Divergence

Exercise3.5

fundamental formula Let $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k},\ r = \sqrt{x^2 + y^2 + z^2}$. Then
(1) $\nabla r = \frac{\boldsymbol{r}}{r}$
(2) $\nabla r^{n} = nr^{n-1}\nabla r = nr^{n-2}\boldsymbol{r}$
(3) $\nabla \cdot(\phi\boldsymbol{A})= (\nabla \phi) \cdot\boldsymbol{A} + \phi \nabla \cdot\boldsymbol{A}$

1.

Find the followings.．

(1) $\nabla \cdot(2x^2 z\boldsymbol{i} - xy^2 z \boldsymbol{j} + 3yz^2 \boldsymbol{k})$(2) $\nabla^2(3x^2 z - y^2 z^3 + 4x^2 y)$

(3) $\nabla(\nabla \cdot\boldsymbol{F}), \ \boldsymbol{F} = (3x^2 y - z)\boldsymbol{i} + (xz^{3} + y^{4})\boldsymbol{j} - 2x^2 z^2 \boldsymbol{k}$

2.

Let $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k},\ r = \vert\boldsymbol{r}\vert$．Find the following scalar．

(1) $\nabla \cdot(r\nabla r^{-3})$(2) $\nabla^{2}\left\{\nabla\cdot\left(\frac{\boldsymbol{r}}{r^2}\right)\right\}$

3.

Let $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k},\ {\bf w}$ be a constant vector. Then show that ．

$$\nabla \cdot({\bf w} \times \boldsymbol{r}) = 0$$

4.

For the scalar fields $\phi, \psi$, prove the followings. (1) $\nabla^2(\phi \psi) = \phi \nabla^2 \psi + 2(\nabla \phi)\cdot(\nabla \psi) + \psi \nabla^2 \phi$

(2) $\nabla \cdot(\phi \nabla \psi) = (\nabla \phi) \cdot(\nabla \psi) + \phi \nabla^2 \psi$

(3) $\nabla \cdot(\phi \nabla \psi - \psi \nabla \phi) = \phi \nabla^2 \psi - \psi \nabla^2 \phi$

5.

Find $\phi = \phi(x,y,z)$ which satisfies $\phi = \phi(x,y,z)$ $\nabla \phi = 2xyz^3 \boldsymbol{i} + x^2 z^3 \boldsymbol{j} + 3x^2 y z^2 \boldsymbol{k}$．provided $\phi(1,-2,2) =4$．

6.

For scalar fields $U,V$, show the following.

$$\nabla \cdot\{(\nabla U) \times (\nabla V)\} = 0$$

7.

For the curve $\boldsymbol{r} = \boldsymbol{r}(t) = x(t)\boldsymbol{i} + y(t)\boldsymbol{j} + z(t)\boldsymbol{k}$ and the scalar field $\phi = \phi(x,y,z)$，show that the derivative $\frac{d\phi(x(t),y(t),z(t))}{dt}$ of $\phi$ along the curve is equal to $\frac{d\boldsymbol{r}}{dt}\cdot\nabla \phi$.

8.

Prove that the derivative of $\phi(t)$ which is composed by putting $x = x(t), y = y(t), z = z(t)$ into $\phi(x,y,z,t)$ is $\frac{d\phi}{dt} = \frac{\partial \phi}{\partial t} + \frac{d\boldsymbol{r}}{dt} \cdot\nabla \phi$．provided， $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z \boldsymbol{k}$.

9.

Find the divergence of $${\boldsymbol{F} = 3xy\:\boldsymbol{i} + x^{2}y\:\boldsymbol{j} + y^{2}z\:\boldsymbol{k}}$$．

10. For $\phi = 3x^2 - yz$, $\boldsymbol{F} = 3xyz^2\:\boldsymbol{i} + 2xy^3\:\boldsymbol{j} -x^2yz\:\boldsymbol{k}$，find the following scalar.

(1)

$\nabla \cdot\boldsymbol{F}$

(2)

$\boldsymbol{F} \cdot\nabla \phi$

(3)

$\nabla \cdot(\phi \boldsymbol{F})$

11. Let $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k},\ r = \vert\boldsymbol{r}\vert$．Prove the following:

(1)

$\nabla \cdot\boldsymbol{r} = 3$,

(2)

$\nabla \cdot(r^{n}\boldsymbol{r}) = (n+3)r^{n}$

(3)

$\nabla \cdot(\frac{\boldsymbol{r}}{r^3}) = 0$

12.

Find $\phi = 3x^2 y - y^3z^2$のとき， $\nabla^2 \phi$．