Calculate the rotational and divergent from each field below and say which are conservative.
a) $\displaystyle F(x,y,z)=(x+y,y-x,xz)$
b) $\displaystyle F(x,y,z)=(x-2z,x+y+z,x-2y)$
c) $\displaystyle F(x,y,z)=(e^xsiny,e^xcosy,z)$
Just use the formulas!
Let $\displaystyle \nabla = \tfrac{\partial}{\partial x}\bold{i} + \tfrac{\partial}{\partial y}\bold{j} + \tfrac{\partial }{\partial z}\bold{k}$.
Then, $\displaystyle \text{div} (\bold{F}) = \nabla \cdot \bold{F}$ and $\displaystyle \text{curl}(\bold{F}) = \nabla \times \bold{F}$.
It is conservative if it a fan of Pat Robertson, just joking, it is conservative if the curl is the vector zero.