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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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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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