Suppose M,N are open subsets of Rn. A smooth map F:MN is conformal iff xMu,vRn:DF(x)u,DF(x)v=h(x)2u,v
Fix x; the mapping B:(u,v)DF(x)u,DF(x)v is bi-linear and symmetric. By the polarization formula we conclude that B(u,v)=h(x)2u,v if and only if B(u,u)=h(x)2u,u.