KNOWPIA
WELCOME TO KNOWPIA

In mathematics, more specifically in multivariable calculus, the **implicit function theorem**^{[a]} is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.

More precisely, given a system of m equations *f _{i}* (

In other words, under a mild condition on the partial derivatives, the set of zeros of a system of equations is locally the graph of a function.

Augustin-Louis Cauchy (1789–1857) is credited with the first rigorous form of the implicit function theorem. Ulisse Dini (1845–1918) generalized the real-variable version of the implicit function theorem to the context of functions of any number of real variables.^{[2]}

If we define the function *f*(*x*, *y*) = *x*^{2} + *y*^{2}, then the equation *f*(*x*, *y*) = 1 cuts out the unit circle as the level set {(*x*, *y*) | *f*(*x*, *y*) = 1}. There is no way to represent the unit circle as the graph of a function of one variable *y* = *g*(*x*) because for each choice of *x* ∈ (−1, 1), there are two choices of *y*, namely .

However, it is possible to represent *part* of the circle as the graph of a function of one variable. If we let for −1 ≤ *x* ≤ 1, then the graph of *y* = *g*_{1}(*x*) provides the upper half of the circle. Similarly, if , then the graph of *y* = *g*_{2}(*x*) gives the lower half of the circle.

The purpose of the implicit function theorem is to tell us the existence of functions like *g*_{1}(*x*) and *g*_{2}(*x*), even in situations where we cannot write down explicit formulas. It guarantees that *g*_{1}(*x*) and *g*_{2}(*x*) are differentiable, and it even works in situations where we do not have a formula for *f*(*x*, *y*).

Let be a continuously differentiable function. We think of as the Cartesian product and we write a point of this product as Starting from the given function *f*, our goal is to construct a function whose graph (**x**, *g*(**x**)) is precisely the set of all (**x**, **y**) such that *f*(**x**, **y**) = **0**.

As noted above, this may not always be possible. We will therefore fix a point (**a**, **b**) = (*a _{1}*, ...,

To state the implicit function theorem, we need the Jacobian matrix of *f*, which is the matrix of the partial derivatives of *f*. Abbreviating (*a*_{1}, ..., *a _{n}*,

where *X* is the matrix of partial derivatives in the variables *x _{i}* and

Let be a continuously differentiable function, and let have coordinates (**x**, **y**). Fix a point (**a**, **b**) = (*a*_{1}, …, *a _{n}*,

is invertible, then there exists an open set containing

Moreover, the partial derivatives of *g* in *U* are given by the matrix product:^{[3]}

If, moreover, *f* is analytic or continuously differentiable *k* times in a neighborhood of (**a**, **b**), then one may choose *U* in order that the same holds true for *g* inside *U*. ^{[4]} In the analytic case, this is called the **analytic implicit function theorem**.

Suppose is a continuously differentiable function defining a curve . Let be a point on the curve. The statement of the theorem above can be rewritten for this simple case as follows:

**Theorem** — If

then for the curve around we can write , where is a real function.

**Proof.** Since *F* is differentiable we write the differential of *F* through partial derivatives:

Since we are restricted to movement on the curve and by assumption around the point (since is continuous at and ). Therefore we have a first-order ordinary differential equation:

Now we are looking for a solution to this ODE in an open interval around the point for which, at every point in it, . Since *F* is continuously differentiable and from the assumption we have

From this we know that is continuous and bounded on both ends. From here we know that is Lipschitz continuous in both *x* and *y*. Therefore, by Cauchy-Lipschitz theorem, there exists unique *y*(*x*) that is the solution to the given ODE with the initial conditions. Q.E.D.

Let us go back to the example of the unit circle. In this case *n* = *m* = 1 and . The matrix of partial derivatives is just a 1 × 2 matrix, given by

Thus, here, the *Y* in the statement of the theorem is just the number 2*b*; the linear map defined by it is invertible if and only if *b* ≠ 0. By the implicit function theorem we see that we can locally write the circle in the form *y* = *g*(*x*) for all points where *y* ≠ 0. For (±1, 0) we run into trouble, as noted before. The implicit function theorem may still be applied to these two points, by writing x as a function of y, that is, ; now the graph of the function will be , since where *b* = 0 we have *a* = 1, and the conditions to locally express the function in this form are satisfied.

