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.


 Apr 9, 2020

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.

 Apr 9, 2020
edited by Tiggsy  Apr 9, 2020

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