Section 17.6 : Divergence Theorem

1. Use the Divergence Theorem to evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}$ where $\vec F = y{x^2}\,\vec i + \left( {x{y^2} - 3{z^4}} \right)\,\vec j + \left( {{x^3} + {y^2}} \right)\,\vec k$ and $S$ is the surface of the sphere of radius 4 with $z \le 0$ and $y \le 0$. Note that all three surfaces of this solid are included in $S$. Solution
2. Use the Divergence Theorem to evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}$ where $\vec F = \sin \left( {\pi x} \right)\,\vec i + z{y^3}\,\vec j + \left( {{z^2} + 4x} \right)\,\vec k$ and $S$ is the surface of the box with $- 1 \le x \le 2$, $0 \le y \le 1$ and $1 \le z \le 4$. Note that all six sides of the box are included in $S$. Solution
3. Use the Divergence Theorem to evaluate $\displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}$ where $\vec F = 2xz\vec i + \left( {1 - 4x{y^2}} \right)\,\vec j + \left( {2z - {z^2}} \right)\,\vec k$ and $S$ is the surface of the solid bounded by $z = 6 - 2{x^2} - 2{y^2}$ and the plane $z = 0$ . Note that both of the surfaces of this solid included in $S$. Solution