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For a transformation, LaTeX: y=f(x), such that LaTeX: f is differentiable, the Jacobian is the determinant, LaTeX: \mbox{det}\left( \nabla f(x) \right). Historically, the number of LaTeX: y-variables equals the number of LaTeX: x-variables, say LaTeX: n, so LaTeX: \nabla f(x) is LaTeX: n \times n. The Jacobian matrix is defined as the matrix, LaTeX: \nabla f(x), allowing the number of functions to be less than the number of variables.

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