Jacobian matrix and determinant

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In vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/,^[1][2][3] /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian.^[4] They are named after Carl Gustav Jacob Jacobi.

The Jacobian matrix is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix.

The Jacobian determinant is fundamentally used for changes of variables in multiple integrals.