Let's derive a formula for the Gaussian curvature of an orthogonal parametrisation of a coordinate neighbourhood at point on a smooth, orientable surface, we have that , hence of the first fundamental form at .
First, we consider the Gauss formula expressed in Christoffel symbols:
How do we compute the Christoffel symbols?
If we assign each point of a natural trihedron given by the vectors , , , and express the derivatives of the vectors , and in the basis ,
And by taking the inner product of the first four relations below with and we get the following:
Using the condition that is an orthogonal parametrisation, that is, , then the above reduces to
We substitute these values into the Gauss formula:
Therefore, simplifying gives the Gaussian curvature