91Ó°ÊÓ

Sketch the vector fields in Exercises 25–32.

$\mathbf{F}(x,y)=2\mathbf{i}+0\mathbf{j}$

${âˆ\neg }_{S}curl\mathbf{F}(x,y,z)·\mathbf{n}dS$, where $S$ is the cap of the unit sphere that lies below the$xy$-plane and inside the cylinder$x^2+y^2=\frac{1}{9}$with outwards-pointing normal vector and whererole="math" localid="1650736891824" $\mathbf{F}(x,y,z)=-y{z}^2\mathbf{i}+x{z}^2\mathbf{j}+3^{-xyz}\mathbf{k}$.

$$\begin{gathered} \mathrm{Compute}\mathrm{the}\mathrm{curl}\mathrm{of}\mathrm{the}\mathrm{vector}\mathrm{fields}\mathrm{in}\mathrm{Exercises}23–28. \\ \mathrm{F}(\mathrm{x},\mathrm{y},\mathrm{z})={\mathrm{xe}}^{\mathrm{yz}}\mathrm{i}+{\mathrm{ye}}^{\mathrm{xz}}\mathrm{j}+{\mathrm{ze}}^{\mathrm{xy}}\mathrm{k} \end{gathered}$$

In Exercises 21–28, evaluate the multivariate line integral of the given function over the specified curve.

$f(x,y,z)=e^{x+y+z}$, with C the straight line segment from the origin to (1,2,3).

In Exercises $25-40$, evaluate the integral

${âˆ\neg }_S\mathbf{F}(x,y,z)·\mathbf{n}dS$

for the specified function $\mathbf{F}(x,y,z)$and the given surface $S$. In each integral, $\mathbf{n}$is the outwards-pointing normal vector.

$\mathbf{F}(x,y,z)=x{y}^2\mathbf{i}+y(z-3x)\mathbf{j}+4xyz\mathbf{k}$, and $S$ is the surface of the region $W$ bounded by the planes $y=0,y=z,z=3,x=0$, and $x=4$.

Find the areas of the given surfaces in Exercises 21–26.

S is the portion of the surface parametrized by $r(u,v)=(3u-v,v+u,v-u)$ whose preimage (the domain in the uv-plane) is the unit square $[0,1]×[0,1]$

Find the divergence and curl of the following vector fields.

$F\left(x,y\right)=x^{2}yi+x{y}^{3}j$

In Exercises 21–28, evaluate the multivariate line integral of the given function over the specified curve.

$g(x,y,z)=xyz$, with C the curve parameterized by $\mathbf{r}\left(t\right)=\left(\frac{\sqrt{2}}{3}{t}^3,t^2,t\right)\mathrm{for}1≤t≤4.$

${âˆ\neg }_{S}curl\mathbf{F}(x,y,z)·\mathbf{n}d\mathbf{S}$, where S is the portion of the hyperbolic paraboloid $z=x^2-y^2$ that lies inside the elliptical cylinder $4x^2+9y^2=36$ with upwards-pointing normal vector and$\mathbf{F}(x,y,z)=(1-yz\sin (xyz\left)\right)\mathbf{i}-$ $(1+xz\sin (xyz\left)\right)\mathbf{j}+(1-xy\sin (xyz\left)\right)\mathbf{k}$.

Compute the curl of the vector fields:

$F\left(x,y\right)=-4x^{2}yi+4x{y}^{2}j$.