Vector field

Exercise3.2

1.

For the scalar field $\phi = x^2 z + e^{y/x},\ \psi = 2z^2 y - xy^2$, find the followings:

(1)

$\nabla \phi, \ \nabla \psi$

(2)

the value $\nabla (\phi\psi)_{P}$ of $\nabla(\phi \psi)$ at P$(1,0,-2)$.

2.

Find the directional derivative in the direction of ${\bf u} = \frac{1}{7}(2\boldsymbol{i} - 3\boldsymbol{j} + 6\boldsymbol{k})$ of $\phi = 4xz^3 - 3x^2yz$ at P$(2,-1,2)$.

3.

Let $\boldsymbol{r}= x\:\boldsymbol{i} + y\:\boldsymbol{j} + z\:\boldsymbol{k}$， $r= \vert\boldsymbol{r}\vert$. Then show that the vector field $\boldsymbol{F} = \frac{\boldsymbol{r}}{r^3}$ has the scalar potential $\phi = \frac{1}{r}$ and find the potential energy at each point in space.

4.

Suppose the force field $\boldsymbol{F}$ has the potential $U$. Prove that the following equation holds when a mass point with mass $m$ moves in this force field and moves from point A to point B.

$$\frac{1}{2}mv_{A}^2 + U(A) = \frac{1}{2}mv_{B}^2 + U(B)$$

Here， $v_{A},v_{B}$ are magnitude of the velocity vectors at $A,B$, respectively．

5.

For any scalar fields $\phi, \psi$, show the followings.

$$\nabla \left(\frac{\phi}{\psi}\right) = \frac{\psi \nabla \phi - \phi \nabla \psi}{\phi^2}$$

6.

Let the distance of P$(x,y,z)$, Q $(\xi, \eta, \zeta)$ be $r$. Then prove the following for the differential operator

$$\nabla_{P} = \boldsymbol{i}\frac{\partial }{\partial x} + \boldsy... ...}\frac{\partial}{\partial \eta} + \boldsymbol{k}\frac{\partial}{\partial \zeta}$$

$${(1)\ \nabla_{Q}r = - \nabla_{P} r\hskip 3cm (2)\ \nabla_{Q}(\frac{1}{r}) = - \nabla_{P}(\frac{1}{r})}$$