Surface integral

Exercise3.4

1.

Let A,B,C be the intersection of the plane and -axis，-axis, and -axis, and the surface 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 -plane be ．Find the surface integral．

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

3.

Find the following surface integrals．

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

4

Find the surface integral．

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

5.

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

6.

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

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

7.

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

8.

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

9.

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

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

10.

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

11.

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

12.