Stokes'theorem

Exercise4.2

1.

Prove the following equation holds for any surface $S$ and its boundary $C$ in the common domain of the scalar fields $\phi, \psi$.．

$$\int_{C}\phi(\nabla \psi)\cdot d\boldsymbol{r} = -\int_{C}\psi(\nabla \phi)\cdot d\boldsymbol{r}$$

2.

Let $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k},\ r = \vert\boldsymbol{r}\vert$ and $\phi$ be a scalar field. Then prove the following equation holds for any surface $S$ and its boundary $C$．

(1)

$\int_{C}\boldsymbol{r} \times d\boldsymbol{r} = 2\int_{S}\boldsymbol{n}dS$

(2)

$\int_{C}r^{k}\boldsymbol{r} \cdot d\boldsymbol{r} = 0$

(3)

$\int_{C}\boldsymbol{r} (\nabla \phi) \cdot d\boldsymbol{r} = \int_{S}(\nabla \phi) \times \boldsymbol{n}dS$

3.

Prove the following holds for any surface $S$ and its boundary $C$ in the common domain of the scalar field $\phi$ and the vector field ${\bf A}$.

$$\int_{S}\phi(\nabla \times {\bf A})\cdot\boldsymbol{n} dS = \int_... ...d\boldsymbol{r} - \int_{S}\{(\nabla \phi)\times {\bf A}\} \cdot\boldsymbol{n}dS$$

4.

Let $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z\boldsymbol{k},\ r = \vert\boldsymbol{r}\vert$ and $\phi$ be a scalar field. Prove that for any surface $S$ and its boundary $C$, the following equation holds.

$$\int_{C}\frac{\boldsymbol{r} \times d\boldsymbol{r}}{r^3} = -\int... ...}\cdot\boldsymbol{n}}{r^5}\boldsymbol{r} - \frac{\boldsymbol{n}}{r^3} \right)dS$$

5.

It is assumed that the vector field ${\bf A}$ is defined in the whole space.．About the border $C$ of any curved surface if

$$\int_{C}{\bf A} \cdot d\boldsymbol{r} = 0$$

${\bf A}$ has a scalar potential. Prove this.

6.

For $${\boldsymbol{F} = -y\boldsymbol{i} + x\boldsymbol{j} + \boldsymbol{k}, \ S: z = (4 - x^2 - y^2)^{1/2}}$$，show that Stokes' theorem holds.

7.

Find $${\int_{C}((2x+yz)\boldsymbol{i} + zx\boldsymbol{j} + xy\boldsymbol{k}) \cdot d\boldsymbol{r}}$$．where $C$ is a curve connecting from the point $(1,0,-1)$ to $(2,-1,3)$ and back to the starting point.．

8.

For any surface $S$ and its boundary $C$ within the common domain of $\phi, \psi$, prove the following equation holds.

$$\iint_{S}\{(\nabla \phi) \times (\nabla \psi)\}\cdot\boldsymbol{n}dS = \int_{C}\phi(\nabla \psi)\cdot d\boldsymbol{r}$$

9. Let $\boldsymbol{r} = x\boldsymbol{i} + y\boldsymbol{j} + z \boldsymbol{k}$．Prove for any surface $S$ and its boundary $C$, the following is true.

(1)

$\int_{C}d\boldsymbol{r} = {\bf0}$

(2)

$\int_{C}\boldsymbol{r}\cdot d\boldsymbol{r} = 0$