If [tex]\nabla f(x,y,z)[/tex] is always parallel to [tex]x \hat i + y \hat j + z \hat k[/tex], show that f must assume equal values at the points (0,0,a) and (0,0,-a).
The Attempt at a Solution
I tried a number of things - inspecting the values arrived at when computing the cross product of the gradient and the position vector, writing [tex]r(t)=(0,0,t)[/tex] and then integrating [tex]\nabla f(r(t))\cdot r'(t)[/tex] from -a to a, nothing is getting me anywhere. I also found this old thread: https://www.physicsforums.com/showthread.php?t=165790 but I don't see what Mathgician is getting at. In particular, f(0,0,a)-f(0,0,-a) is not necessarily equal to -2ak as the poster mentions toward the bottom.