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While studying Landau-Lifshitz equation following term appears,

$-m \times (m \times \Delta m) = \Delta m + |\nabla m|^2 m$

In above equation m is a vector quantity. It will be great if someone can point out what the symbol $\Delta$ here stands for and how these two sides are equal.

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@alekhine: I was just about to work on a fuller solution when I realized that the right-hand side does not really make sense. – Haskell Curry Jan 6 '13 at 9:49
This is called "Vector Laplacian". In orthonormal coordinates it is just the Laplacian applied termwise. – Giuseppe Negro Jan 6 '13 at 11:18
up vote 2 down vote accepted

By definition, $$ \Delta = \sum_{i=1}^{n} \frac{\partial^{2}}{\partial x_{i}^{2}}. $$

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It is the Laplace Operator, or the divergence-of-the-gradient operator (not as catchy, though).

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