v = dot(s1, v) * s1_vec + dot(s2,v) * s2_vec

Doesn't look right geometrically because it maps the unknown vector V onto the basis vectors in Euclidian sense, instead of a curvilinear sense.

Is the solution to this to use the metric tensor defined by Jtranspose J and use that to replace the Cartesian dot product?

Thanks for any help