Implicit function theorem


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 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 each yi ) 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.[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 edit

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 edit

The unit circle can be specified as the level curve f(x, y) = 1 of the function f(x, y) = x2 + y2. 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 f(x, y) = x2 + y2, 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 = g1(x) provides the upper half of the circle. Similarly, if  , then the graph of y = g2(x) gives the lower half of the circle.

The purpose of the implicit function theorem is to tell us that functions like g1(x) and g2(x) almost always exist, even in situations where we cannot write down explicit formulas. It guarantees that g1(x) and g2(x) are differentiable, and it even works in situations where we do not have a formula for f(x, y).

Definitions edit

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.

Statement of the theorem edit

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]

Higher derivatives edit

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.

Proof for 2D case edit

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.

The circle example edit

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 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:


Application: change of coordinates edit

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 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 coordinates edit

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 edit

Banach space version edit

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 × 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 functions edit

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 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.

Collapsing manifolds edit

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]

See also edit

Notes edit

  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.

References edit

  1. ^ Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (3rd ed.). McGraw-Hill. pp. 204–206. ISBN 0-07-010813-7.
  2. ^ Krantz, Steven; Parks, Harold (2003). The Implicit Function Theorem. Modern Birkhauser Classics. Birkhauser. ISBN 0-8176-4285-4.
  3. ^ de Oliveira, Oswaldo (2013). "The Implicit and Inverse Function Theorems: Easy Proofs". Real Anal. Exchange. 39 (1): 214–216. arXiv:1212.2066. doi:10.14321/realanalexch.39.1.0207. S2CID 118792515.
  4. ^ Fritzsche, K.; Grauert, H. (2002). From Holomorphic Functions to Complex Manifolds. Springer. p. 34. ISBN 9780387953953.
  5. ^ Lang, Serge (1999). Fundamentals of Differential Geometry. Graduate Texts in Mathematics. New York: Springer. pp. 15–21. ISBN 0-387-98593-X.
  6. ^ Edwards, Charles Henry (1994) [1973]. Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. pp. 417–418. ISBN 0-486-68336-2.
  7. ^ Kudryavtsev, Lev Dmitrievich (2001) [1994], "Implicit function", Encyclopedia of Mathematics, EMS Press
  8. ^ Jittorntrum, K. (1978). "An Implicit Function Theorem". Journal of Optimization Theory and Applications. 25 (4): 575–577. doi:10.1007/BF00933522. S2CID 121647783.
  9. ^ Kumagai, S. (1980). "An implicit function theorem: Comment". Journal of Optimization Theory and Applications. 31 (2): 285–288. doi:10.1007/BF00934117. S2CID 119867925.
  10. ^ Cao, Jianguo; Ge, Jian (2011). "A simple proof of Perelman's collapsing theorem for 3-manifolds". J. Geom. Anal. 21 (4): 807–869. arXiv:1003.2215. doi:10.1007/s12220-010-9169-5. S2CID 514106.

Further reading edit

  • 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.