91Ó°ÊÓ

Find the divergence and curl of the following vector fields.

$F\left(x,y,z\right)=\left(y^2-z^2\right)i+\left(z^2-x^2\right)j+\left(x^2-y^2\right)k$

Evaluate the multivariate line integral of the given function over the specified curve.

$f(x,y,z)=e^{x^2+y^2+z^2}$, with $C$the circular helix of radius $1$, centered about the $y$-axis, and parametrized by$r\left(t\right)=(\cos t,t,\sin t)$for$\mathrm{π}≤y≤3\mathrm{π}$.

Compute the curl of the vector fields:

$G\left(x,y,z\right)=\left(2y+3z\right)i+2x{y}^{2}j+\left(zx-y\right)k$.

${∫}_C\mathbf{F}(x,y,z)×d\mathbf{r}$, where C is the curve on the paraboloid $z=x^2+y^2$that lies above the unit circle, traversed counterclockwise with respect to the outwards-pointing normal vector, and where

$\mathbf{F}(x,y,z)=(3x+y−z)\mathbf{i}+(4y−2z)\mathbf{j}+(x−3z)\mathbf{k}$

Integrate the given function over the accompanying surface in Exercises 27–34.
$f(x,y,z)=x^2-y+3z$, where Sis the portion of the plane with equation $2x-6y+3z=1$whose preimage in the xz plane is the region bounded by the coordinate axes and the lines with equations z = 4 and x = z.

$\mathbf{F}(x,y,z)=x^3\mathbf{i}+y^3\mathbf{j}+z^3\mathbf{k}$, and $S$is the sphere of radius 3 and centered at the origin.

Sketch the vector fields in Exercises 25–32.

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

Evaluate each of the vector field line integrals over the indicated curves.

$\mathbf{F}(x,y)=\mathbf{i}-\mathbf{j}$ with $C$ the curve with equation$y-x^2=1$for$5≤x≤10$.

Integrate the given function over the accompanying surface in Exercises 27–34.
$f(x,y,z)=xy{z}^2$, where S is the portion of the cone $\sqrt{3}z=\sqrt{x^2+y^2}$ that lies within the sphere of radius 4 and centered at the origin.

$\mathbf{F}(x,y,z)=15x{z}^2\mathbf{i}+15y{x}^2\mathbf{j}+15y^{2}z\mathbf{k}$, and $s$ is the surface of the lower half of the unit sphere, along with the unit circle in the plane.