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)

REFERENCES:

Gradshteyn, I. S. and Ryzhik, I. M. "Jacobian Determinant." §14.313 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1068-1069, 2000.

Kaplan, W. Advanced Calculus, 3rd ed. Reading, MA: Addison-Wesley, pp. 98-99, 123, and 238-245, 1984.

Simon, C. P. and Blume, L. E. Mathematics for Economists. New York: W. W. Norton, 1994.