University Mathematics Handbook (2015)

VII. Differential Calculus of Multivariable Functions

Chapter 9. Inverse Functions System

a.  Given a system of functions of variables:

  (*)

The determinant

is called the Jacobian of (*) system at .

b.  If all functions of (*) system belong to class of the neighborhood of and , then there exists a neighborhood of when, , where there is defined a unique system of functions

  (**)

inverse to functions of (*), and belonging to class in the neighborhood of .

c.  If is a Jacobian of system (**), then .