Jun. 29th, 2006

letters!

Jun. 29th, 2006 08:29 am
synchcola: (Default)
A neat proof of Lame's formula for the Laplacian.

Let u be a compactly supported test function.
\int u\,\Delta^2 f\, d\xi = -\int {\partial u \over \partial \xi^i} {\partial f \over \partial \xi^i}d\xi=-\int{ \partial u\over\partial x_a}{\partial x_a\over\partial\xi_i}{\partial x_b\over\partial \xi_i}{\partial f\over\partial x_b} \bigg| {d\xi \over dx}\bigg|dx

The xi are orthogonal, so ∂xa/&partξi ∂xb/∂ξi is gaa if a = b, and zero otherwise. Also, the Jacobian is √(g11 g22 g33) = √g.
-\int {g^{aa} \over \sqrt g} {\partial u \over \partial x_a} {\partial f \over \partial x_a} dx = \int u {\partial \over \partial x_a} \bigg({g^{aa} \over \sqrt g} {\partial f \over \partial x_a}\bigg) dx = \int u \sqrt g {\partial \over \partial x_a} \bigg({g^{aa} \over \sqrt g} {\partial v \over \partial x_a}\bigg) d\xi

So,
\Delta^2 f = \sqrt g {\partial \over \partial x_a} \bigg( {g^{aa} \over \sqrt g} {\partial f \over \partial x_a} \bigg)

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