Laplacian Identity Question

Question

Is the following identity true? $$\nabla^2(A+B) =? \nabla^2A +\nabla^2B$$ My intuition says yes because $\nabla^2A = \nabla \cdot \nabla A $ where both grad and divergence have this property.

Answer 1

Yes, the Laplacian is a linear operator, and this is inherited from the the gradient and divergence as you suspected.
$$\begin{align}\nabla^2(A+B)&=\nabla\cdot\nabla(A+B)\\&=\nabla\cdot(\nabla A+\nabla B)\\&=\nabla\cdot \nabla A+\nabla\cdot\nabla B\\&=\nabla^2A+\nabla^2B\end{align}$$