Laplacian of two scalar fields

Question

I am trying to find $$\nabla^2 (gh)$$ so far I have $$\nabla^2 (gh) = \nabla\cdot(\nabla(gh)) = \nabla\cdot(g\nabla h+h\nabla g) = \nabla\cdot(g\nabla h)+\nabla\cdot(h\nabla g)$$

I am not sure the last step is correct.

I am trying to show that $$\nabla^2 (gh) = g(\nabla^2 h) + 2(\nabla g)\cdot(\nabla h) + h(\nabla^2 g)$$

In particular, I am not sure how to treat $$\nabla\cdot(g\nabla h)$$ is there a rule that I am missing? thanks

Answer 1

Here is the correct solution:$$\nabla^2(gh)$$

$$= \nabla\cdot\nabla(gh)$$ $$=\nabla\cdot(g\nabla h+h\nabla g)$$ $$=\nabla\cdot (g\nabla h)+\nabla\cdot(h\nabla g)$$ $$=(\nabla g)\cdot\nabla h +g(\nabla\cdot\nabla h)+(\nabla h)\cdot\nabla g + h(\nabla\cdot\nabla g)$$ $$=g\nabla^2h +2(\nabla g)\cdot(\nabla h)+h\nabla^2g$$

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