Help with the Laplacian Formula Can someone please give me the formula of the Laplacian $∆$$_{x,y}$
where $x$ and $y$ are both in $R^n$?
The Laplacian operator is defined as :

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

$$\Delta_x = \frac{\partial}{\partial x}(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2})$$
$$\implies \Delta_x = \frac{\partial^3}{\partial x^3} + \frac{\partial^3}{\partial x \partial y^2}$$

Similarly,

$$\implies \Delta_{x,y} = \frac{\partial^4}{\partial x^3 \partial y} + \frac{\partial^4}{\partial x \partial y^3}$$
The Laplacian operator is defined as :

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

$$\Delta_x = \frac{\partial}{\partial x}(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2})$$
$$\implies \Delta_x = \frac{\partial^3}{\partial x^3} + \frac{\partial^3}{\partial x \partial y^2}$$

Similarly,

$$\implies \Delta_{x,y} = \frac{\partial^4}{\partial x^3 \partial y} + \frac{\partial^4}{\partial x \partial y^3}$$