Question

Show that the function f(x,y)=arctan(y/x) satisfies Laplace's equation:

Answer 1

$\Delta f = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0$

So let's find second derivatives:

$\frac{\partial f}{\partial x} = (arctan(\frac{y}{x}))^\prime_x = -\frac{y}{x^2 + y^2} \\ \frac{\partial^2 f}{\partial x^2} = ( -\frac{y}{x^2 + y^2})^{\prime}_x = \frac{2xy}{(x^2 + y^2)^2}\\ \frac{\partial f}{\partial y} = (arctan(\frac{y}{x}))^\prime_y = \frac{x}{x^2 + y^2} \\ \frac{\partial^2 f}{\partial y^2} = ( \frac{x}{x^2 + y^2} )^{\prime}_y =-\frac{2xy}{(x^2 + y^2)^2}$

And now substitute in the formula:

$\Delta f = \frac{2xy}{(x^2 + y^2)^2} - \frac{2xy}{(x^2 + y^2)^2} = 0$

So $f(x,y) = arctan(\frac{y}{x})$ satisfies Laplace's equation