Question

Determine whether 𝐹 is conservative vector field. If so, find a potential

function for it is

f(x,y,z)=x^yi+5xy^2j

Answer 1

$f\left( x,y,z \right) =x^yi+5xy^2j\\rot\left( f \right) =\left| \begin{matrix} i& j& k\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ x^y& xy^2& 0\\\end{matrix} \right|=\left( y^2-x^y\ln x \right) k\ne 0$

The field is not conservative.