt{x^{2}+y^{2}+z^{2}}$.

Write $\Delta u = \operatorname {div} \operatorname {grad} u$ :

A medium rotates as a rigid body about an axis with constant angular velocity $\omega$. Let $\mathbf{v}$ be the field of linear velocities of points at a fixed instant of time. Find $\operatorname {rot} \mathbf{v}$ (use cylindrical coordinates).

In the simplest case, the system of Maxwell's equations for the electromagnetic field has the form

Here $\mathbf{E}$ and $\mathbf{H}$ are the vector fields of electric and magnetic intensity, $\varepsilon , \mu , c =$ const $> 0$. Assuming all functions sufficiently smooth, prove that $\mathbf{E}$ and $\mathbf{H}$ satisfy the wave equation

Let an orthogonal system of curvilinear coordinates $(\xi ; \eta ; \zeta )$ be introduced in the region $\Omega$ :

where the right-hand sides are continuously differentiable functions. Let $\mathbf{e}_{\xi }, \mathbf{e}_{\eta }, \mathbf{e}_{\zeta }$ be the unit vectors of this system (vectors tangent to the coordinate lines and directed toward increasing coordinates, $\mathbf{e}_{\xi } \perp \mathbf{e}_{\eta }, \mathbf{e}_{\eta } \perp \mathbf{e}_{\zeta }, \mathbf{e}_{\zeta } \perp \mathbf{e}_{\xi }$)$^{*}$. Let $H_{\xi }, H_{\eta }, H_{\zeta }$ be the Lame coefficients, i.e. $H_{\xi } = \sqrt{\left(\frac{\partial x}{\partial \xi }\right)^{2}+\left(\frac{\partial y}{\partial \xi }\right)^{2}+\left(\frac{\partial z}{\partial \xi }\right)^{2}}$ etc. Prove that:

Here, for orthogonal curvilinear coordinates $(\xi ; \eta ; \zeta )$ with unit vectors $\mathbf{e}_{\xi }, \mathbf{e}_{\eta }, \mathbf{e}_{\zeta }$ and Lame coefficients $H_{\xi }, H_{\eta }, H_{\zeta }$ (see problem 66), formulas (39)-(41) are

Using formulas (39)-(41), derive expressions for $\operatorname {grad} u, \operatorname {div} \mathbf{a}, \operatorname {rot} \mathbf{a}$ :

Prove formula (24).

Formula (24): let $\mathbf{a}$ be a continuously differentiable vector field in a region $\Omega$, $M \in \Omega$, and consider the regions $G \subset \Omega$ containing $M$ for which the Gauss--Ostrogradsky formula (23)

holds ($G$ is a bounded region, $\bar{G} \subset \Omega$, with piecewise smooth boundary $\partial G$ oriented by the outward normal $\mathbf{n}$). Let $d(G)$ be the diameter and $\mu G$ the volume of $G$. Then

Let the field a be continuously differentiable in $\Omega , G$ an arbitrary region with piecewise smooth boundary, $\bar{G} \subset \Omega$. Prove that the flux of $\operatorname {rot} \mathbf{a}$ through $\partial G$ equals zero.

Prove that

Let $u$ and a be continuously differentiable fields in $\Omega$, $G$ a region of $\Omega , \bar{G} \subset \Omega , \partial G$ a piecewise smooth surface, oriented by the outward normal. Prove that

Let $u$ and $\mathbf{a}$ be continuously differentiable fields in $\Omega$, $M \in \Omega$. Consider the collection of regions $G \subset \Omega$ containing $M$ for which formula (23) holds.

Formula (23), the Gauss--Ostrogradsky formula: if $G$ is a bounded region, $\bar{G} \subset \Omega$, with piecewise smooth boundary $\partial G$ oriented by the outward normal $\mathbf{n}$, then $\iint_{\partial G} (\mathbf{a}, \mathbf{n}) d S = \iiint_{G} \operatorname {div} \mathbf{a} d V = \iiint_{G} (\nabla , \mathbf{a}) d V$.

Let $d(G)$ be the diameter, $\mu (G)$ the volume of $G$. Prove that:

Prove that the potential $u$ of a continuously differentiable field a satisfies the equation $\Delta u = \operatorname {div} \mathbf{a}$.

Let $u$ be a field twice continuously differentiable in $\Omega$, $\mathbf{a}$ and $\mathbf{b}$ differentiable fields in $\Omega , \mathbf{a} = \mathbf{b}+\operatorname {grad} u$. Prove that, for the field $\mathbf{b}$ to be solenoidal, it is necessary and sufficient that the field $u$ satisfy the equation $\Delta u = \operatorname {div} \mathbf{a}$.

From the equations of electrostatics

where $\mathbf{E}$ is the electric field intensity, $\rho$ the charge density, $\varepsilon_{0} =$ const $> 0$, derive Gauss's law

on the proportionality of the flux of the intensity through the boundary of the region $G$ (with outward normal $\mathbf{n}$) and the total charge $Q$ located in this region.