Determinant of Hessian approximation (SURF)
I think $w$ is a factor that makes the ratio of  $D_{xy}$ to $D_{yy}$ the same as $L_{xy}$ to $L_{yy}$. This is so the value of the determinant for the simplified kernel roughly matches that of the continuous version.
You could write it that way. But if you leave it as $wD_{xy}$ then if used in other expressions you don't have to muck around with adding square roots. e.g. $(wD_{xx})^3$ expressed your suggested way would be $\sqrt{w}^3D_{xx}^3$