We can parametrise an open set of the plane using Cartesian coordinates, , where . Then and
How about in polar coordinates? We can parametrise an open set of the plane with where this time.
Then and and the coefficients of the first fundamental form are Considering the trihedron given by the vectors , taking inner product with , , we have the following system of equations:
(for reference, see fifth note on computing Gaussian curvature)
To find the Christoffel symbols, we let
And since ,
In fact, we can check that the Gaussian curvature of the open set of the plane is zero , by substituting the Christoffel symbols above in the formula for Gauss Curvature: