§ Theorem Egregium / Gauss's theorem (Integrating curvature in 2D) [TODO ]
- Let be a 2 dimensional surface.
- Gauss Rodriguez map map: . The derivative of this map goes from .
- Since surfaces are parametric, we can think of it as a map from .
- For gauss, the curvature of the surface at is . This tells us how small areas (on the tangent plane of ) is distorted (on the tangent plane of , because it's the determinant / jacobian of the map. Thus, heuristically, it is the ratio of the area around at to the area around at
- To show that this normal curvature view really is curvature, let's compute for a normal paraboloid. Wildberger says that all surfaces are like normal paraboloids upto second order.
- This fits with one of our views of curvature of a curve: one way was one over the osculating circle, the other was
- We had a formula like was a change in angle. Similarly, in our case, we see that if we consider , we get the area of the image of , because infinitesimally is the ratio of areas.
- In particular if the surface is homeomorphic to a sphere, then we get the total area of the sphere, .. This is the 2D analogue of the fact that if we integrate the curvature of a closed curve, we get . [area of a circle ]. This is by green's theorem.