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implicit function theorem

Implicit function theorems

Idea

Implicit function theorems give sufficient conditions for the existence of a differentiable inverse of a germ f pf_p of a differentiable map f:MNf\colon M \to N of smooth manifolds at a point pp. The invertibility is trivially equivalent to the statement that the germ is a local diffeomorphism of some neighborhood of pp to some neighborhood of f(p)f(p). If it is invertible, then we can consider the tangent map? T pf:T pMT f(p)NT_p f\colon T_p M \to T_{f(p)}N. If ff is locally invertible with differentiable inverse, then for all yy in some neighborhood of yy the functoriality of TT implies that Id T y=T y(f 1f)=T f(y)f 1T yfId_{T_y} = T_y (f^{-1} \circ f) = T_{f(y)} f^{-1} \circ T_y f and alike for ff 1f \circ f^{-1} at f(y)f(y), demonstrating that T yfT_y f must then be invertible. The inverse function theorem? says that the invertibility of T pfT_p f is in fact sufficient for the invertibility of the germ, which is then automatically differentiable.

In R n\mathbf{R}^n

Let UR nU \subset \mathbf{R}^n be an open set in a cartesian space, aUa\in U, f:UR nf\colon U \to \mathbf{R}^n a map of class C 1C^1 and det(f ix j(a))0det\left(\frac{\partial f_i}{\partial x_j}(a)\right)\neq 0. Then there are open sets VaV \ni a, Wf(a)W \ni f(a), VUV \subset U such that f V:VWf|_V\colon V \to W is a diffeomorphism and for all yWy \in W and (f 1)(y)=(f[f 1(y)]) 1(f^{-1})'(y) = (f'[f^{-1}(y)])^{-1}.

Local statement on manifolds

This is the theorem stated in the Idea section; the differentiable germ is assumed to be of class C 1C^1 (continuously differentiable). The statement is local, so one can consider it in charts, hence the proof reduces to the case of R n\mathbf{R}^n.

Global statement on manifolds

Let f:MNf\colon M \to N be a smooth map of smooth manifolds. A point qNq \in N is a regular value of ff if for every point pf 1(q)p \in f^{-1}(q) the differential T pf:T pMT qNT_p f\colon T_p M \to T_q N is an epimorphism. The implicit function theorem asserts that Q=f 1(p)Q = f^{-1}(p) is a smooth submanifold of MM and the tangent bundle TM NT M|_N globally splits as TM NTNR nT M|_N \cong T N \oplus \mathbf{R}^n where n=dimNn = dim N.

More generally, if WNW \subset N is a submanifold, we say that the map ff is transversal along WW if for every point xf 1(W)x\in f^{-1}(W) there is an equality

T f(x)N=T f(x)W+(T xf)(T xX). T_{f(x)} N = T_{f(x)}W + (T_x f)(T_x X) .

In particular, ff is transveral along every regular value pNp \in N. The implicit function theorem asserts that the preimage f 1(W)f^{-1}(W) is a smooth submanifold of MM, the normal bundle ν(f 1(W)M)\nu(f^{-1}(W) \subset M) is isomorphic to f *(ν(WN))f^*(\nu(W\subset N)), and the differential TfT f exhibits the fiberwise isomorphism ν(f 1(W)M)ν(WN)\nu(f^{-1}(W)\subset M)\to \nu(W\subset N).

References

  • L. H. Loomis, S. Sternberg, Advanced calculus, 1968, 1990 (3.11 in 1990 edition)

  • S. Lang, Analysis I

Various applications and related theorems can be found in chapter 5: Local and tangential properties of

  • T. Bröcker, K. Jänich, C. B. Thomas, M. J. Thomas, Introduction to differentiable topology, 1982 (translated from German 1973 edition; \exists also 1990 German 2nd edition)

An invariant global statement on manifolds is at page 44 of

  • А. С. Мищенко, Векторные расслоенния и их применения, Moscow, Nauka 1984

Elementary course notes of the case in R n\mathbf{R}^n (mainly lots of examples):

  • Frank Jones, Implicit function theorem, pdf

Revised on November 4, 2011 22:40:54 by Toby Bartels (64.89.53.95)