- #1

William Crawford

- 40

- 36

- TL;DR Summary
- I'm trying to work out different product rules for the Laplacian and I've gotten stuck on the Laplacian of a cross product.

Suppose ##A = A_i\mathbf{\hat{e}}_i## and ##B = B_i\mathbf{\hat{e}}_i## are vectors in ##\mathbb{R}^3##. Then

\begin{align}

\Delta\left(A\times B\right)

&= \epsilon_{ijk}\Delta\left(A_jB_k\right)\mathbf{\hat{e}}_i \\

&= \epsilon_{ijk}\left[A_j\Delta B_k + 2\partial_mA_j\partial_mB_k + B_k\Delta A_j\right]\mathbf{\hat{e}}_i \\

&= A\times\left(\Delta B\right) + 2\epsilon_{ijk}\partial_mA_j\partial_mB_k\mathbf{\hat{e}}_i + \left(\Delta A\right)\times B

\end{align}

I can't identify the term ##2\epsilon_{ijk}\partial_mA_j\partial_mB_k\mathbf{\hat{e}}_i## on the last line with anything more familiar in terms of standard vector calculus operations. This feels somewhat odd to me, as the additional two terms can be written neatly in terms of "standard" vector calculus operators (namely the cross product and the laplacian) but the third term can't. I hope someone can help me shed light on this matter.

\begin{align}

\Delta\left(A\times B\right)

&= \epsilon_{ijk}\Delta\left(A_jB_k\right)\mathbf{\hat{e}}_i \\

&= \epsilon_{ijk}\left[A_j\Delta B_k + 2\partial_mA_j\partial_mB_k + B_k\Delta A_j\right]\mathbf{\hat{e}}_i \\

&= A\times\left(\Delta B\right) + 2\epsilon_{ijk}\partial_mA_j\partial_mB_k\mathbf{\hat{e}}_i + \left(\Delta A\right)\times B

\end{align}

I can't identify the term ##2\epsilon_{ijk}\partial_mA_j\partial_mB_k\mathbf{\hat{e}}_i## on the last line with anything more familiar in terms of standard vector calculus operations. This feels somewhat odd to me, as the additional two terms can be written neatly in terms of "standard" vector calculus operators (namely the cross product and the laplacian) but the third term can't. I hope someone can help me shed light on this matter.