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) 
Taking the differential

(5) 
shows that is the Determinant of the Matrix
, and therefore gives the ratios of D
volumes (Contents) in and ,

(6) 
The concept of the Jacobian can also be applied to functions in more than variables. For example, considering
and , the Jacobians
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) 
See also Change of Variables Theorem, Curvilinear Coordinates, Implicit Function Theorem
References
Kaplan, W. Advanced Calculus, 3rd ed. Reading, MA: AddisonWesley, pp. 9899, 123, and 238245, 1984.
Simon, C. P. and Blume, L. E. Mathematics for Economists. New York: W. W. Norton, 1994.
© 19969 Eric W. Weisstein
19990525