Given a set 
of 
equations in 
variables 
, ..., 
, written explicitly as
![y=[f_1(x); f_2(x); |; f_n(x)],](/images/equations/Jacobian/NumberedEquation1.svg) |
(1)
|
or more explicitly as
 |
(2)
|
the Jacobian matrix, sometimes simply called "the Jacobian" (Simon and Blume 1994) is defined by
![J(x_1,...,x_n)=[(partialy_1)/(partialx_1) ... (partialy_1)/(partialx_n); | ... |; (partialy_n)/(partialx_1) ... (partialy_n)/(partialx_n)].](/images/equations/Jacobian/NumberedEquation3.svg) |
(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)
|
