Calculating Riemann Curvature Tensor: Faster Methods?

[HilbertSpace]

I've been trying to calculate the Riemann Curvature Tensor for a certain manifold in 3-dimensional Euclidean Space using Christoffel Symbols of the second kind, and so far everything has gone well however...

It is extremely tedious and takes a very long time; there is also a high probability of making silly mistakes (like misplacing a variable). Are there any faster methods (not necessarily simpler) or is there no other alternative?

 

[lavinia]

The equations simplify with respect to an orthonormal basis.

For a surface, if dx and dy are a local orthonormal basis for the 1 forms, then

dx = w_{12}^dy and dy = -w_{12}^dx

dw_{12} = -KdV where K is the Gauss curvature and dV is the volume element of the metric.