Implicit function theorem

The set of the implicit function is an important sentence in the analysis. It contains a relatively simple criterion when an implicit equation or a system of equations ( local) can be clearly resolved.

The sentence says when implicitly by an equation or a system of equations defines a function, applies. Such a function can be found locally in an environment of a place in general.

Next important is the implicit differentiation, so that the derivation can be determined (as a function of and ) without explicit knowledge of.

• 2.1 statement
• 2.2 Example
• 2.3 evidence approach
• 2.4 Summary

Definition

An implicitly defined function (short implicit function ) is a function that is not represented by an explicit assignment rule, but the function values ​​are implicitly defined by an equation. This is a vector-valued function that contains the same number of individual functions, such as components having. Is fixed, then a system of equations results in with as many equations as unknowns. The theorem on implicit function describes conditions under which the following statement is true:

This statement of the theorem now allows to define a function that assigns to each parameter vector just the solution vector, so that this function satisfies the equation on its domain of definition. The set of the implicit function also ensures that this mapping to under certain conditions and limitations, and is well-defined, in particular, that it is unique.

Example

Substituting, the equation describes the unit circle in the plane. The unit circle can not be written as a graph of a function as to each of the interval, there are two ways, viz.

However, it is possible to display part of the circle as a function of the graph. The upper semi-circle you get as graph of the function

The bottom as a graph of

The set of the implicit function gives criteria for the existence of features such as or. He also guaranteed that these functions are differentiable.

Set of the implicit function

Statement

Be open and quantities and

A continuously differentiable mapping. The Jacobian matrix

Then consists of two sub-arrays

And

The latter is square.

The set of the implicit function now states:

Satisfies the equation and is the second submatrix in point invertible, then there exist open neighborhoods and and a unique continuously differentiable mapping

With so that

Applies to all.

Example

Turning now this set on the first example given equation of the circle: this, the partial derivatives are to be considered according to the variables. (In this case, so this results in a matrix, or simply a real function ): The partial derivative of the function to be implicit. This derivation has an inverse if and only if is. This one concludes with the sentence, that this equation is solvable locally by when. The case occurs only at the locations or on. So these are the problem areas. In fact, one can see that the formula branches in all of these problem areas in a positive and negative solution. In all other respects, the resolution is locally unique.

Evidence approach

The classical approach is to look for the solution of the equation the initial value problem of ordinary differential equation

There is in inverted, this is also in a small environment of the case, ie for small vectors exists, the differential equation and its solution for everyone. The solution of the implicit equation is now by

Given the above properties of this solution are derived from the properties of the solutions of parameter-dependent differential equations.

The modern approach formulated the system of equations by means of the simplified Newton's method as a fixed point problem and apply it to the fixed point theorem of Banach. For the corresponding fixed point mapping is the inverse of the submatrix of the Jacobian matrix of the given solution point is formed. At the Figure

One can now show that for parameter vectors close to this figure on a neighborhood of contractionary. This follows from the fact that is continuously differentiable and is valid.

Summary

The advantage of the set is that you do not need to know explicitly the function, and can still deduce a statement about their existence and uniqueness. In many cases, the equation also not solvable with formulas, but only with numerical methods. It is interesting that the convergence of such methods are usually the same or similar condition as the set of the implicit function requires ( the invertibility of the matrix of derivations ).

Another valuable conclusion of the theorem is that the function is differentiable if it is what is required when applying the implicit function theorem. The derivative can even be specified explicitly by deriving the equation for the multidimensional chain rule:

And then dissolving:

A similar reasoning applies to higher derivatives. If we replace the condition " continuously differentiable " with " is - times continuously differentiable " (or as often or analytically ), we can conclude that - times differentiable (or as often or analytical) is.

Set of the inverse mapping

A useful corollary to the theorem of the implicit function is the set of the inverse map. He gives an answer to the question whether one can find a (local) inverse function:

Be open and

A continuously differentiable mapping. Be and. The Jacobian matrix is invertible. Then there is an open neighborhood of and an open neighborhood of b, so that the amount of bijective maps and the inverse function

Is continuously differentiable, or in short: is a diffeomorphism. It is