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[T] Use a CAS to find the flux of vector field $\text{F}(x,y,z)=z\text{i}+z\text{j}+\sqrt{{x}^{2}+{y}^{2}}\text{k}$ across the portion of hyperboloid ${x}^{2}+{y}^{2}={z}^{2}+1$ between planes $z=0$ and $z=\frac{\sqrt{3}}{3},$ oriented so the unit normal vector points away from the z-axis.

[T] Use a CAS to find the flux of vector field $\text{F}(x,y,z)=({e}^{y}+x)\text{i}+(3\,\cos (xz)-y)\text{j}+z\text{k}$ through surface S, where S is given by ${z}^{2}=4{x}^{2}+4{y}^{2}$ from $0\le z\le 4,$ oriented so the unit normal vector points downward.

[T] Use a CAS to compute ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=x\text{i}+y\text{j}+2z\text{k}$ and S is the boundary of a part of solid sphere ${x}^{2}+{y}^{2}+{z}^{2}\le 2$ with $0\le z\le 1.$

Evaluate ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=bx{y}^{2}\text{i}+b{x}^{2}y\text{j}+({x}^{2}+{y}^{2}){z}^{2}\text{k}$ and S is the boundary of the solid cylinder ${x}^{2}+{y}^{2}\le {a}^{2}$ and $0\le z\le b.$

[T] Use a CAS to calculate the flux of $\text{F}(x,y,z)=({x}^{3}+y\,\sin \,z)\text{i}+({y}^{3}+z\,\sin \,x)\text{j}+3z\text{k}$ across surface S, where S is the boundary of the solid bounded by hemispheres $z=\sqrt{4-{x}^{2}-{y}^{2}}$ and $z=\sqrt{1-{x}^{2}-{y}^{2}},$ and plane $z=0.$

Use the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=xy\text{i}-\frac{1}{2}{y}^{2}\text{j}+z\text{k}$ and S is the surface consisting of three pieces: $z=4-3{x}^{2}-3{y}^{2},1\le z\le 4$ on the top; ${x}^{2}+{y}^{2}=1,0\le z\le 1$ on the sides; and $z=0$ on the bottom.

[T] Use a CAS and the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=(2x+y\,\cos \,z)\text{i}+({x}^{2}-y)\text{j}+{y}^{2}z\text{k}$ and S is sphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ orientated outward.

Use the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k}$ and S is the boundary of the solid enclosed by paraboloid $y={x}^{2}+{z}^{2}-2,$ cylinder ${x}^{2}+{z}^{2}=1,$ and plane $x+y=2,$ and S is oriented outward.

For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, $\text{F}=-k\nabla T,$ which means that heat energy flows hot regions to cold regions. The constant $k>0$ is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux ${\iint }_{S}\text{F}\cdot \text{N}dS=-k{\iint }_{S}\nabla T\cdot \text{N}dS$ across the boundary S of D, where $k=1.$

$T(x,y,z)=100+x+2y+z;$ $D=\{(x,y,z):0\le x\le 1,0\le y\le 1,0\le z\le 1\}$

For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, $\text{F}=-k\nabla T,$ which means that heat energy flows hot regions to cold regions. The constant $k>0$ is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux ${\iint }_{S}\text{F}\cdot \text{N}dS=-k{\iint }_{S}\nabla T\cdot \text{N}dS$ across the boundary S of D, where $k=1.$

$T(x,y,z)=100+{e}^{-z};$ $D=\{(x,y,z):0\le x\le 1,0\le y\le 1,0\le z\le 1\}$

$T(x,y,z)=100{e}^{-{x}^{2}-{y}^{2}-{z}^{2}};$ D is the sphere of radius a centered at the origin.

What property of a vector field does the Divergence Theorem relate to flux?

What property of a vector field does Stokes' Theorem relate to circulation?

Stokes' Theorem is the spatial version of what other theorem?

In Exercises 5–8, a closed surface $\mathcal{S}$ enclosing a domain $D$ and a vector field $\vec F$ are given. Verify the Divergence Theorem on $\mathcal{S}$; that is, show $\iint_\mathcal{S} \vec F\cdot \vec n\ dS = \iiint_D\operatorname{div} \vec F\ dV$.

$\mathcal{S}$ is the surface bounding the domain $D$ enclosed by the plane $z=2-x/2-2y/3$ and the coordinate planes in the first octant; $\vec F = \langle x^2,y^2,x\rangle$.

In Exercises 5–8, a closed surface $\mathcal{S}$ enclosing a domain $D$ and a vector field $\vec F$ are given. Verify the Divergence Theorem on $\mathcal{S}$; that is, show $\iint_\mathcal{S} \vec F\cdot \vec n\ dS = \iiint_D\operatorname{div} \vec F\ dV$.

$\mathcal{S}$ is the surface bounding the domain $D$ enclosed by the cylinder $x^2+y^2=1$ and the planes $z=-3$ and $z=3$; $\vec F = \langle -x,y,z\rangle$.

In Exercises 5–8, a closed surface $\mathcal{S}$ enclosing a domain $D$ and a vector field $\vec F$ are given. Verify the Divergence Theorem on $\mathcal{S}$; that is, show $\iint_\mathcal{S} \vec F\cdot \vec n\ dS = \iiint_D\operatorname{div} \vec F\ dV$.

$\mathcal{S}$ is the surface bounding the domain $D$ enclosed by $z=xy(3-x)(3-y)$ and the plane $z=0$; $\vec F = \langle 3x,4y,5z+1\rangle$.

In Exercises 5–8, a closed surface $\mathcal{S}$ enclosing a domain $D$ and a vector field $\vec F$ are given. Verify the Divergence Theorem on $\mathcal{S}$; that is, show $\iint_\mathcal{S} \vec F\cdot \vec n\ dS = \iiint_D\operatorname{div} \vec F\ dV$.

$\mathcal{S}$ is the surface composed of $\mathcal{S}_1$, the paraboloid $z=4-x^2-y^2$ for $z\geq 0$, and $\mathcal{S}_2$, the disk of radius 2 centered at the origin; $\vec F = \langle x,y,z^2\rangle$.

In Exercises 9–12, a closed curve $C$ that is the boundary of a surface $\mathcal{S}$ is given along with a vector field $\vec F$. Verify Stokes' Theorem on $C$; that is, show $\oint_C \vec F\cdot d\vec r = \iint_\mathcal{S}\big(\operatorname{curl} \vec F\,\big)\cdot\vec n\ dS$.

$C$ is the curve parametrized by $\vec r(t) = \langle \cos t, \sin t, 1\rangle$ and $\mathcal{S}$ is the portion of $z=x^2+y^2$ enclosed by $C$; $\vec F = \langle z,-x,y\rangle$.

In Exercises 9–12, a closed curve $C$ that is the boundary of a surface $\mathcal{S}$ is given along with a vector field $\vec F$. Verify Stokes' Theorem on $C$; that is, show $\oint_C \vec F\cdot d\vec r = \iint_\mathcal{S}\big(\operatorname{curl} \vec F\,\big)\cdot\vec n\ dS$.

$C$ is the curve parametrized by $\vec r(t) = \langle \cos t, \sin t, e^{-1}\rangle$ and $\mathcal{S}$ is the portion of $z=e^{-x^2-y^2}$ enclosed by $C$; $\vec F = \langle -y,x,1\rangle$.