Inverse function theorem

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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,

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

where J denotes the corresponding Jacobian matrix.

Assume that the inverse function theorem holds at F(p). Let $G (p) = F - 1 (p)$ .

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

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

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

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

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

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

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 of y₁,...,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.

[edit] 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 $u = ln x$ and restrict the domain to x > 0. Then

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

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

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

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