Multivariable Calculus - Basic Approach      Implicit Function Theorem - Explanation
How to start applying the implicit function theorem  In order to apply the implicit function theorem we need to find at least one point a which solves the given equation f(x)=h. Finding such a point is often a matter of "eagle-eyed inspection of the given equation". Often one can succeed by setting all but one variable equal to 0.
 
From one solution point to a whole manifold of solutions  Once we have a single solution, the derivative of the function in question allows us to determine the existence of more solutions. The underlying principle is the following:
  1. The derivative is a linear function from n-space to the real numbers. It is represented by a matrix consisting of a single row with n entries. In our case, this row is given by the gradient vector of f at a.
  2. If this linear function is onto, then each its of level sets is a hyperplane, and this hyperplane is perpendicular to the gradient vector of f at a.
  3. The key thing is now: Near a the level sets of f are like the level sets of its derivative - just up to some bending. For example, if n = 2, a level set of the derivative is a line. So the level set of f near a is a bent piece of that line; i.e. a curve. If n = 3, a level set of the derivative is a plane. So the level set of f near a is a bent piece of that plane; i.e. a surface.
What's so special about the n-th partial derivative of f?  The requirement that the n-th partial derivative of f at a be non-zero is only to simplify the exposition. All that is needed is that at least one partial derivative of f does not vanish. For example, to apply the theorem if the i-th partial derivative is not zero, simply interchange the i-th and the n-th variable.
 
On the function g  The function g appearing in the formulation of the implicit function theorem is said to be implicitly defined by the equation f(x)=h. This reflects the fact that, near a, every solution of this equation is of the form (x1,...,xn-1,g(x1,...,xn-1)). Thus all our interest should be directed towards writing down some formula which helps us to compute the values of g. Unfortunately, it is in the nature of non-linear equations that we will rarely succeed in doing so. If we do get lucky enough to succeed, we have then an explicit expression for g.
 
In the majority of cases, however, the implicit function theorem merely gives us the existence of g, and we have absolutely no additional information on how to compute it. The insight that we do gain is of qualitative nature: Near a the solutions of the given equation form an (n-1)-dimensional manifold.
 
Here is another way visualizing the level set of f at level h near a: We just said that all such points are of the form (x1,...,xn-n-1,g(x1,...,xn-1)); i.e. they belong to the graph of g.
 
The derivative of g  We have just learned that we can only rarely hope to write down an explicit formula for the implicitly defined function g. Therefore it is all the more remarkable that we can always compute the partial derivatives of g at a. This is accomplished by a process called implicit differentiation.

Return to the main document on the implicit function theorem.