The implicit derivative of *y* with respect to *x*, and that of *x* with respect to *y*, can be found by totally differentiating the implicit function and equating to 0:

giving

and

Suppose we have an m-dimensional space, parametrised by a set of coordinates . We can introduce a new coordinate system by supplying m functions each being continuously differentiable. These functions allow us to calculate the new coordinates of a point, given the point's old coordinates using . One might want to verify if the opposite is possible: given coordinates , can we 'go back' and calculate the same point's original coordinates ? The implicit function theorem will provide an answer to this question. The (new and old) coordinates are related by *f* = 0, with

Now the Jacobian matrix of

where I

As a simple application of the above, consider the plane, parametrised by polar coordinates (*R*, *θ*). We can go to a new coordinate system (cartesian coordinates) by defining functions *x*(*R*, *θ*) = *R* cos(*θ*) and *y*(*R*, *θ*) = *R* sin(*θ*). This makes it possible given any point (*R*, *θ*) to find corresponding Cartesian coordinates (*x*, *y*). When can we go back and convert Cartesian into polar coordinates? By the previous example, it is sufficient to have det *J* ≠ 0, with

Since det

Based on the inverse function theorem in Banach spaces, it is possible to extend the implicit function theorem to Banach space valued mappings.^{[5]}^{[6]}

Let *X*, *Y*, *Z* be Banach spaces. Let the mapping *f* : *X* × *Y* → *Z* be continuously Fréchet differentiable. If , , and is a Banach space isomorphism from *Y* onto *Z*, then there exist neighbourhoods *U* of *x*_{0} and *V* of *y*_{0} and a Fréchet differentiable function *g* : *U* → *V* such that *f*(*x*, *g*(*x*)) = 0 and *f*(*x*, *y*) = 0 if and only if *y* = *g*(*x*), for all .

Various forms of the implicit function theorem exist for the case when the function *f* is not differentiable. It is standard that local strict monotonicity suffices in one dimension.^{[7]} The following more general form was proven by Kumagai based on an observation by Jittorntrum.^{[8]}^{[9]}

Consider a continuous function such that . There exist open neighbourhoods and of *x*_{0} and *y*_{0}, respectively, such that, for all *y* in *B*, is locally one-to-one *if and only if* there exist open neighbourhoods and of *x*_{0} and *y*_{0}, such that, for all , the equation
*f*(*x*, *y*) = 0 has a unique solution

where

- Inverse function theorem
- Constant rank theorem: Both the implicit function theorem and the inverse function theorem can be seen as special cases of the constant rank theorem.

**^**Also called**Dini's theorem**by the Pisan school in Italy. In the English-language literature, Dini's theorem is a different theorem in mathematical analysis.

**^**Chiang, Alpha C. (1984).*Fundamental Methods of Mathematical Economics*(3rd ed.). McGraw-Hill. pp. 204–206. ISBN 0-07-010813-7.**^**Krantz, Steven; Parks, Harold (2003).*The Implicit Function Theorem*. Modern Birkhauser Classics. Birkhauser. ISBN 0-8176-4285-4.**^**de Oliveira, Oswaldo (2013). "The Implicit and Inverse Function Theorems: Easy Proofs".*Real Anal. Exchange*.**39**(1): 214–216. doi:10.14321/realanalexch.39.1.0207. S2CID 118792515.**^**Fritzsche, K.; Grauert, H. (2002).*From Holomorphic Functions to Complex Manifolds*. Springer. p. 34. ISBN 9780387953953.**^**Lang, Serge (1999).*Fundamentals of Differential Geometry*. Graduate Texts in Mathematics. New York: Springer. pp. 15–21. ISBN 0-387-98593-X.**^**Edwards, Charles Henry (1994) [1973].*Advanced Calculus of Several Variables*. Mineola, New York: Dover Publications. pp. 417–418. ISBN 0-486-68336-2.**^**Kudryavtsev, Lev Dmitrievich (2001) [1994], "Implicit function",*Encyclopedia of Mathematics*, EMS Press**^**Jittorntrum, K. (1978). "An Implicit Function Theorem".*Journal of Optimization Theory and Applications*.**25**(4): 575–577. doi:10.1007/BF00933522. S2CID 121647783.**^**Kumagai, S. (1980). "An implicit function theorem: Comment".*Journal of Optimization Theory and Applications*.**31**(2): 285–288. doi:10.1007/BF00934117. S2CID 119867925.

- Allendoerfer, Carl B. (1974). "Theorems about Differentiable Functions".
*Calculus of Several Variables and Differentiable Manifolds*. New York: Macmillan. pp. 54–88. ISBN 0-02-301840-2. - Binmore, K. G. (1983). "Implicit Functions".
*Calculus*. New York: Cambridge University Press. pp. 198–211. ISBN 0-521-28952-1. - Loomis, Lynn H.; Sternberg, Shlomo (1990).
*Advanced Calculus*(Revised ed.). Boston: Jones and Bartlett. pp. 164–171. ISBN 0-86720-122-3. - Protter, Murray H.; Morrey, Charles B. Jr. (1985). "Implicit Function Theorems. Jacobians".
*Intermediate Calculus*(2nd ed.). New York: Springer. pp. 390–420. ISBN 0-387-96058-9.