What is the significance of rotation in equations?

[wumple]

This is sort of a homework question but I'm not looking for an answer. I'm just trying to understand exactly what's going on. It says "Among all the equations of the form [the general second order linear homogeneous partial differential equation], show that the only ones that are unchanged under all rotations (rotationally invariant) havce the form a(uxx + uyy) + bu =0.

What exactly does it mean for an equation to be rotated? I don't understand what's going on here very well.

 

[AlephZero]

I would guess it means if you rotate the coordinate system through an arbitrary angle, the form of the equation stays the same, i.e. you don't get a uxy term.

 

[HallsofIvy]

Let $x= x'cos(\theta)+ y' sin(\theta)$, $y= -x'sin(\theta)+ y'cos(\theta)$, so that $x'= xcos(\theta)- ysin(\theta)$ and $y'= xsin(\theta)+ ycos(\theta)$, and use the chain rule to replace $u_{xx}$ and $u_{yy}$ with derivatives in terms of x' and y' rather than x and y.

For example, $u_x= u_x'(x'_x)+ u_y'(y'_x)= cos(\theta)u_x'+ sin(\theta)u_y'$.