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 ${\displaystyle {\int }_{C}Pdx+Qdy=0}.$

If $\text{curl}\text{F}=0,$ then $\text{F}$ is a conservative vector field.

$\mathbf{\text{F}}(x,y)=\frac{<\mathit{y}\mathrm{,}\mathrm{3}\mathit{x}>}{\sqrt{x^2+y^2}}$

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

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

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

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

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

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

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

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. Use 9.8 m/s² as the acceleration due to gravity.

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 $\rho \left(x,y,z\right)=x+y+z.$