What does ∇∇:Q mean?

Question

I'm reading a paper from the 1960's on Electromagnetism, and I wanted to know what the symbol ∇∇:Q (where Q is the Electric Quadrupole) means. Is this just the Laplacian? See Below .

Answer 1

The expression is $\nabla \nabla : Q$. I suspect that $:$ refers to the dyadic "double dot-product", so that $$ \nabla \nabla = \pmatrix{ \frac{\partial^2}{\partial x^2} & \frac{\partial^2}{\partial x\partial y} & \frac{\partial^2}{\partial x\partial z}\\ \frac{\partial^2}{\partial y\partial z} & \frac{\partial^2}{\partial y^2} & \frac{\partial^2}{\partial y\partial z}\\ \frac{\partial^2}{\partial z\partial x} & \frac{\partial^2}{\partial z\partial y} & \frac{\partial^2}{\partial z^2} }, $$ and $\nabla \nabla : Q = \sum_{i,j = 1}^3 \frac{\partial^2Q_{ij}}{\partial x_i\partial x_j}$, where $x_1 = x, x_2 = y, x_3 = z$.