For a surface in $\mathbb{R}^3$, the mean curvature can be determined as $H = - \nabla\cdot \mathbf{n}$, where $\mathbf{n}$ is the unit normal to the surface. This is also written sometimes with an additional factor of $\frac12$. This definition is especially useful in fluid dynamics communities, since avoiding the metric tensor and related stuff is quite helpful when one doesn't have a background in differential geometry.
Is there an equivalent relation for the Gaussian curvature (again restricting ourselves to a surface in 3-D)? Something that equates the Gaussian curvature with the derivatives of the normal field.