Gradient and directional derivative

Exercise3.1

1.

Show that the gradient through the point $${(x_{0},y_{0},z_{0})}$$ is orthogonal to the level surface through the point $${(x_{0},y_{0},z_{0})}$$.

2.

Find the unit normal vector orthogonal to the surface $${z^{2} = x^{2} + 2y^{2}}$$ at $(1,2,3)$ and find the directional derivative in the direction of $(1,3,-1)$ and the equation of tangent plane.

3.

Find the streamline of the vector field $${\boldsymbol{F}(x,y) = -y\:\boldsymbol{i} + x\:\boldsymbol{j}}$$．

4.

Let the position vector of ${\rm P}(x,y,z)$ be $\boldsymbol{r}= x\:\boldsymbol{i} + y\:\boldsymbol{j} + z\:\boldsymbol{k}$，and vector field be $${\boldsymbol{F} = -\frac{\boldsymbol{r}}{\vert\boldsymbol{r}\vert}}$$. Then show that this vector field is a conservative field in any area except at origin and

$$f(x,y,z) = \vert\boldsymbol{r}\vert = (x^{2}+y^{2}+z^{2})^{1/2}$$

is the scalar potential of $\boldsymbol{F}$.

5.

For the vector field $\boldsymbol{r}= x\:\boldsymbol{i} + y\:\boldsymbol{j} + z\:\boldsymbol{k}$ and the scalar field $r = \vert\boldsymbol{r}\vert = \sqrt{x^2 + y^2 + z^2}$, show the followings:

(1)

$\nabla r = \frac{\boldsymbol{r}}{r}$

(2)

$\nabla r^{n} = nr^{n-2}\boldsymbol{r}$

6.

Let $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k},\ r = \vert\boldsymbol{r}\vert$. Then find ${\bf A, B}$.．

(1)

$\boldsymbol{A} = \nabla\left(2r^2 - 4\sqrt{r} + \frac{6}{3\sqrt{r}}\right)$

(2)

$\boldsymbol{B} = \nabla(r^2 e^{-r})$

7.

Find the unit normal vector ${\bf n}$ of $x^2y + 2xz = 4$ at P$(2,-2,3)$．

8.

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

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