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This shows how an integral in the Cartesian coordinate system is transformed into an integral in the polar coordinate system: The partial derivatives of all these functions (if they exist) can be organized in an m-by- n matrix, the Jacobian matrix J of F, as follows:: Such a function is given by m real-valued component functions, y 1( x 1., x n). Suppose F : R n → R m is a function from Euclidean n-space to Euclidean m-space. In vector calculus, the Jacobian matrix ( rarely /jɨˈkoʊbiən/) is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector.
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