I've encountered a really hard question as follows:
I've tried to find the divergence of the vector field, but this gives me only one equation for three unknowns.
How would I solve this question?
Thanks in advance.
The field is conservative if curl F is zero.
The first two components turn out to be zero, the third is 2Bx - Ax and will be zero if A = 2B.
For the gradient, integrate the first compont of F partially wrt x, the second partially wrt y and the third partially wrt z.
The results are Ax^2y/2 +f(y,z), Bx^2y + Cy^/2 - 3yz^2 + g(x,z) and -3yz^2 - z^2 + h(x,y), respectively.
All that you need to do after that is to deduce expressions for f(y,z), g(x,z) and h(x,y) so that the three are identical.