Multiplying Gradients in Vector Calculus

Question

What happens when you multiply two gradients of two scalar fields together? So: $$ \vec{\nabla}A\cdot\vec{\nabla}B $$ Using Einstein summation convention I get: $$ (\hat{e}_{i}\partial_{i}A)\cdot(\hat{e}_{j}\partial_{j}B) = (\hat{e}_{i}\cdot\hat{e}_{j})(\partial_{i}A\partial_{j}B)=\delta_{ij}(\partial_{i}A\partial_{j}B)=\partial_{i}A\partial_{i}B $$ Which is correct. But in my notes it says that this also equals: $$ \vec{\nabla}A\cdot\vec{\nabla}B=\frac{1}{2}(\nabla^2(AB)-A\nabla^2B-B\nabla^2A) $$ How can this be? Thanks!

Answer 1

Apply the product rule:

$$ \nabla (AB)= B \nabla A + A\nabla B $$

Then apply the product rule again.