The Implicit Function Theorem
A specific solution of this equation is a point in D, and the collection of all such solution points forms an object in D.
There is an incredible variety of objects that can arise in this way, making a conclusive study of them next to impossible. However, in most situations which we encounter these level sets will turn out to be manifolds, and manifolds can be studied effectively with the tools of calculus. The foundation for such an study is provided by the implicit function theorem, formulated below. As a preparation, we discuss consequences of the relationship between the domain dimension n and the target dimension m of the function f.
General information about level sets In most cases, we already get information about a level set of a function by comparing the number of variables (i.e. the dimension of D) to the number of equations in our system (i.e. the dimension m of the target space):
- n < m In this situation, there are more equations than variables. Consequently, the given equation will normally have no solutions. - One calls such a system overdetermined.
- n = m This time, the number of equations is equal to the number of variables. Consequently, if the given equation has a solution x, then there will normally be no other solution near x. - One says that such a system has solutions which are locally unique, and we call such an equation exactly determined. This is the content of the inverse function theorem.
- n > m This is the most interesting situation because, normally, such a system of equations will have infinitely many solutions. Moreover these solutions will normally form a manifold of dimension n-m. Infinitely many solutions arise because, in the present situation, the number of variables is greater than the number of equations. Such a system of equations is also called underdetermined. This is the content of the implicit function theorem..
The general implicit function theorem Let us now turn to the general situation where f takes values in an m-dimensional space. This means that we now have m equations in n unknowns.
The inverse function theorem The general implicit function theorem has a special case which is frequently useful: If the dimensions of the domain and the target space are equal, then it gives the following:
Examples
interactive explorer of level sets of cubic polynomials- Implicit Function Theorem - case n = 2
- Implicit Function Theorem - case n = 3
- Implicit differentiation
- Inverse Function Theorem - case n = 2
- Inverse Function Theorem - case n = 3
- Inverse Function Theorem - local invertibility of the polar coordinate transformation
- Inverse Function Theorem - local invertibility of the cylindrical coordinate transformation
- Inverse Function Theorem - local invertibility of the spherical coordinate transformation
- Finding the tangent plane to a level set.
- Finding implicit derivatives.
- Finding the derivative of the inverse of a function.
Move on to extrema.