# Commutative property of partial derivatives

1. Dec 20, 2012

### yayyyymath

Hi everyone,

I am working on simplifying a differential equation, and I am trying to figure out if a simplification is valid. Specifically, I'm trying to determine if:

$\frac{\del^2 p(x)}{\del p(x) \del x} = \frac{\del^2 p(x)}{\del x \del p(x)}$

where p(x) is a function of x. Both p(x) and x are assumed to be continuous.

From what I found on wikipedia at http://en.wikipedia.org/wiki/Partial_derivatives (at the bottom of the subsection "Formal definition"), it appears that all partial derivatives have this commutative property if the functions are continuous.

However, a reputable colleague of mine said that this is not the case here. He said that the commutative property doesn't hold since p(x) is being differentiated with respect to itself. He did not have time to explain it thoroughly or give a proof.

Can anyone give a proof (or a strong argument) showing that the commutative property is/is not valid here?

Thank you very much for your help!

2. Dec 20, 2012

### Vorde

$$\frac{\partial^2 p(x)}{\partial p(x) \partial x} = \frac{\partial^2 p(x)}{\partial x \partial p(x)}$$

I'm not really equipped to answer your question with a rigorous proof, but to expedite other people helping you I fixed your latex.

3. Dec 20, 2012

### yayyyymath

Thanks Vorde

4. Dec 20, 2012

### symbolipoint

You want your tags to say "tex", not "latex". The typesetting will then work.

5. Dec 21, 2012

### Stephen Tashi

I think you should state the un-simplified equation. I have no idea what the expression $\frac{\partial^2 p(x)}{ \partial p(x) \partial x}$ would mean. How did you end up with an expression like that?

6. Dec 21, 2012

### amiras

Equation on the right:

$$\frac{\partial^2 p(x)}{\partial x \partial p(x)}=\frac{\partial}{\partial x}(\frac{\partial p(x)}{\partial p(x)})=\frac{\partial}{\partial x}1=0$$

Maybe you meant something like this:

$$\frac{\partial^2 f(p(x))}{\partial x \partial p(x)}=\frac{\partial^2 f(p(x))}{\partial p(x) \partial x }$$

7. Dec 29, 2012

### yayyyymath

Oops, sorry for the silence...I assumed I would get an email for every reply and thought the thread died, but I guess not. I'm new to this, so I apologize. Anyway, thank you all very much for your help. I still haven't found a good solution, so additional insight would be appreciated.

The expression comes from the optimality conditions (a set of PDEs analogous to the Euler-Lagrange equations) derived for a specific PDE constrained optimization problem which is used for an optimal control application. It's kind of complicated and I think it's irrelevant to what I'm trying to figure out here, so I didn't bother to include the details.

In one of the optimality conditions, I end up with the (unsimplified) term,
$$\frac{\del^2 p(x)}{\del p(x) \del x}$$
What I'm trying to figure out is if this term will equal 0, which would allow me to cancel out that term and simplify the equation. As amiras noted, this would be the case if the order of the partial derivatives can indeed be switched to get,
$$\frac{\del^2 p(x)}{\del x \del p(x)}$$
However, I'm not sure if switching the derivatives like that is valid since p(x) is being differentiated by itself. I'm looking for a proof or argument that would say whether switching the derivatives is/is not valid.

I know it's a strange expression, but it's definitely correct. I did not mean to write,
$$\frac{\del^2 f[p(x)]}{\del p(x) \del x}$$

Hopefully that makes more sense. Thanks again!

8. Dec 29, 2012

### yayyyymath

Sorry, still trying to figure out the latex syntax used here...I guess you have to use the another command for del. Anyway, here are the corrected equations from my post above:

$$\frac{∂^2 p(x)}{∂p(x) ∂x}$$

$$\frac{∂^2 p(x)}{∂x ∂p(x)}$$

$$\frac{∂^2 f[p(x)]}{∂p(x) ∂x}$$

9. Dec 29, 2012

### yayyyymath

Actually, is there a post or resource available that explains the correct typesetting format? Is it just standard tex syntax (if such a thing exists)? I am only familiar with latex. Thanks.