TITLE: Why do mathematicians use $\Delta$ instead of $\nabla^2$?
QUESTION [5 upvotes]: I often hear that, when writing PDEs, $\nabla^2$ is the convention among physicists and engineers, while mathematicians write $\Delta$ instead.
To me, the physicists' notation seems like it is objectively better: first you $\nabla$, then you $\nabla$ again.  Thus, $\nabla^2$.  Makes sense, right?
When mathematicians resist conventions from other disciplines, they generally have a good, concrete reason for doing so. So, what is the reason that mathematicians use $\Delta$?

REPLY [4 votes]: Let $f$ be a scalar field. 
Then, $\nabla^2$  is the Hessian, i.e. the matrix $\nabla^2 f=\dfrac{\partial ^2f}{\partial x_i \partial x_j}$ , while $\Delta$ is the Laplacian, i.e. the scalar $\Delta f=\Sigma_{i}\dfrac{\partial ^2f}{\partial x_i^2 }$