In Riemannian geometry the statement of the coarea forumula involves an expression $J(f,x)$ ($f$ is a smooth map between Riemannian manifolds) which is defined as the "Jacobian" of a linear isomorphism between different vector spaces (first I thought it just means "determinant"). Although both of these vector spaces carry a scalar product (the aforementioned linear isomorphism is the restriction of a differential of $f$) I cannot make any sense out of this terminology.

Can somebody else?