1) How can I know if the field is gradient?
2) How can I know if the fiels is conservative ?
3) How can I find the function potential of vector field ?
Since you only have a two dimensional vector field, say
$\displaystyle
{\bf F} = <P,Q>
$
then this would be conservative if $\displaystyle P_y = Q_x$ and if $\displaystyle P = e^x \cos y$ and $\displaystyle Q = - e^x \sin y$ then $\displaystyle P_y = -e^x \sin y$ and $\displaystyle Q_x = -e^x \sin y$ which are identical. So the vector field is conservative and an $\displaystyle f $exists such that
$\displaystyle f_x = P = e^x \cos y$ and $\displaystyle f_y = Q = - e^x\sin y$ which gives $\displaystyle f = e^x \cos y + c$
First since the vector field is conservative you know an $\displaystyle f$ exists, i.e. $\displaystyle f$ exists such that $\displaystyle f_x = P = e^x \cos y \;{\bf and} \;f_y = Q = - e^x\sin y$
If $\displaystyle f_x = e^x \cos y$ then integrating wrt to $\displaystyle x$ gives $\displaystyle f = e^x \cos y + g(y)$. Now subs. into $\displaystyle f_y = - e^x\sin y$ which shows that $\displaystyle - e^x\sin y + g'(y) = - e^x\sin y$ which gives $\displaystyle g'(y) = 0$ which gives $\displaystyle g(y)=c$, thus our $\displaystyle f.$