Question

5. Let D be the first-octant portion of the unit ball, pictured below. D is bounded by four surfaces: Ary: blue quarter-disk in the xy-plane Azz: green quarter-disk in the x z-plane Ayz: red quarter-disk in the yz-plane As : transparent mesh eight-sphere Z y х Let F(x, y, z) = (0,0,z), and let all normal vectors point outward. (a) Explain why | Loud Ē. ñds = 0. Do not work out the integral.

Answer 1

Here,

$\vec F (x,y,z)=<0,0,z>$

(a) According to the Gauss divergence Theorem,

$\int \int_{S} (\vec F . \vec n) dS = \int \int \int_{V} (\Delta . \vec F ) dV$

here, $\int \int \int_{V} (\Delta . \vec F ) dV= \int \int \int_{V} \frac{\partial }{\partial z} (z) dV =\int \int \int_{V} dV =0$

Since, the volume of the given surface is 0.

(b) Now, The unit normal vector of the green region  $\mathcal{A}_{xz}$ is $\vec n=<0,1,0>$

Then, $\vec F . \vec n = <0,0,z> . <0,1,0>=0$

Hence,

$\int \int_{\mathcal{A}_{xz}} \vec F . \vec n dS=0$

(c)

Now, The unit normal vector of the green region  $\mathcal{A}_{yz}$ is $\vec n=<1,0,0>$

Then, $\vec F . \vec n = <0,0,z> . <1,0,0>=0$

Hence,

$\int \int_{\mathcal{A}_{yz}} \vec F . \vec n dS=0$