Jacobian matrix and determinant

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${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
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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. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature.^[4]