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. Euclid, detail from The School of Athens by Raphael. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ...

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 mathematics, a total derivative is a combination of partial derivatives. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...

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, In calculus, the chain rule is a formula for the derivative of the composition of two functions. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

where J denotes the corresponding Jacobian matrix.

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

where I is the identity transformation. This is often expressed more clearly as the useful single-variable formula, In mathematics, an identity function, also called identity map or identity transformation, is a function which does not have any effect: it always returns the same value that was used as its argument. ...

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, This article is in need of attention. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...

(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. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

If the derivative of F is an isomorphism at all points p in M then the map F is a local diffeomorphism. In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f|U : U → f(U) is a 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 u = lnx and restrict the domain to x > 0. Then The natural logarithm is the logarithm to the base e, where e is equal to 2. ... The exponential function is one of the most important functions in mathematics. ...

For more general logarithms, we see that In mathematics, if two variables of bn = x are known, the third can be found. ...

A similar approach can be used to differentiate an inverse trigonometric function. Let u = tanx. Then In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...