Question

Determine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y, z) = 6xy2z2i + 4x2yz3j + 6x2y2z2k

Answer 1

A conservative vector field $\mathbf f$ has curl $\nabla\times\mathbf f=\mathbf0$. In this case,

$\nabla\times\mathbf f=12xy^2z(1-z)\,\mathbf j+4yz^2(2z-3x)\,\mathbf k\neq\mathbf 0$

so the vector field is not conservative.