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Implicit function theorem

In mathematics, more specifically in multivariable calculus, the implicit function theorem[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 fi(x1, ..., xn, y1, ..., ym) = 0, 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 yis) at a point, the m variables yi are differentiable functions of the xj in some neighborhood of the point. As these functions can generally not be expressed in closed form, they are implicitly defined by the equations, and this motivated the name of the theorem.[2]

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.

Contents

HistoryEdit

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.[3]

First exampleEdit

 
The unit circle can be specified as the level curve f(x, y) = 1 of the function   Around point A, y can be expressed as a function y(x). In this example this function can be written explicitly as   in many cases no such explicit expression exists, but one can still refer to the implicit function y(x). No such function exists around point B.

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).

DefinitionsEdit

Let f : Rn+mRm be a continuously differentiable function. We think of Rn+m as the Cartesian product Rn × Rm, and we write a point of this product as (x, y) = (x1, ..., xn, y1, ..., ym). Starting from the given function f, our goal is to construct a function g: RnRm 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) = (a1, ..., an, b1, ..., bm) which satisfies 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 Rn containing a, an open set V of Rm containing b, and a function g : UV 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 (a1, ..., an, b1, ..., bm) to (a, b), the Jacobian matrix is

 

where X is the matrix of partial derivatives in the variables xi and Y is the matrix of partial derivatives in the variables yj. The implicit function theorem says that if 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 theoremEdit

Let f: Rn+mRm be a continuously differentiable function, and let Rn+m have coordinates (x, y). Fix a point (a, b) = (a1, ..., an, b1, ..., bm) with f(a, b) = 0, where 0Rm is the zero vector. If the Jacobian matrix Jf, y(a, b) = [(∂fi / ∂yj)(a, b)] (this is the right-hand panel of the Jacobian matrix shown in the previous section) is invertible, then there exists an open set U of Rn containing a such that there exists a unique continuously differentiable function g: URm such that

 

and

 

Moreover, the partial derivatives of g in U are given by the matrix product[4]

 

Higher derivativesEdit

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. [5] In the analytic case, this is called the analytic implicit function theorem.

The circle exampleEdit

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 2b; 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, 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 coordinatesEdit

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 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 Im 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 coordinatesEdit

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.

GeneralizationsEdit

Banach space versionEdit

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

Let X, Y, Z be Banach spaces. Let the mapping f : X × YZ be continuously Fréchet differentiable. If  ,  , and   is a Banach space isomorphism from Y onto Z, then there exist neighbourhoods U of x0 and V of y0 and a Fréchet differentiable function g : UV 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 functionsEdit

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.[7] The following more general form was proven by Kumagai[8] based on an observation by Jittorntrum.[9]

Consider a continuous function   such that  . If there exist open neighbourhoods   and   of x0 and y0, respectively, such that, for all y in B,   is locally one-to-one then there exist open neighbourhoods   and   of x0 and y0, such that, for all  , the equation f(x, y) = 0 has a unique solution

 ,

where g is a continuous function from B0 into A0.

See alsoEdit

NotesEdit

  1. ^ 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.
  2. ^ See Chiang 1984, pp. 204–206.
  3. ^ Krantz, Steven; Parks, Harold (2003). The Implicit Function Theorem. Modern Birkhauser Classics. Birkhauser. ISBN 0-8176-4285-4.
  4. ^ de Oliveira 2013, pp. 214–216.
  5. ^ Fritzsche & Grauert 2002, p. 34.
  6. ^ See Lang 1999, pp. 15–21 and Edwards 1994, pp. 417–418.
  7. ^ See Kudryavtsev 2001.
  8. ^ See Kumagai 1980, pp. 285–288.
  9. ^ See Jittorntrum 1978, pp. 575–577.

ReferencesEdit

  • Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (3rd ed.). McGraw-Hill.
  • de Oliveira, Oswaldo (2013). "The Implicit and Inverse Function Theorems: Easy Proofs". Real Anal. Exchange. 39 (1).
  • Edwards, Charles Henry (1994) [1973]. Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. ISBN 978-0-486-68336-2.
  • Jittorntrum, K. (1978). "An Implicit Function Theorem". Journal of Optimization Theory and Applications. 25 (4). doi:10.1007/BF00933522.
  • Kumagai, S. (1980). "An implicit function theorem: Comment". Journal of Optimization Theory and Applications. 31 (2). doi:10.1007/BF00934117.