Chapter 16.9, Problem 1PT

To determine

To choose: The appropriate option that will complete the statement “Let $F\left(x,y,z\right)=Pi+Qj+Rk$ have continuous first partial derivatives and E be a simple solid with boundary surface S that has a positive orientation. By the Divergence theorem, ${\displaystyle \underset{S}{\iint }F\cdot dS}=$”.

Answer to Problem 1PT

The correct option is $\underset{\_}{\text{c)}}$.

Explanation of Solution

Theorem used:

Divergence theorem:

Let E be a simple solid whose boundary surface S has positive orientation. Let $F\left(x,y,z\right)=Pi+Qj+Rk$, where P,Q and R are component functions with continuous first partial derivatives on an open region containing E. Then ${\displaystyle \underset{S}{\iint }F\cdot dS}={\displaystyle \underset{E}{\iiint }divFdV}$

Calculation:

Given that, $F\left(x,y,z\right)=Pi+Qj+Rk$ have continuous first partial derivatives and E be a simple solid with boundary surface S that has a positive orientation.

From divergence theorem, it can be written that,

${\displaystyle \underset{S}{\iint }F\cdot dS}={\displaystyle \underset{E}{\iiint }divFdV}$

Thus, the value of the integral ${\displaystyle \underset{S}{\iint }F\cdot dS}$ is ${\displaystyle \underset{E}{\iiint }divFdV}$.

Therefore, the correct option is $\underset{\_}{\text{c)}}$.