$ \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = \left( \frac{\partial^2 f}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2} + \frac{1}{r}\frac{\partial f}{\partial r} \right) $
I have seen this and was told by someone random that it is called laplacian... something and since then I couldn't figure it out. Does anyone know what it would be called?
Thanks
Yes it is called Laplacian. This will be important in mechanics, wave theory, quantum mechanics, and more. It is defined as the divergence (denoted by $\nabla\cdot$) of the gradient (denoted by $\nabla f$). Visit http://en.wikipedia.org/wiki/Laplace_operator to learn more. I am just going to write a couple of formulas for determining Laplacian.
Two dimensions:
The formula for the Laplacian in polar coordinates is: $$\nabla^2 f=\frac{1}{r}\frac{\partial}{\partial r}+\frac{\partial^2 f}{\partial r^2}+\frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2}$$
In Cartesian Coordinates, given a scalar field $f(x, y)$, the Laplacian of $f$ is: $$\nabla^2 f=\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}$$
Three dimensions:
In Cartesian Coordinates, given a scalar field $f(x, y, z)$, the Laplacian of $f$ is: $$\nabla^2 f=\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2}$$
In cylindrical coordinates, given a scalar field $f(\rho, \theta, z)$, the Laplacian is: $$\nabla^2 f=\frac{1}{\rho} \frac{\partial}{\partial \rho}\left(\rho \frac{\partial f}{\partial \rho}\right)+\frac{1}{\,\rho^2}\frac{\partial^2 f}{\partial \varphi^2}+\frac{\partial^2 f}{\partial z^2}$$
In spherical coordinates, given a scalar field $f(r, \theta, \varphi)$, the Laplacian is: $$\nabla^2 f=\frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial f}{\partial \theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \varphi^2}$$
They are fairly complicated formulas, but they are there in case you need them. Thank you for reading.