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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?

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Isn't it just the Jacobian of the map $f$ at $x$? – Mariano Suárez-Alvarez May 14 '11 at 19:47
I suggest you ask on This is a basic question not suitable for here. Or you could just try to figure out why a linear isomorphism between two different vector spaces with inner products has a well-defined determinant (here, "Jacobian" really does mean determinant). – Deane Yang May 14 '11 at 21:20
@Deane: isn't it only well-defined up to sign? – Qiaochu Yuan Jun 11 '11 at 23:21

In essence differentiability is about the existence of linear approximations to mappings from vector spaces into vector spaces.

Mariano Suárez-Alvarez is absolutely right: the Jacobian matrix consists of the partial derivatives dFi(X)/dXj (i, j = 1...N). It is the matrix of the linear map that is the first-order approximation to the actual map F. And the linear approximation of a linear map is the map itself.

Ciao: Johan E. Mebius

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