# Commutativity of differentiation in a special case

1. Aug 24, 2011

### HotMintea

Problem

I'd like to prove $\frac{d}{dt}[\frac{\partial}{\partial{x}}f(x(t),y(t),t)]=\frac{\partial}{\partial{x}}[\frac{d}{dt}f(x(t),y(t),t)]$.

Attempt
$$\begin{equation*}\begin{split} \frac{d}{dt}[\frac{\partial}{\partial x}f(x(t),y(t),t)]=\frac{d}{dt}\lim_{\epsilon\to 0}\frac{f(x(t)+\epsilon,y(t),t)-f(x(t),y(t),t)}{\epsilon}\\ =\lim_{\delta\to 0}\lim_{\epsilon\to 0} \frac{[f(x(t+\delta)+\epsilon,y(t+\delta),t+\delta)-f(x(t)+\epsilon,y(t),t)]-[f(x(t+\delta),y(t+\delta),t+\delta)-f(x(t),y(t),t)]}{\epsilon\delta}\end{split}\end{equation*}$$

So if my previous steps are correct, I need to show that the 2 limits are commutative, which I have no idea...

2. Aug 24, 2011

### hunt_mat

Did you know you can express the total derivative as a sum of partial derivatives?

3. Aug 24, 2011

### HallsofIvy

Staff Emeritus
Are you required to go all the way back to the definition of derivative? If not, use the chain rule as hunt mat suggests.

4. Aug 24, 2011

### HotMintea

Thanks for the hints!

$\frac{d}{dt}\partial_{x}f(x(t), y(t), t)=(\sum\dot{x}_k \partial_{x_k}+\partial_t)\partial_{x}f=\sum\dot x_k\partial_{x_k}\partial_{x}f+\partial_t\partial_{x}f$.

At this point, I need to show the "equality of mixed partials". I found the proof of $f_{xy}=f_{yx}$ for $f(x, y)$ here: http://www.sju.edu/~pklingsb/clairaut.pdf

My function is f(x(t), y(t), t) so a bit different.

For proving $f_{x(t)\hspace{1mm}t} = f_{t\hspace{1 mm}x(t)}$ for f(x(t), t), I think the proof in the note works if I substitute c=x(a) and d=x(b).

I also think the theorem can be extended to $f(x_1(t), x_2(t), ..., x_n(t), t)$ if I imagine slicing the n dimentional space by a plane parallel to $x_i x_j$ plane (or $x_i\hspace{1 mm}t$ plane).

So I have $f_{x_i x_j}=f_{x_j x_i}$ and $f_{x_i\hspace{1 mm}t}=f_{t\hspace{1 mm}x_i}$ for $f(x_1(t), ..., x_n(t), t)$. Is that okay?

By the way, I don't understand why the author of the note wrote $f_{xy}(x,y)-f_{yx}(x,y)\ge \frac{h}{2}$. Is that arbitrary? Can I use h/3 instead for example?