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For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral ${\iint }_{S}^{}\text{F}\cdot \text{N}ds$ for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector.

[T] $\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k};$ S is the surface of cube $0\le x\le 1,0\le y\le 1,0<z\le 1.$

[T] $\text{F}(x,y,z)=(\cos \,yz)\text{i}+{e}^{xz}\text{j}+3{z}^{2}\text{k};$ S is the surface of hemisphere $z=\sqrt{4-{x}^{2}-{y}^{2}}$ together with disk ${x}^{2}+{y}^{2}\le 4$ in the xy-plane.

[T] $\text{F}(x,y,z)=({x}^{2}+{y}^{2}-{x}^{2})\text{i}+{x}^{2}y\text{j}+3z\text{k};S$ is the surface of the unit cube $0\le x\le 1,0\le y\le 1,0\le z\le 1$ excluding the face $z=0$.

[T] $\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k};$ S is the surface of the solid bounded by the parabola $z={x}^{2}+{y}^{2}\,$ and the plane $z=9$.

For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral ${\iint }_{S}^{}\text{F}\cdot \text{N}ds$ for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector.

[T] $\text{F}(x,y,z)={x}^{2}\text{i}+{y}^{2}\text{j}+{z}^{2}\text{k};$ S is the surface of sphere ${x}^{2}+{y}^{2}+{z}^{2}=4.$

[T] $\text{F}(x,y,z)=x\text{i}+y\text{j}+({z}^{2}-1)\text{k};$ S is the surface of the solid bounded by cylinder ${x}^{2}+{y}^{2}=4$ and planes $z=0\,\text{and}\,z=1.$

[T] $\text{F}(x,y,z)=x{y}^{2}\text{i}+y{z}^{2}\text{j}+{x}^{2}z\text{k};$ S is the surface of the solid bounded above by sphere $\rho =2$ and below by cone $\varphi =\frac{\pi }{4}$ in spherical coordinates. (Think of S as the surface of an “ice cream cone.”)

[T] $\text{F}(x,y,z)={x}^{3}\text{i}+{y}^{3}\text{j}+3{a}^{2}z\text{k}\,\text{(constant}\,a>0);$ S is the surface of the solid bounded by cylinder ${x}^{2}+{y}^{2}={a}^{2}$ and planes $z=0\,\text{and}\,z=1.$

[T] Surface integral ${\iint }_{S}\text{F}\cdot d\text{S},$ where S is the surface of the solid bounded by paraboloid $z={x}^{2}+{y}^{2}$ and plane $z=4,$ and $\text{F}(x,y,z)=(x+{y}^{2}{z}^{2})\text{i}+(y+{z}^{2}{x}^{2})\text{j}+(z+{x}^{2}{y}^{2})\text{k}$

Use the divergence theorem to calculate surface integral ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=({e}^{{y}^{2}})\text{i}+(y+\sin ({z}^{2}))\text{j}+(z-1)\text{k}$ and S is the surface of the solid bounded by the sphere ${x}^{2}+{y}^{2}+{z}^{2}=1,$ and below by the plane $z=0$.

Use the divergence theorem to calculate surface integral ${\iint }_{S}\text{F}\cdot ds,$ where $\text{F}(x,y,z)={x}^{4}\text{i}-{x}^{3}{z}^{2}\text{j}+4x{y}^{2}z\text{k}$ and $S$ is the surface bounded by cylinder ${x}^{2}+{y}^{2}=1$ and planes $z=x+2$ and $z=0.$

Use the divergence theorem to calculate surface integral ${\iint }_{S}\text{F}\cdot d\text{S}$ when $\text{F}(x,y,z)={x}^{2}{z}^{3}\text{i}+2xy{z}^{3}\text{j}+x{z}^{4}\text{k}$ and S is the surface of the box with vertices $(\pm 1,\pm 2,\pm 3).$

Use the divergence theorem to calculate surface integral ${\iint }_{S}\text{F}\cdot d\text{S}$ when $\text{F}(x,y,z)=z\,{\tan }^{-1}({y}^{2})\text{i}+{z}^{3}\ln ({x}^{2}+1)\text{j}+z\text{k}$ and S is the surface of the solid bounded by the paraboloid ${x}^{2}+{y}^{2}+z=2$ and the plane $z=1$.

[T] Use a CAS and the divergence theorem to calculate flux ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=({x}^{3}+{y}^{3})\text{i}+({y}^{3}+{z}^{3})\text{j}+({z}^{3}+{x}^{3})\text{k}$ and S is a sphere with center (0, 0, 0) and radius 2.

Use the divergence theorem to compute flux integral ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=y\text{j}-z\text{k}$ and S consists of the union of paraboloid $y={x}^{2}+{z}^{2},0\le y\le 1,$ and disk ${x}^{2}+{z}^{2}\le 1,y=1,$ oriented outward. What is the flux through just the paraboloid?

Use the divergence theorem to compute flux integral ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=x+y\text{j}+{z}^{4}\text{k}$ and S is a part of cone $z=\sqrt{{x}^{2}+{y}^{2}}$ beneath top plane $z=1,$ oriented downward.

Use the divergence theorem to calculate surface integral ${\iint }_{S}\text{F}\cdot d\text{S}$ for $\text{F}(x,y,z)={x}^{4}\text{i}-{x}^{3}{z}^{2}\text{j}+4x{y}^{2}z\text{k},$ where S is the surface inside the cylinder ${x}^{2}+{y}^{2}=1$ between the planes $z=x+2\,\text{and}\,z=0.$

Consider $\text{F}(x,y,z)={x}^{2}\text{i}+xy\text{j}+(z+1)\text{k}.$ Let E be the solid enclosed by paraboloid $z=4-{x}^{2}-{y}^{2}$ and plane $z=0$ with normal vectors pointing outside E. Compute flux F across the boundary of E using the divergence theorem.

For the following exercises, use a CAS along with the divergence theorem to compute the net outward flux for the fields across the given surfaces S.

[T] $\text{F}=\langle x,-2y,3z\rangle ;$ S is sphere $\{(x,y,z):{x}^{2}+{y}^{2}+{z}^{2}=6\}.$

[T] $\text{F}=\langle x,2y,z\rangle ;$ S is the boundary of the tetrahedron in the first octant formed by plane $x+y+z=1.$