Say I have a map S from R^2 -> R^3 defined by
f(u,v)=(u.cos(v), u.sin(v))

How do I find the matrix of the Weingarten map, g^-1.x.a (where a denotes the 2nd fundemental form) with respect to the partial derivatives df/du, df/dv?

I sort of know how to get the matrix in general from text books but not g^-1.x.a w.r.t....which seem complicated to me :(

is anyone able to help me solve this? or help me get started at least?

Any help is greatly appreciated!

Thanks guys