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On converting relations to functions of several real variables From Wikipedia, the free encyclopedia

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}* (

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 that functions like *g*_{1}(*x*) and *g*_{2}(*x*) almost always exist, 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 , our goal is to construct a function whose graph is precisely the set of all such that .

As noted above, this may not always be possible. We will therefore fix a point which satisfies , and we will ask for a that works near the point . In other words, we want an open set containing , an open set containing , and a function such that the graph of satisfies the relation on , and that no other points within do so. In symbols,

To state the implicit function theorem, we need the Jacobian matrix of , which is the matrix of the partial derivatives of . Abbreviating to , the Jacobian matrix is

where is the matrix of partial derivatives in the variables and is the matrix of partial derivatives in the variables . The implicit function theorem says that if is an invertible matrix, then there are , , and as desired. Writing all the hypotheses together gives the following statement.

Let be a continuously differentiable function, and let have coordinates . Fix a point with , 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 such that there exists a unique function such that , and . Moreover, is continuously differentiable and, denoting the left-hand panel of the Jacobian matrix shown in the previous section as:
the Jacobian matrix of partial derivatives of in is given by the matrix product:^{[3]}

If, moreover, is analytic or continuously differentiable times in a neighborhood of , then one may choose in order that the same holds true for inside . ^{[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 in a neighbourhood of the point 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 *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.

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.

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 . 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}.

Perelman’s collapsing theorem for 3-manifolds, the capstone of his proof of Thurston's geometrization conjecture, can be understood as an extension of the implicit function theorem.^{[10]}

- 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.

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