Let $F=(y{e}^x+siny)i(e^x+xcosy)j.$

$($a$)$ Verify that $F$ is conservative.

$($b$)$ Determine a potential $$ for $F.$

Let $F=(2x+y,x).$

$($a$)$ Verify that $F$ is conservative, and find a potential $$ for $F.$

$($b$)$ Calculate ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F$dr$,$ where $C$ is the straight line path from $(1,0)$ to $(0,-1).$

$($c$)$ Calculate ${\displaystyle \underset{C_2}{\overset{\phantom{}}{}}}F$dr$,$ where $C$ is the path from $(1,0)$ to $(0,-1)$ counterclockwise along the circle $x^2+y^2=1.$

$($d$)$ Calculate $(0,-1)-(1,0),$ and compare it to your answers from parts $($b$)$ and $($c$).$ What do you notice?

Let $F=(yz,xz,xy2z),$

$($a$)$ Verify that $F$ is conservative.

$($b$)$ Verify that $(x,y,z)=xyz+z^2$ is a potential for $F.$

$($c$)$ Evaluate the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F$dr where $C$ is the line memery from $(1,0,1)$ to $(0,-1,2).$

$($d$)$ Calculate $(0,-1,2)-$

$(1,0,1),$ and compare this to your answer to part $($c$).$

Corsider the wector field

$F(x,y,z)=y^{2}i+(2xy+e^{3x})j+3y{e}^{3x}k$

$($a$)$ Demonstrate that $F$ is conservative.

$($b$)$ Find a potential $$ for $F.$