### Review Exercises

*True or False?* Justify your answer with a proof or a counterexample.

For vector field $\text{F}(x,y)=P(x,y)\text{i}+Q(x,y)\text{j},$ if ${P}_{y}(x,y)={Q}_{x}(x,y)$ in open region $D,$ then ${\int}_{\partial D}Pdx+Qdy=0}.$

If $\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}=0,$ then $\text{F}$ is a conservative vector field.

Draw the following vector fields.

$\text{F}\left(x,y\right)=\sqrt{\frac{y\text{i}+3x\text{j}}{{x}^{2}+{y}^{2}}}$

Are the following the vector fields conservative? If so, find the potential function $f$ such that $\text{F}=\nabla f.$

$\text{F}\left(x,y\right)=\left(6xy\right)\text{i}+\left(3{x}^{2}-y{e}^{y}\right)\text{j}$

$\text{F}\left(x,y,z\right)=\left(2xy+{z}^{2}\right)\text{i}+\left({x}^{2}+2yz\right)\text{j}+\left(2xz+{y}^{2}\right)\text{k}$

$\text{F}(x,y,z)=\left({e}^{x}y\right)\text{i}+\left({e}^{x}+z\right)\text{j}+\left({e}^{x}+{y}^{2}\right)\text{k}$

Evaluate the following integrals.

$\underset{C}{\int}{x}^{2}dy+\left(2x-3xy\right)dx,$ along $C:y=\frac{1}{2}x$ from (0, 0) to (4, 2)

$\underset{C}{\int}ydx+x{y}^{2}dy,$ where $C:x=\sqrt{t},y=t-1,0\le t\le 1$

Find the divergence and curl for the following vector fields.

$\text{F}(x,y,z)=3xyz\text{i}+xy{e}^{z}\text{j}-3xy\text{k}$

Use Green’s theorem to evaluate the following integrals.

$\underset{C}{\int}3xydx+2x{y}^{2}dy,$ where *C* is a square with vertices (0, 0), (0, 2), (2, 2) and (2, 0)

${\oint}_{C}3ydx+\left(x+{e}^{y}\right)dy,$ where *C* is a circle centered at the origin with radius 3

Use Stokes’ theorem to evaluate $\int {\displaystyle {\int}_{S}\text{curl}\phantom{\rule{0.2em}{0ex}}\text{F}\xb7dS}}.$

$\text{F}(x,y,z)=y\text{i}-x\text{j}+z\text{k},$ where $S$ is the upper half of the unit sphere

$\text{F}(x,y,z)=y\text{i}+xyz\text{j}-2zx\text{k},$ where $S$ is the upward-facing paraboloid $z={x}^{2}+{y}^{2}$ lying in cylinder ${x}^{2}+{y}^{2}=1$

Use the divergence theorem to evaluate $\int {\displaystyle {\int}_{S}\text{F}\xb7dS}}.$

$\text{F}(x,y,z)=\left({x}^{3}y\right)\text{i}+\left(3y-{e}^{x}\right)\text{j}+\left(z+x\right)\text{k},$ over cube $S$ defined by $\mathrm{-1}\le x\le 1,$ $0\le y\le 2,$ $0\le z\le 2$

$\text{F}(x,y,z)=\left(2xy\right)\text{i}+\left(\text{\u2212}{y}^{2}\right)\text{j}+\left(2{z}^{3}\right)\text{k},$ where $S$ is bounded by paraboloid $z={x}^{2}+{y}^{2}$ and plane $z=2$

Find the amount of work performed by a 50-kg woman ascending a helical staircase with radius 2 m and height 100 m. The woman completes five revolutions during the climb.

Find the total mass of a thin wire in the shape of a semicircle with radius $\sqrt{2,}$ and a density function of $\rho \left(x,y\right)=y+{x}^{2}.$

Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for $z\ge 0$ with a density function $\rho \left(x,y,z\right)=x+y+z.$

Use the divergence theorem to compute the value of the flux integral over the unit sphere with $\text{F}(x,y,z)=3z\text{i}+2y\text{j}+2x\text{k}.$