I recently got to grips with simple 2D curvature, and I'm aware that formulae exist for Gauss curvatures of surfaces defined in terms of some $z=f(x,y)$, but I'm finding little progress making sense of these. Is there a nice way to get at the curvature of such defined surfaces?
In particular I'd like to find the curvatures of the saddles defined by $z=xy$ and $z=x^y\;(0^0:=1)$ (intuitively, the latter "feels" more negative than the former, but by how much?).
Here is cooking recipe for computing Gaussian and Mean curvatures.
1. First you want to parametrise the surface, naive $\phi(u,v)=(u,v,uv)$ will do, or equivalently $\phi(u,v)=(u,v,u^2-v^2)$ for same surface. (Similiarly $\psi(u,v)=(u,v,u^v)$ for surface defined by $z=x^y$ with saddle point at $\psi(1,0)$.)
2. Compute first partial derivatives: $\partial_u\phi, \partial_v\phi$.
3. While you are at it compute second partial derivatives: $\partial^2_{uu}\phi, \partial^2_{uv}\phi,\partial^2_{vv}\phi$.
4. Compute coefficients of first fundamental form by
$g_{11}=\langle\partial_u\phi,\partial_u\phi \rangle$,
$g_{12}=g_{21}=\langle\partial_u\phi,\partial_v\phi \rangle$
$g_{22}=\langle\partial_v\phi,\partial_v\phi \rangle$
,where $\langle ,\rangle$ denotes standard scalar product of $\mathbb R^3$
5. Now you need a normal unit field, which you can construct from 2.: $$N=\frac{\partial_u\phi \times \partial_v\phi}{\Vert \partial_u\phi \times \partial_v\phi \Vert} $$ where $\times$ denotes cross product.
6. Compute coefficients of second fundamental form by
$h_{11}=\langle\partial^2_{uu}\phi,N \rangle$,
$h_{12}=h_{21}=\langle\partial^2_{uv}\phi,N \rangle$
$h_{22}=\langle\partial^2_{vv}\phi,N \rangle$
7. Now you are ready to compute Gaussian curvature: $$ K=\frac{h_{11}\cdot h_{22}-h_{12}^2}{g_{11}\cdot g_{22}-g_{12}^2}$$ and mean Curvature $$H=\frac{1}{2}\frac{h_{11}\cdot g_{22}-2\cdot h_{12}\cdot g_{12}+h_{22}\cdot g_{11}}{g_{11}\cdot g_{22}-g_{12}^2} $$
If done right, you should get $K(\phi(0,0))=-4$ for $z=xy$ surface.