TITLE: Does the symbol $\nabla^2$ has the same meaning in Laplace Equation and Hessian Matrix?
QUESTION [12 upvotes]: we know that the Laplace Equation can be written in the form:
$$\nabla^2 \Phi=0$$
while in this equation,the symbol $\nabla^2 \Phi$ stand for $\sum_{i=1}^n\frac{\partial^2 \Phi}{\partial x_i^2}$.
At the same time, the Hessian Matrix is also denoted as
$$H=\nabla^2f$$
My question is what does $\nabla^2$ really mean?

REPLY [8 votes]: As pointed out by the other answer, the notation $\nabla^2$ is somewhat ambiguous (though it most commonly refers to the Laplacian). The reason is that $\nabla^2$ could be interpreted either as
\begin{align}
\nabla^2
&= \nabla \cdot \nabla \\
&= \left( \sum_i \mathbf{e}_i \nabla_i \right) \cdot \left( \sum_i \mathbf{e}_i \nabla_i \right) \\
&= \sum_i \nabla_i \nabla_i \\
\end{align}
where $\cdot$ is the inner product, which is the Laplacian (scalar), or as
\begin{align}
\nabla^2
&= \nabla \otimes \nabla \\
&= \left( \sum_i \mathbf{e}_i \nabla_i \right) \otimes \left( \sum_i \mathbf{e}_i \nabla_i \right) \\
&= \sum_i \sum_j \mathbf{e}_i \mathbf{e}_j \nabla_i \nabla_j
\end{align}
where $\otimes$ is the outer product, which is the Hessian (matrix).