Implicit Function Theorem
Given
if the Jacobian
then
, 
, and 
can be solved for in terms of 
, 
, and 
and Partial Derivatives of , , with respect to , , and can be found by differentiating implicitly.More generally, let 
be an Open Set in 
and let 
be a 
Function. Write
in the form 
, where and are elements of 
and 
. Suppose that (
, 
) is apoint in such that 
and the Determinant of the 
Matrix whose elements are theDerivatives 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
Then there exists a Neighborhood
of in and a unique Function
such that 
and 
for all 
.See also Change of Variables Theorem, Jacobian
References
Munkres, J. R. Analysis on Manifolds. Reading, MA: Addison-Wesley, 1991.
- Perspectivity
- Pesin Theory
- Petersen Graphs
- Petersen-Shoute Theorem
- Peters Projection
- Peter-Weyl Theorem
- Petrie Polygon
- Petrov Notation
- Pfaffian Form
- p-Good Path
- p-Group
- Phase
- Phase Space
- Phase Transition
- Phasor
- Phi Curve
- Phi Number System
- Phragmén-Lindêlöf Theorem
- Phyllotaxis
- Pi
- Piano Mover's Problem
- Picard's Existence Theorem
- Picard's Little Theorem
- Picard's Theorem
- Picard Variety