Given a set of equations in variables , ..., , written explicitly as
(1)
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or more explicitly as
(2)
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the Jacobian matrix, sometimes simply called "the Jacobian" (Simon and Blume 1994) is defined by
(3)
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The determinant of is the Jacobian determinant (confusingly, often called "the Jacobian" as well) and is denoted
(4)
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The Jacobian matrix and determinant can be computed in the Wolfram Language using
JacobianMatrix[f_List?VectorQ, x_List] := Outer[D, f, x] JacobianDeterminant[f_List?VectorQ, x_List] := Det[JacobianMatrix[f, x]]
Taking the differential
(5)
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shows that is the determinant of the matrix , and therefore gives the ratios of -dimensional volumes (contents) in and ,
(6)
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It therefore appears, for example, in the change of variables theorem.
The concept of the Jacobian can also be applied to functions in more than variables. For example, considering and , the Jacobians
(7)
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(8)
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can be defined (Kaplan 1984, p. 99).
For the case of variables, the Jacobian takes the special form
(9)
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where is the dot product and is the cross product, which can be expanded to give
(10)
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