I was reading this question and I'm a bit confused by how the potential is found. I'm finding it hard to understand why it is not -2kxy for a field $E=ky\hat{x}+kx\hat{y}$.
My thinking is as follows: first, let there be just a field $E=ky\hat{x}$
From the origin, we move along the y-axis to $(0,y)\Rightarrow$no work done (motion was perpendicular to the field). From $(0,y)$, we move to $(x,y)$ along a line parallel to the x-axis ie-along the field.
Now the potential is $-kxy$
Now, let there be just the field $E=kx\hat{y}$
Through the same reasoning as above (moving along the x-axis, and then parallel to the y-axis ), the potential at $(x,y)$ is $-kxy$
So when the vector field is the sum of the two individual fields ($E=ky\hat{x}+kx\hat{y}$), shouldn't the potential be the sum of the two individual potentials, and hence be $-2kxy$? The only problem is, this doesn't give the gradient it should.
I know I'm doing something wrong, but I can't find the error.
Some help?