Editing Laplace operator (section)

==Euclidean invariance==
The Laplacian is invariant under all [[Euclidean transformation]]s: [[rotation]]s and [[Translation (geometry)|translations]].  In two dimensions, for example, this means that:
<math display="block">\Delta ( f(x\cos\theta - y\sin\theta + a, x\sin\theta + y\cos\theta + b)) = (\Delta f)(x\cos\theta - y\sin\theta + a, x\sin\theta + y\cos\theta + b)</math>
for all ''θ'', ''a'', and ''b''.  In arbitrary dimensions,
<math display="block">\Delta (f\circ\rho) =(\Delta f)\circ \rho</math>
whenever ''ρ'' is a rotation, and likewise:
<math display="block">\Delta (f\circ\tau) =(\Delta f)\circ \tau</math>
whenever ''τ'' is a translation. (More generally, this remains true when ''ρ'' is an [[orthogonal transformation]] such as a [[reflection (mathematics)|reflection]].)

In fact, the algebra of all scalar linear differential operators, with constant coefficients, that commute with all Euclidean transformations, is the polynomial algebra generated by the Laplace operator.