Divergence and curl (rot) of vector fields

Find $\operatorname {div}(\operatorname {grad} u)$.

Find $\operatorname {div}\left[\operatorname {grad} f(r)\right]$, where $r=\sqrt{x^{2}+y^{2}+z^{2}}$. Under what condition is $\operatorname {div}\left[\operatorname {grad} f(r)\right]=0$?

Compute: (1) $\operatorname {div}\overrightarrow {r}$; (2) $\operatorname {div}\dfrac {\overrightarrow {r}}{r}$.

Compute $\operatorname {div}\left[f(r)\overrightarrow {c}\right]$, where $\overrightarrow {c}$ is a constant vector.

Find $\operatorname {div}\left[f(r)\overrightarrow {r}\right]$. Under what condition is the divergence of this vector equal to zero?

Find: (1) $\operatorname {div}(u\operatorname {grad} u)$; (2) $\operatorname {div}(u\operatorname {grad} v)$.

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.

Find the divergence of the force field produced by a finite system of gravitational centers.

Find the expression for the divergence of a plane vector $\overrightarrow {a}=\overrightarrow {a}(r,\varphi )$ represented in polar coordinates $r$ and $\varphi$.

Let

express $\operatorname {div}\overrightarrow {a}(x,y,z)$ in orthogonal curvilinear coordinates $u$, $v$, $w$.

As special cases, find the expressions for $\operatorname {div}\overrightarrow {a}$ in cylindrical coordinates and in spherical coordinates.

Prove:

Find: (1) $\operatorname {rot}\overrightarrow {r}$; (2) $\operatorname {rot}\left[f(r)\overrightarrow {r}\right]$.

Find: (1) $\operatorname {rot}\overrightarrow {c}f(r)$, (2) $\operatorname {rot}\left[\overrightarrow {c}\times f(r)\overrightarrow {r}\right]$ (where $\overrightarrow {c}$ is a fixed vector).

Prove that $\operatorname {div}(\overrightarrow {a}\times \overrightarrow {b}) = \overrightarrow {b}\cdot \operatorname {rot}\overrightarrow {a}-\overrightarrow {a}\cdot \operatorname {rot}\overrightarrow {b}$.

Find: (1) $\operatorname {rot}(\operatorname {grad} u)$; (2) $\operatorname {div}(\operatorname {rot}\overrightarrow {a})$.

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.