Rescaling Variables HW: Get \frac{\partial \hat{f}(z,x)}{\partial z}

• Nusc
In summary, the conversation discusses a given expression and the desired outcome, and includes equations and an attempt at a solution. The focus is on simplifying a derivative and verifying a change of variable. The conversation also addresses a factor of x2 that appears in the equations and seeks clarification on where it comes from.
Nusc

Homework Statement

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

I want to obtain:

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

Homework Equations

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

The Attempt at a Solution

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

is this possible?

Where does the x2 come from?

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

mfb said:
Where does the x2 come from?Assuming H and r0 are constant, you can simplify the derivative.
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?

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

What do you mean?

I am trying to show

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

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)##

What does "Rescaling Variables" mean?

"Rescaling Variables" refers to the process of changing the scale or units of measurement of a variable in a dataset. This can be done to make the data more easily interpretable or to prepare it for analysis using a particular statistical method.

Why is it important to rescale variables?

Rescaling variables is important because it allows for more accurate and meaningful comparison between different variables. It also ensures that the statistical analysis performed on the data is not affected by the differences in scale or units of measurement.

What is the purpose of the partial derivative in the expression "Get \frac{\partial \hat{f}(z,x)}{\partial z}"?

The partial derivative in this expression represents the rate of change of the function \hat{f}(z,x) with respect to the variable z. It allows for the calculation of the sensitivity of the function to changes in z, which is useful for understanding the relationship between the variables in the function.

How do you compute the partial derivative in the expression "Get \frac{\partial \hat{f}(z,x)}{\partial z}"?

The partial derivative can be computed using the rules of differentiation, where all other variables in the function \hat{f}(z,x) are treated as constants. This results in a new function that represents the rate of change of the original function with respect to the variable z.

Can rescaling variables affect the outcome of the partial derivative in the expression "Get \frac{\partial \hat{f}(z,x)}{\partial z}"?

Yes, rescaling variables can affect the outcome of the partial derivative. This is because the sensitivity of the function to changes in z may be different when the variable is rescaled. Thus, it is important to carefully consider the scale of the variables when computing partial derivatives.

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