Let’s consider the function . The collection of points so that defines a curve in the plane: the unit circle. Unfortunately, this relation is not a function. Neither is defined as a function of , nor is defined as a function of by this curve. However, if we consider a point on the curve (that is, with ), then near this point we usually do have a graph of as a function of (except for a few isolated points). That is, as we move near the value then we have to adjust to maintain the relation . There is some function defined “implicitly” in a neighborhood of satisfying the relation .
We want to generalize this situation. Given a system of functions of variables
we consider the collection of points in -dimensional space satisfying .
If this were a linear system, the rank-nullity theorem would tell us that our solution space is (generically) dimensional. Indeed, we could use Gauss-Jordan elimination to put the system into reduced row echelon form, and (usually) find the resulting matrix starting with an identity matrix, like
This makes finding solutions to the system easy. We put our variables into a column vector and write
and from this we find
Thus we can use the variables as parameters on the space of solutions, and define each of the as a function of the .
But in general we don’t have a linear system. Still, we want to know some circumstances under which we can do something similar and write each of the as a function of the other variables , at least near some known point .
The key observation is that we can perform the Gauss-Jordan elimination above and get a matrix with rank if and only if the leading matrix is invertible. And this is generalized to asking that some Jacobian determinant of our system of functions is nonzero.
Specifically, let’s assume that all of the are continuously differentiable on some region in -dimensional space, and that is some point in where , and at which the determinant
where both indices and run from to to make a square matrix. Then I assert that there is some -dimensional neighborhood of and a uniquely defined, continuously differentiable, vector-valued function so that and .
That is, near we can use the variables as parameters on the space of solutions to our system of equations. Near this point, the solution set looks like the graph of the function , which is implicitly defined by the need to stay on the solution set as we vary . This is the implicit function theorem, and we will prove it next time.