Suppose X~(α,β) is a reparameterization of X(u,v) with first fundamental form given by:
Then X(u,v)=X~(α(u,v),β(u,v)). Compute E~,F~,G~ from E,F,G.
We get that:
Hence the dual vectors are given by:
Then since Edu2+2Fdudv+Gdv2=ds2=E~dα2+2F~dαdβ+G~dβ2 , we obtain that:
Thus:
Giving us finally: