§ The implicit and inverse function theorem
I keep forgetting the precise conditions of these two theorems. So here
I'm writing it down as a reference for myself.
§ Implicit function: Relation to function
If we have a function , we can write this as .
This can be taken as an implicit function . We then
want to recover the explicit version of such that
. That is, we recover the original explicit formulation
of in a way that satisfies .
- Example 1: to .
§ The 1D linear equation case
In the simplest possible case, assume the relationship between and
is a linear one, given implicitly. So we have .
Solving for , one arrives at: .
- Note that the solution exists iff .
- Also note that the the existence of the solution is equivalent to asking that .
§ The circle case
In the circle case, we have . We can write
. These are two solutions, not one, and hence
a relation, not a function.
- We can however build two functions by taking two parts. ; .
- In this case, we have , which changes sign for the two solutions. If , then . Similarly for the negative case.
§ Assuming that a solution for exists
Let us say we wish to solve . Let's assume
that we have a solution around the point . Then we
must have: . Differentiating by ,
we get: . This gives
us the condition on the derivative:
The above solution exists if . This quantity is again
§ Application to economics
- We have two inputs which are purchaed as units of input 1, units of input .
- The price of the first input is . That of the second input is .
- We produce an output which is sold at price .
- For a given units of input, we can produce units of output where . The Coob-douglas function .
- The profit is going to be .
- We want to select to maximize profits.
- Assume we are at break-even: .
- The implicit function theorem allows us to understand how any variable changes with respect to any other variable. It tells us that locally, for example, that the number of units of the first input we buy ( ) is a function of the price . Moreover, we can show that it's a decreasing function of the price.
§ Inverse function: Function to Bijection
- Given a differentiable function , at a point , we will have a continuous inverse if the derivative is locally invertible.
- The intuition is that we can approximate the original function with a linear function. . Now since is locally invertible, we can solve for . implies that . This gives us the pre-image .
One perspective we can adopt is that of Newton's method. Recall that Newton's
method allows us to find for a fixed such that . It follows
the exact same process!
- The fact that is non-zero is the key property. This generalizes in multiple dimensions to saying that is invertible.
- We start with some .
- We then find the tangent .
- We draw the tangent at the point as .
- To find the we set .
- This gives us .
- Immediately generalizing, we get .
§ Idea of proof of implicit function theorem (from first principles)
- Let , point be the point where we wish to implicitize.
- To apply implicit fn theorem, take . Say WLOG that , since is assumed to be continuously differentiable.
- Since is continuous and positive at , it's positive in a nbhd of by continuity.
- Consider as a single variable function of . Its derivative with respect to is positive; it's an increasing function in terms of .
- Since and is an increasing function of , we must have two values such that is positive, and is negative.
- Since is zero and continuous, we have that for all near , that and . We have released into a wild in the neighbourhood!
- Now pick some near . Since we have that and , there exists a unique (by MEAN VALUE THEOREM) that
- Since the is unique, we found a function: .
- We are not done. We need to prove the formula for where is the implicit mapping.
- by defn of . Apply chain rule!
§ Idea of proof of inverse function theorem (from implicit function theorem)
- Given the implicit function, say we want to locally invert .
- Pick the implicit function . If we consider the level set , the implicit function theorem grants us a such that . That is, we get , or .
- To show that it's also a right inverse, consider . Since , we have that .
- Hence, and are both left and right inverses, and hence bijections.
§ Idea of proof of implicit function theorem (from inverse function theorem)
- We are given a function . We wish to find a formula such that .
- The idea is to consider a function .
- Since this is invertible, we get a local inverse function such that . That is, .
- Now, for a given , set . This gives us a such that . That is, we get a such that and this a bijection from to since is a bijection.
§ Idea of proof of inverse function theorem (from newton iterates)
We know that where is the jacobian,
is error term. Upto first order, this works. We take iterates of this
process to get the full inverse.