Surface integral

Exercise3.4

1.

Let A,B,C be the intersection of the plane $2x + 2y + z = 2$ and $x$-axis，$y$-axis, and $z$-axis, and the surface $S$ be the $\triangle$ ABC. Find the following surface integrals.

(1)

$\int_{S}fdS, \ f = x^2 + 2y + z -1$

(2)

$\int_{S}{\bf A}\cdot\boldsymbol{n}dS, \ {\bf A} = x^2 \boldsymbol{i} + z \boldsymbol{k}$

2.

Let the region $x \geq 0, y \geq 0, x^2 + y^2 \leq a^2$ on the $xy$-plane be $S$．Find the surface integral．

$$\int_{S}{\bf A} \times \boldsymbol{n}dS, \ {\bf A} = x\boldsymbol{i} + (x-y)\boldsymbol{j} + (\log{xy})\boldsymbol{k}$$

3.

Find the following surface integrals．

$$\iint_{S}(3x{\bf i} + 4z{\bf j} + 2y{\bf k}) \cdot {\bf n}dS, S:y... ...eq x \leq 3, y \geq 0, z \geq 0, ({\bf n}\mbox{の}z\mbox{component is positive}$$

4

Find the surface integral．

$$\iint_{S}(x{\bf i} + y{\bf j} - 2z{\bf k}) \cdot {\bf n}dS, S:x^2 + y^2 = a^2, 0 \leq z \leq 1$$

5.

Let the vector field be $\boldsymbol{F} = y\:\boldsymbol{j} + z\:\boldsymbol{k}$，surface be $${S : x^2 + y^2 = 4 - z , \ z \geq 0}$$．Find the surface integral of $${\iint_{S} \boldsymbol{F} \cdot\boldsymbol{n} dS }$$

6.

Find the surface integral of the scalar field $f$ on the paraboloid $${S : x^2 + y^2 + z = 4}$$ where $z \geq 0$.

$$f(x,y,z) = \frac{2y^2 + z}{(4x^2 + 4y^2 + 1)^{1/2}}$$

7.

Find $${\iint_{S}x y^2 dS}$$, where $S : x+y+z = 1, x \geq 0, y \geq 0, z \geq 0$．

8.

Let $A,B,C$ be the intersection of the plane $2x + 2y + z = 2$ and $x$-axis，$y$-axis, $z$-axis and let the $\triangle$ABC be the surface $S$. Find $\int_{S}fdS, \ f = x^2 + 2y + z -1$．

9.

Let $S$ be the sphere of the radius $a$ with the center Ｏ．Let $\boldsymbol{r} = \overrightarrow{\rm OP}$ be the position vector of $P$．Then by taking the unit normal vector $\boldsymbol{n}$ of the sperical surface $S$ outward, prove the followings.

$$\int_{S} \frac{\boldsymbol{r}}{r^3}\cdot\boldsymbol{n}\;dS = 4\pi,\ \ r = \vert\boldsymbol{r}\vert$$

10.

Let the vector field be $\boldsymbol{F} = x\:\boldsymbol{i} + y\:\boldsymbol{j} - 2z\:\boldsymbol{k}$ , and the surface be $${S : x^2 + y^2 = a^2 , \ 0 \leq z \leq 1}$$ ．Then find the surface integral of $${\iint_{S} \boldsymbol{F} \cdot\boldsymbol{n} dS }$$.

11.

$\boldsymbol{F} = (2x-z)\boldsymbol{i} + x^2 y\boldsymbol{j} -x^2 z\boldsymbol{k}$，Let the surface $S$ be a bounded part of the plane $x = 0, x = 1, y = 0, y = 1, z = 0, z = 1$．Then find the surface integral $\iint_{S} \boldsymbol{F}\cdot\boldsymbol{n}dS$.

12.

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

(1)

$\nabla^2 r^{n} = n(n+1)r^{n-2}$

(2)

$\nabla^2 (\frac{1}{r}) = 0$

13.

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

(1)

Prove that $${\nabla^2 f(r) = \frac{d^{2}f}{dr^2} + \frac{2}{r}\frac{df}{dr}}$$よ．

(2)

Find $f(r)$ so that $${\nabla^2 f(r) = 0}$$.．