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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.

L(f(x+2)) = f(x+2) + x f'(x+2)
but
L(f)(x+2) = f(x+2) + (x+2) f'(x+2).

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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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 12 Aug 1998 --- Last modified: 18 Aug 1998 23:59:59
Comments to: CVM@maa.org