inverse function theorem
Let be a continuously differentiable, vector-valued function mapping the open set to and let . If, for some point , the Jacobian, , is non-zero, then there is a uniquely defined function and two open sets and such that
is continuously differentiable on and for all .
0.0.1 Simplest case
When , this theorem becomes: Let be a continuously differentiable, real-valued function defined on the open interval . If for some point , , then there is a neighbourhood of in which is strictly monotonic. Then is a continuously differentiable, strictly monotonic function from to . If is increasing (or decreasing) on , then so is on .
|Title||inverse function theorem|
|Date of creation||2013-03-22 12:58:30|
|Last modified on||2013-03-22 12:58:30|
|Last modified by||azdbacks4234 (14155)|