Revision of How to use implicit function theorem to prove smoothness from Fri, 24/04/2009 - 23:44

Quick description

Suppose you have a function ${\mathbb R}^n\rightarrow {\mathbb R}^m$ and you would like to see that $f$ is $C^r$ . One way to show this is to find a $C^r$ function ${\mathbb R}^n \times {\mathbb R}^m \rightarrow {\mathbb R}^m$ such that $\det(D_y F) \neq 0$ and

$F(x,f(x)) = c.$

The implicit function theorem will imply that $f$ is $C^r$ .

Prerequisites

Calculus

Example 1

General discussion

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Comments

are there other examples of the argument in example one?

Sat, 25/04/2009 - 07:11 — devin

The above example has a more sophisticated method than the one I tried to describe in the quick description section. In the generalization of the method of example 1, there would be a function $f$ and the goal would be to improve our understanding of its smoothness. To that end, we use $f$ itself to define another function

${\mathbb R}^n \times {\mathbb R}^{m\times n} \rightarrow {\mathbb R}^{m\times n}$

such that $\Phi(x,Df(x) ) = c$ , $\det(D_y\Phi) \neq 0$ and $\Phi$ is as smooth as $f$ . Now the implicit function theorem says $Df$ is as smooth as $\Phi$ and hence $f$ . The argument is continued as many steps as possible. Does anyone know of another application of this argument, or an argument similar to this?

Notation

Thu, 07/05/2009 - 00:41 — Brendan Murphy

The notation " $D_y\,F$ " isn't very clear. My first thought was: "partial derivative with respect to $y$ ". After this, I figured you meant the operator on ${\mathbb R}^m$ given by the decomposition of $D\, F$ (the Jacobian of $F$ ), since this is the condition required by the theorem. Still, the notation gives rise the possibility $D_y\,\Phi=\Phi_y$ .

What does the notation $x^*$ mean in equation 2?

Instead of using $L_y$ , why not use $L_{\dot x}$ ?

Notation

Sat, 09/05/2009 - 15:18 — devin

Thanks for the comment. I like the $D_y$ notation and frequently use it because it consisely states everything related to the operation (we are taking a derivative with respect to a variable.) The one you suggest ( $\Phi_y$ ) is also good I think. As for $L_{\dot{x}}$ . This is common notation in many books, including Fleming and Rishel. I don't prefer it because I think it is confusing to use the same symbol to mean two entirely different things on the same page. In this case, if we use the $L_{\dot{x}}$ notation, $\dot{x}$ will mean the derivative of the function $x$ with respect to time and also the name of a free variable.

$x^*$ is a is a free variable representing a function satisfying the Euler-Lagrange equation. It probably is not a good choice of notation because as you point out upon reading it one thinks it must be related to the $x$ in its context. I changed it to $x$ . This I think causes an abuse of notation, but hopefully not a confusing one.