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| - A Jacobian matrix, sometimes simply called a Jacobian, is a matrix of first order partial derivatives (in some cases, the term "Jacobian" also refers to the determinant of the Jacobian matrix). For a function , the Jacobian is the following matrix: or, in Einstein notation, Note that in some conventions, the Jacobian is the transpose of the above matrix.
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| abstract
| - A Jacobian matrix, sometimes simply called a Jacobian, is a matrix of first order partial derivatives (in some cases, the term "Jacobian" also refers to the determinant of the Jacobian matrix). For a function , the Jacobian is the following matrix: or, in Einstein notation, Note that in some conventions, the Jacobian is the transpose of the above matrix. Jacobians where are square matrices, and are commonly used when changing coordinates, especially when taking multiple integrals and determining whether complex functions are holomorphic. For example, a Jacobian representing a change in variables from to and to in two dimensions is represented as A Jacobian matrix is what is usually meant by the derivative of higher-dimensional functions; indeed, differentiability in the components of a Jacobian guarantees differentiability in the function itself. In the case of a multivariable function , the Jacobian matrix with respect to the input variables is simply the gradient of the function. The Jacobian is also related to the Hessian matrix by
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