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Invariance of the Laplacian

Intuitively, invariance means that the Laplacian is measuring something geometric, something about the shape of the membrane.

Formally, this means that D(f(gx)) = (Df)(gx)

This might not look like much of a requirement, so let's look at an example where it fails. Suppose L represents the operation of multiplying a function by x and then taking the derivative of the result with respect to x.

So that operator is not translation invariant. It doesn't measure anything geometric about the function. This is one of the many reasons the Laplacian is special.

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