# Rescaling Variables

1. May 19, 2015

### Nusc

1. The problem statement, all variables and given/known data

Suppose I have the following expression:
$$(v \frac {\partial}{\partial r} ) f(r,v)$$

I want to obtain:

$$\frac {\partial \hat{f}(z,x)}{\partial z}$$

2. Relevant equations

$$x \rightarrow v/v0$$
$$z \rightarrow (r-r0)/H$$
$$H \rightarrow \frac{k_{b} T} {m g }$$
$$\hat{f}(z,x) = x^2 f(r,v)$$

3. The attempt at a solution

$$x v0 \frac {\partial}{\partial (zH+r0)} \frac{1}{x^2} \hat{f}(z,x)$$

is this possible?

2. May 19, 2015

### Staff: Mentor

Where does the x2 come from?

Assuming H and r0 are constant, you can simplify the derivative.

3. May 19, 2015

### Nusc

It was just a given a change of variable. I am trying to verify it.
$$f(r,v) = f( zH+r0, xv0)$$

Can you show me explicitly?

4. May 19, 2015

### Staff: Mentor

The equations look good apart from that x2. Where does it come from?

5. May 20, 2015

### Nusc

What do you mean?

I am trying to show

$$(v \frac {\partial}{\partial r} ) f(r,v) \rightarrow \frac {\partial \hat{f}(z,x)}{\partial z}$$

6. May 21, 2015

### Staff: Mentor

There is an x2 in your equations. Why? What did you calculate that let this factor appear in the equation?

Edit: This one: $\hat{f}(z,x) = x^2 f(r,v)$