How to compute the Gaussian curvature of a surface defined by a parametric equation?

I'm studying differential geometry and I need to compute the Gaussian curvature of a parametric surface $\mathbf{r}(u, v)$.

Given the first fundamental form:

And the second fundamental form:

The Gaussian curvature is:

Can someone walk through this computation for a specific surface like a torus or a helicoid?