# Implicit function theorem

In mathematics, more specifically in multivariable calculus, the **implicit function theorem**[lower-alpha 1] 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}* (

*x*

_{1}, ...,

*x*,

_{n}*y*

_{1}, ...,

*y*) = 0,

_{m}*i*= 1, ...,

*m*(often abbreviated into

*F*(

**x**,

**y**) = 0), the theorem states that, under a mild condition on the partial derivatives (with respect to the

*y*s) at a point, the m variables

_{i}*y*are differentiable functions of the

_{i}*x*in some neighborhood of the point. As these functions can generally not be expressed in closed form, they are

_{j}*implicitly*defined by the equations, and this motivated the name of the theorem.[1]

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.

## History

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]

## First example

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*).

## Definitions

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}*, ...,

*a*,

_{n}*b*

_{1}, ...,

*b*) which satisfies

_{m}*f*(

**a**,

**b**) =

**0**, and we will ask for a

*g*that works near the point (

**a**,

**b**). In other words, we want an open set containing

**a**, an open set containing

**b**, and a function

*g*:

*U*→

*V*such that the graph of

*g*satisfies the relation

*f*=

**0**on

*U*×

*V*, and that no other points within

*U*×

*V*do so. In symbols,

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}*,

*b*

_{1}, ...,

*b*) to (

_{m}**a**,

**b**), the Jacobian matrix is

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

*Y*is the matrix of partial derivatives in the variables

*y*. The implicit function theorem says that if

_{j}*Y*is an invertible matrix, then there are

*U*,

*V*, and

*g*as desired. Writing all the hypotheses together gives the following statement.

## Statement of the theorem

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

*b*

_{1}, …,

*b*) with

_{m}*f*(

**a**,

**b**) =

**0**, where is the zero vector. If the Jacobian matrix (this is the right-hand panel of the Jacobian matrix shown in the previous section):

is invertible, then there exists an open set containing **a** such that there exists a unique continuously differentiable function such that , and .

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

## Proof for 2D case

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:

- 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. ∎

## The circle example

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

## Application: change of coordinates

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 *f* at a certain point (*a*, *b*) [ where ] is given by

where I_{m} denotes the *m* × *m* identity matrix, and *J* is the *m* × *m* matrix of partial derivatives, evaluated at (*a*, *b*). (In the above, these blocks were denoted by X and Y. As it happens, in this particular application of the theorem, neither matrix depends on *a*.) The implicit function theorem now states that we can locally express as a function of if *J* is invertible. Demanding *J* is invertible is equivalent to det *J* ≠ 0, thus we see that we can go back from the primed to the unprimed coordinates if the determinant of the Jacobian *J* is non-zero. This statement is also known as the inverse function theorem.

### Example: polar coordinates

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 *J* = *R*, conversion back to polar coordinates is possible if *R* ≠ 0. So it remains to check the case *R* = 0. It is easy to see that in case *R* = 0, our coordinate transformation is not invertible: at the origin, the value of θ is not well-defined.

## Generalizations

### Banach space version

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 .

### Implicit functions from non-differentiable functions

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 *g* is a continuous function from *B*_{0} into *A*_{0}.

## See also

- 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.

## Notes

- 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.

## References

- 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.

## Further reading

- 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.