91Ó°ÊÓ

In Exercises$21-24$, compute the divergence of the given vector field.

$\mathbf{F}(x,y,z)=x{e}^{xyz}\mathbf{i}+y{e}^{xyz}\mathbf{j}+z{e}^{xyz}\mathbf{k}$

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

$f(x,y)=x^2+y^2$, with C the unit circle traversed counterclockwise.

Find the outward flux of the given vector field through the specified surface.

$\mathbf{F}(x,y,z)=x^2\mathbf{i}+y^2\mathbf{j}+z^2\mathbf{k}$and S is the top half of the unit sphere centered at the origin.

In Exercises $21-24$, compute the divergence of the given vector field.

$\mathbf{F}(x,y,z)=\cos x\mathbf{i}-y\sin zy\mathbf{j}+\cos z\mathbf{k}$

${∫}_C\mathbf{F}(x,y,z)·d\mathbf{r}$, where $C$ is the boundary of the region in the plane $z=2x-y+10$ and that lies above the curves $x=1,x=2$, and $y=e^x$, and where

In Exercises 17–24, find a potential function for the given vector field.

$\mathbf{G}(x,y,z)=(y{e}^{xy+z},x{e}^{xy+z},e^{xy+z})$

$$\begin{gathered} ComputethecurlofthevectorfieldsinExercises23–28. \\ \mathrm{G}(\mathrm{x},\mathrm{y},\mathrm{z})=({\mathrm{x}}^2+\mathrm{y}−\mathrm{z})\mathrm{i}−2\mathrm{y}\cos \mathrm{zj}+{\mathrm{e}}^{{\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2}\mathrm{k} \end{gathered}$$

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

$f(x,y)=x^2+y^2$, with C the unit circle traversed clockwise.

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

Sis the portion of the surface determined by $x=9-y^2-z^2$ that lies on the positive side of the yzplane (i.e., where $xâ‰\yen 0$)

Find the divergence and curl of the following vector fields.

$F\left(x,y\right)=\left(3x+4y\right)i+\left(x-5y\right)j$