Jacobian

Given a set of equations in variables , ..., , written explicitly as

(1)

or more explicitly as

(2)

the Jacobian matrix, sometimes simply called "the Jacobian" (Simon and Blume 1994) is defined by

(3)

The determinant of is the Jacobian determinant (confusingly, often called "the Jacobian" as well) and is denoted

(4)

The Jacobian matrix and determinant can be computed in the Wolfram Language using

  JacobianMatrix[f_List?VectorQ, x_List] :=
    Outer[D, f, x] /; Equal @@ (Dimensions /@ {f, x})
  JacobianDeterminant[f_List?VectorQ, x_List] :=
    Det[JacobianMatrix[f, x]] /;
      Equal @@ (Dimensions /@ {f, x})

Taking the differential

(5)

shows that is the determinant of the matrix , and therefore gives the ratios of -dimensional volumes (contents) in and ,

(6)

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)
			(8)

can be defined (Kaplan 1984, p. 99).

For the case of variables, the Jacobian takes the special form

(9)

where is the dot product and is the cross product, which can be expanded to give

(10)

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