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### Section 6-6 : Divergence Theorem

1. Use the Divergence Theorem to evaluate $$\displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}$$ where $$\vec F = \left( {3x - z{x^2}} \right)\,\vec i + \left( {{x^3} - 1} \right)\,\vec j + \left( {4{y^2} + {x^2}{z^2}} \right)\,\vec k$$ and $$S$$ is the surface of the box with $$0 \le x \le 1$$, $$- 3 \le y \le 0$$ and $$- 2 \le z \le 1$$. Note that all six sides of the box are included in $$S$$.
2. Use the Divergence Theorem to evaluate $$\displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}$$ where $$\vec F = 4x\,\vec i + \left( {1 - 6y} \right)\,\vec j + {z^3}\,\vec k$$ and $$S$$ is the surface of the sphere of radius 2 with $$z \ge 0$$, $$y \le 0$$ and $$x \ge 0$$. Note that all four surfaces of this solid are included in $$S$$.
3. Use the Divergence Theorem to evaluate $$\displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}$$ where $$\vec F = - xy\,\vec i + \left( {z - 1} \right)\,\vec j + {z^3}\,\vec k$$ and $$S$$ is the surface of the solid bounded by $$y = 4{x^2} + 4{z^2} - 1$$ and the plane $$y = 7$$. Note that both of the surfaces of this solid included in $$S$$.
4. Use the Divergence Theorem to evaluate $$\displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}$$ where $$\vec F = \left( {4x - {z^2}} \right)\,\vec i + \left( {x + 3z} \right)\,\vec j + \left( {6 - z} \right)\,\vec k$$ and $$S$$ is the surface of the solid bounded by the cylinder $${x^2} + {y^2} = 36$$ and the planes $$z = - 2$$ and $$z = 3$$ . Note that both of the surfaces of this solid included in $$S$$.