I've found lots of examples that show various functions that are harmonic but I still can't figure something out. Does a function have to hold true for all (x,y) to be considered harmonic or is it enough for it to hold along a certain line or at a certain point?
For example, I'm currently working with the function v(x,y) = x^3 + y^3 which, when you solve for the double derivatives of x and y and place it into the Laplace equation yields 6x + 6y = 0. Now, this is true along the line y=-x.
Is that enough to show that the function is harmonic or does the Laplace equation have to hold true for all (x,y) and not just along a single line?