Implicit function theorem

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.

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 ± 1 − x 2 .

However, it is possible to represent part of the circle as the graph of a function of one variable. If we let g 1 ( x ) = 1 − x 2 for −1 < x < 1, then the graph of y = g 1 ( x ) provides the upper half of the circle. Similarly, if g 2 ( x ) = − 1 − x 2 , 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 f : R^n+m → R^m be a continuously differentiable function. We think of R^n+m as the Cartesian product Rⁿ × R^m, and we write a point of this product as (x, y) = (x₁, ..., x_n, y₁, ..., y_m). Starting from the given function f, our goal is to construct a function g: Rⁿ → 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) = (a₁, ..., a_n, b₁, ..., b_m) 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 R^m 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 (a₁, ..., a_n, b₁, ..., b_m) to (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_j. 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 theorem

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) = (a₁, ..., a_n, b₁, ..., b_m) with f(a, b) = c, where c ∈ R^m. If the Jacobian matrix J_{f, y}(a, b) = [(∂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 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. 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 f ( x , y ) = x 2 + y 2 − 1 . 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, but writing x as a function of y, that is, x = h ( y ) ; now the graph of the function will be ( h ( y ) , y ) , 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 x 2 + y 2 − 1 and equating to 0:

giving

and

Application: change of coordinates

Suppose we have an m-dimensional space, parametrised by a set of coordinates ( x 1 , … , x m ) . We can introduce a new coordinate system ( x 1 ′ , … , x m ′ ) by supplying m functions h 1 … h m . These functions allow us to calculate the new coordinates ( x 1 ′ , … , x m ′ ) of a point, given the point's old coordinates ( x 1 , … , x m ) using x 1 ′ = h 1 ( x 1 , … , x m ) , … , x m ′ = h m ( x 1 , … , x m ) . One might want to verify if the opposite is possible: given coordinates ( x 1 ′ , … , x m ′ ) , can we 'go back' and calculate the same point's original coordinates ( x 1 , … , x m ) ? The implicit function theorem will provide an answer to this question. The (new and old) coordinates ( x 1 ′ , … , x m ′ , x 1 , … , x m ) are related by f = 0, with

Now the Jacobian matrix of f at a certain point (a, b) [ where a = ( x 1 ′ , … , x m ′ ) , b = ( x 1 , … , x m ) ] 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 ( x 1 , … , x m ) as a function of ( x 1 ′ , … , x m ′ ) 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.

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 ( x 0 , y 0 ) ∈ X × Y , f ( x 0 , y 0 ) = 0 , and y ↦ D f ( x 0 , y 0 ) ( 0 , y ) 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 : U → V such that f(x, g(x)) = 0 and f(x, y) = 0 if and only if y = g(x), for all ( x , y ) ∈ U × V .

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. The following more general form was proven by Kumagai based on an observation by Jittorntrum.

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

where g is a continuous function from B₀ into A₀.