It is common to define a new coordinate system to help with mathematical manipulation of various kinds. But if the original system was subject to the Laplace Equation, how would that be maintained in the new coordinate system?
As an example, let's say you define a 2D skew coordinate system $\bar{x} = ax + by + c$ and $\bar{y} = dx + ey + f$. How would I obtain the Laplacian Operator for this coordinate system in terms of partial derivatives with respect to $\bar{x}, \bar{y}$?
For $u(x, y)$, $$\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$$ For $v(r, \theta)$, $$\nabla^2 v= \frac{1}{r} \frac{\partial}{\partial r} \left[ r \frac{\partial v}{\partial r} \right] + \frac{1}{r^2} \frac{\partial^2 v}{\partial \theta^2}$$ For $w(\bar{x}, \bar{y})$, $$\nabla^2 w = ?$$
I knew it could be found with something along the lines of the chain rule, but was unsure how. See my answer below for the details.
Let $ u(x, y) = w(\bar{x}, \bar{y}) $: $$ u_x = w_{\bar{x}} \frac{\partial \bar{x}}{\partial x} + w_y \frac{\partial \bar{y}}{\partial x} = a w_{\bar{x}} + d w_{\bar{y}} $$
$$ u_{xx} = a \left( w_{\bar{x}} \right)_x + d \left( w_{\bar{y}} \right)_x = a \left[ w_{\bar{x} \bar{x}} \frac{\partial \bar{x}}{\partial x} + w_{\bar{x} \bar{y}} \frac{\partial \bar{y}}{\partial x} \right] + d \left[ w_{\bar{y} \bar{x}} \frac{\partial \bar{x}}{\partial x} + w_{\bar{y} \bar{y}} \frac{\partial \bar{y}}{\partial x} \right] $$ $$ u_{xx} = a \left[ a w_{\bar{x} \bar{x}} + d w_{\bar{x} \bar{y}} \right] + d \left[ a w_{\bar{y} \bar{x}} + d w_{\bar{y} \bar{y}} \right] $$
By symmetry: $$ u_{yy} = b \left[ b w_{\bar{x} \bar{x}} + e w_{\bar{x} \bar{y}} \right] + e \left[ b w_{\bar{y} \bar{x}} + e w_{\bar{y} \bar{y}} \right] $$
$$ u_{xx} + u_{yy} = (a^2 + b^2) w_{\bar{x}\bar{x}} + 2 (ad + be) w_{\bar{x} \bar{y}} + (d^2 + e^2) w_{\bar{y}\bar{y}} $$