Inverse function theorem

In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its ___domain. The theorem can be generalized to maps defined on manifolds, and on infinite dimensional Banach spaces.

The theorem states that if the total derivative of a function F : Rⁿ → Rⁿ is invertible at a point p (i.e., the Jacobian determinant of F at p is nonzero), and F is continuously differentiable near p, then it is an invertible function near p. That is, an inverse function to F exists in some neighborhood of F(p). In the infinite dimensional case it is required that the Frechet derivative have a bounded inverse near p.

The Jacobian matrix of F⁻¹ at F(p) is then the inverse of the Jacobian of F, evaluated at p. This can be understood as a special case of the chain rule, which states that for linear transformations F and G,

${\displaystyle J_{G\circ F}(p)=J_{G}(F(p))\cdot J_{F}(p)}$

where J denotes the corresponding Jacobian matrix.

Assume that the inverse function theorem holds at F(p). Let ${\displaystyle G(p)=F^{-1}(p)}$.

${\displaystyle J_{F^{-1}\circ F}(p)=J_{F^{-1}}(F(p))\cdot J_{F}(p)}$

${\displaystyle J_{I}(p)\cdot (J_{F}(p))^{-1}=J_{F^{-1}}(F(p))\cdot J_{F}(p)\cdot (J_{F}(p))^{-1}}$

${\displaystyle I\cdot (J_{F}(p))^{-1}=J_{F^{-1}}(F(p))\cdot I}$

${\displaystyle (J_{F}(p))^{-1}=J_{F^{-1}}(F(p))}$

where I is the identity transformation. This is often expressed more clearly as the useful single-variable formula,

${\displaystyle f'(x)={{1} \over {(f^{-1})'(f(x))}}.}$

The conclusion of the theorem is that the system of n equations y_i = f_j(x₁,...,x_n) can be solved for x₁,...,x_n in terms ofy₁,...,y_n if we restrict x and y to small enough neighborhoods of p.

The inverse function theorem can be generalized to differentiable maps between differentiable manifolds. In this context the theorem states that for a differentiable map F : M → N, if the derivative of F,

(DF)_p : T_pM → T_F(p)N

is a linear isomorphism at a point p in M then there exists an open neighborhood U of p such that

F|_U : U → F(U)

is a diffeomorphism. Note that this implies that M and N must have the same dimension.

If the derivative of F is an isomorphism at all points p in M then the map F is a local diffeomorphism.

Examples

Several functions exist for which differentiating the inverse is much easier than differentiating the function itself. Using the inverse function theorem, a derivative of a function's inverse indicates the derivative of the original function. Perhaps the most well-known example is the method used to compute the derivative of the natural logarithm, whose inverse is the exponential function. Let ${\displaystyle u=\ln x}$  and restrict the ___domain to x > 0. Then

${\displaystyle {\frac {d}{dx}}\ln x={{1} \over {{\frac {d}{du}}e^{u}}}={{1} \over {e^{u}}}={{1} \over {e^{\ln x}}}={{1} \over {x}}.}$ 

For more general logarithms, we see that ${\displaystyle {\frac {d}{dx}}\log _{b}(x)={\frac {1}{x\ln(b)}}={\frac {\log _{b}(e)}{x}}.}$ 

A similar approach can be used to differentiate an inverse trigonometric function. Let ${\displaystyle u=\arctan x.}$  Then

${\displaystyle {\frac {d}{dx}}\arctan x={{1} \over {{\frac {d}{du}}\tan u}}=\cos ^{2}{u}=\cos ^{2}{\arctan x}=\left({{1} \over {\sqrt {1+x^{2}}}}\right)^{2}={{1} \over {1+x^{2}}}.}$