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Use the divergence theorem to compute the value of flux integral ${\iint }_{S}\text{F}\cdot d\text{S},$ where $\text{F}(x,y,z)=({y}^{3}+3x)\text{i}+(xz+y)\text{j}+[z+{x}^{4}\cos ({x}^{2}y)]\text{k}$ and S is the surface of the solid bounded by ${x}^{2}+{y}^{2}=1,x\ge 0,y\ge 0,\,\text{and}\,0\le z\le 1.$

Evaluate ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)={x}^{2}\text{i}+xy\text{j}+{x}^{3}{y}^{3}\text{k}$ and S is the surface consisting of all faces of the tetrahedron bounded by plane $x+y+z=1$ and the coordinate planes, with outward unit normal vector N.

Let E be the solid bounded by the xy-plane and paraboloid $z=4-{x}^{2}-{y}^{2}$ so that S is the surface of the paraboloid piece together with the disk in the xy-plane that forms its bottom. If $\text{F}(x,y,z)=(xz\,\sin (yz)+{x}^{3})\text{i}+\cos (yz)\text{j}+(3z{y}^{2}-{e}^{{x}^{2}+{y}^{2}})\text{k},$ find ${\iint }_{S}\text{F}\cdot d\text{S}$ using the divergence 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 $C$; $\vec F = \langle -y,x,1\rangle$.

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 that follows the triangle with vertices at $(0,0,2)$, $(4,0,0)$ and $(0,3,0)$, traversing the the vertices in that order and returning to $(0,0,2)$, and $\mathcal{S}$ is the portion of the plane $z=2-x/2-2y/3$ enclosed by $C$; $\vec F = \langle y,-z,y\rangle$.

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 whose $x$ and $y$ coordinates follow the parabola $y=1-x^2$ from $x=1$ to $x=-1$, then follow the line from $(-1,0)$ back to $(1,0)$, where the $z$ coordinates of $C$ are determined by $f(x,y) = 2x^2+y^2$, and $\mathcal{S}$ is the portion of $z=2x^2+y^2$ enclosed by $C$; $\vec F = \langle y^2+z,x,x^2-y\rangle$.

In Exercises 13–16, a closed surface $\mathcal{S}$ and a vector field $\vec F$ are given. Find the outward flux of $\vec F$ over $\mathcal{S}$ either through direct computation or through the Divergence Theorem.

$\mathcal{S}$ is the surface formed by the intersections of $z=0$ and $z=(x^2-1)(y^2-1)$; $\vec F = \langle x^2+1,yz,xz^2\rangle$.

In Exercises 13–16, a closed surface $\mathcal{S}$ and a vector field $\vec F$ are given. Find the outward flux of $\vec F$ over $\mathcal{S}$ either through direct computation or through the Divergence Theorem.

$\mathcal{S}$ is the surface formed by the intersections of the planes $z=\frac12(3-x)$, $x=1$, $y=0$, $y=2$ and $z=0$; $\vec F = \langle x,y^2,z\rangle$.

In Exercises 13–16, a closed surface $\mathcal{S}$ and a vector field $\vec F$ are given. Find the outward flux of $\vec F$ over $\mathcal{S}$ either through direct computation or through the Divergence Theorem.

$\mathcal{S}$ is the surface formed by the intersections of the planes $z=2y$, $y=4-x^2$ and $z=0$; $\vec F = \langle xz,0,xz\rangle$.

In Exercises 13–16, a closed surface $\mathcal{S}$ and a vector field $\vec F$ are given. Find the outward flux of $\vec F$ over $\mathcal{S}$ either through direct computation or through the Divergence Theorem.

$\mathcal{S}$ is the surface formed by the intersections of the cylinder $z=1-x^2$ and the planes $y=-2$, $y=2$ and $z=0$; $\vec F = \langle 0,y^3,0\rangle$.

In Exercises 17–20, a closed curve $C$ that is the boundary of a surface $\mathcal{S}$ is given along with a vector field $\vec F$. Find the circulation of $\vec F$ around $C$ either through direct computation or through Stokes' Theorem.

$C$ is the curve whose $x$- and $y$-values are determined by the three sides of a triangle with vertices at $(-1,0)$, $(1,0)$ and $(0,1)$, traversed in that order, and the $z$-values are determined by the function $z=xy$; $\vec F = \langle z-y^2,x,z\rangle$.

In Exercises 17–20, a closed curve $C$ that is the boundary of a surface $\mathcal{S}$ is given along with a vector field $\vec F$. Find the circulation of $\vec F$ around $C$ either through direct computation or through Stokes' Theorem.

$C$ is the curve whose $x$- and $y$-values are given by $\vec r(t) = \langle 2\cos t,2\sin t\rangle$ and the $z$-values are determined by the function $z=x^2+y^3-3y+1$; $\vec F = \langle -y,x,z\rangle$.

In Exercises 17–20, a closed curve $C$ that is the boundary of a surface $\mathcal{S}$ is given along with a vector field $\vec F$. Find the circulation of $\vec F$ around $C$ either through direct computation or through Stokes' Theorem.

$C$ is the curve whose $x$- and $y$-values are given by $\vec r(t) = \langle \cos t,3\sin t\rangle$ and the $z$-values are determined by the function $z=5-2x-y$; $\vec F = \langle -\frac13y,3x,\frac23y-3x\rangle$.

In Exercises 17–20, a closed curve $C$ that is the boundary of a surface $\mathcal{S}$ is given along with a vector field $\vec F$. Find the circulation of $\vec F$ around $C$ either through direct computation or through Stokes' Theorem.

$C$ is the curve whose $x$- and $y$-values are sides of the square with vertices at $(1,1)$, $(-1,1)$, $(-1,-1)$ and $(1,-1)$, traversed in that order, and the $z$-values are determined by the function $z=10-5x-2y$; $\vec F = \langle 5y^2,2y^2,y^2\rangle$.