# Implicit function theorem

In multivariable calculus, the **implicit function theorem**, also known, especially in Italy, as **Dini's theorem**, is a tool that allows relations to be converted to functions of several real variables. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is 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.

The theorem states that if the equation *R*(*x*, *y*) = 0 satisfies some mild conditions on its partial derivatives, then one can in principle (though not necessarily with an analytic expression) express *y* in terms of *x* as *f*(*x*), at least in some disk. Then this implicit function *f*(*x*),^{[1]}^{:204-206} implied by *R*(*x*, *y*)=0, is such that geometrically the locus defined by *R*(*x*, *y*) = 0 will coincide locally (that is in that disk) with the graph of *f*.

## Contents

## First example

If we define the function , 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 provides the upper half of the circle. Similarly, if , then the graph of gives the lower half of the circle.

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

## Statement of the theorem

Let *f* : **R**^{n+m} → **R**^{m} be a continuously differentiable function. We think of **R**^{n+m} as the Cartesian product **R**^{n} × **R**^{m}, and we write a point of this product as (**x**, **y**) = (*x _{1}*, ...,

*x*,

_{n}*y*, ...,

_{1}*y*). Starting from the given function

_{m}*f*, our goal is to construct a function

*g*:

**R**

^{n}→

**R**

^{m}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

*U*of

**R**

^{n}containing

**a**, an open set

*V*of

**R**

^{m}containing

**b**, and a function

*g*:

*U*→

*V*such that the graph of

*g*satisfies the relation

*f*= 0 on

*U*×

*V*. 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.

- Let
*f*:**R**^{n+m}→**R**^{m}be a continuously differentiable function, and let**R**^{n+m}have coordinates (**x**,**y**). Fix a point (**a**,**b**) = (*a*_{1}, ...,*a*,_{n}*b*_{1}, ...,*b*) with_{m}*f*(**a**,**b**) =**c**, where**c**∈**R**^{m}. If the matrix [(∂*f*/∂_{i}*y*)(_{j}**a**,**b**)] is invertible, then there exists an open set*U*containing**a**, an open set*V*containing**b**, and a unique continuously differentiable function*g*:*U*→*V*such that

### Regularity

It can be proven that whenever we have the additional hypothesis that *f* is continuously differentiable up to *k* times inside *U* × *V*, then the same holds true for the explicit function *g* inside *U* and

Similarly, if *f* is analytic inside *U* × *V*, then the same holds true for the explicit function *g* inside *U*.^{[2]} This generalization is called the **analytic implicit function theorem**.

## 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 iff *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, but 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 . These functions allow 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 1_{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.

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 it holds in one dimension.^{[3]} The following more general form was proven by Kumagai^{[4]} based on an observation by Jittorntrum.^{[5]}

Consider a continuous function such that . **If** there exist open neighbourhoods and of *x*_{0} and *y*_{0}, respectively, such that, for all *y* in *B*, is locally one-to-one **then** 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

- 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

- ↑ Chiang, Alpha C.
*Fundamental Methods of Mathematical Economics*, McGraw-Hill, third edition, 1984 - ↑ K. Fritzsche, H. Grauert (2002), "From Holomorphic Functions to Complex Manifolds", Springer-Verlag, page 34.
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## References

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