Implicit function theorem
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 generally cannot 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.
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) = x2 + y2, then the equation f(x, y) = 1 cuts out the unit circle as the level set {(x, y) | f(x, y) = 1}.
However, it is possible to represent part of the circle as the graph of a function of one variable.
, then the graph of y = g2(x) gives the lower half of the circle.
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).
To state the implicit function theorem, we need the Jacobian matrix of
is continuously differentiable and, denoting the left-hand panel of the Jacobian matrix shown in the previous section as:
Now we are looking for a solution to this ODE in an open interval around the point
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.
, since where b = 0 we have a = 1, and the conditions to locally express the function in this form are satisfied.
Suppose we have an m-dimensional space, parametrised by a set of 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 implicit function theorem now states that we can locally express
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.
As a simple application of the above, consider the plane, parametrised by polar coordinates (R, 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?
Since det J = R, conversion back to polar coordinates is possible if 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.
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
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.
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.
