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Analysis Crib Sheet Implicit Function Theorem

Summary: Analysis Crib Sheet
Implicit Function Theorem
Theorem 1. Let f(x, y) be a C1
mapping of an open set Rn+m
into Rn
, such that f(x0, y0) = 0 for some point (x0, y0) .
Set A = Dxf(x0, y0), B = Dyf(x0, y0), and assume that A = Dxf(x0, y0)
is invertible.
Then there exist open sets U Rn+m
and W Rm
, with (x0, y0) U
and y0 W, and a C1
function g(y) from W to Rn
having the property
that for every y W, (y, g(y)) is the unique point in U such that
f(g(y), y) = 0.
Morevoer, it holds that
Dyg(y0) = -A-1
Inverse Function Theorem


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics