More generally, let be an Open Set in
be a Function. Write
in the form , where and are elements of and . Suppose that (, ) is a
point in such that and the Determinant of the Matrix whose elements are the
Derivatives of the component Functions of with respect to the
variables, written as , evaluated at , is not equal to zero. The latter may be rewritten as
See also Change of Variables Theorem, Jacobian
Munkres, J. R. Analysis on Manifolds. Reading, MA: Addison-Wesley, 1991.