Is it possible to manipulate partial derivatives in this manner?

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In summary, the chain rule can be applied to the new variables x and t to get:##\dfrac{\partial A_y}{\partial ct^\prime (x, ct)} = \dfrac{\partial A_y}{\partial ct} \dfrac{\partial}{\partial ct^\prime} (\gamma ct^\prime + \gamma \beta x^\prime) + \dfrac{\partial A_y}{\partial x} \dfrac{\partial}{\partial ct^\prime} (\gamma x^\prime + \gamma \beta ct^\prime)##
  • #1
SiennaTheGr8
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If I have an expression like ##\dfrac{\partial x}{\partial y}##, and I know that ##x = a + bc## and ##y = f + gh##, then I have:

##\dfrac{\partial (a + bc)}{\partial (f + gh)}##.

I'm wondering what kind of maneuvers are "legal" here. Say that ##b## and ##g## are constants, and the other symbols are variables. Could I, for example, do something like this?

##\dfrac{\partial a + b \, \partial c}{\partial f + g \, \partial h}##

And then even, say, split up the "fraction" and do something like this to each part?:

##\dfrac{\partial a}{\partial f + g \, \partial h} = \dfrac{\partial a \, \partial f - g \, \partial a \, \partial h}{(\partial f)^2 - (g \, \partial h)^2} = \dfrac{\partial a \, \partial f}{\partial f \left( \partial f - \frac{(g \, \partial h)^2}{\partial f} \right) } - \dfrac{g \, \partial a \, \partial h}{\partial h \left( \frac{(\partial f)^2}{\partial h} - g^2 \, \partial h \right) } = \dfrac{\partial a}{\partial f} [+] \dfrac{\partial a}{g \, \partial h}##,

where in the last step I set the only remaining 2nd-order partials to zero.

I'm asking because I had occasion to try using such "methods," and much to my surprise I ended up with the answer I was looking for. I just don't know whether that was coincidence or if this is all actually kosher.

[edit: changed - to +, indicated by square brackets]
 
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  • #2
Your notation is a bit confusing are a, b, f, g, constant and c and h variable? In changing variables you should use the well know "chain rule".
 
  • #3
No, ##b## and ##g## are the only constants.
 
  • #4
If a,c,f, and h are independent variables and y does not depend and a and c and x does not depend on f and h then x is not a function of y.
BTW the last expression should have a plus sign.

Is this question a part of a larger problem?
 
  • #5
gleem said:
BTW the last expression should have a plus sign.

Yes, thank you—fixed.

gleem said:
Is this question a part of a larger problem?

I'm trying to use the transformation of the four-potential ##(\phi, A_x, A_y, A_z)## under a Lorentz boost (along the ##x##-axis) to derive the transformation formulas for the electric and magnetic field components. As an example, I end up with:

##E_y^\prime = - \left( \dfrac{\partial \phi^\prime}{\partial y^\prime} + \dfrac{\partial A_y^\prime}{\partial (ct)^\prime} \right) = - \left( \dfrac{\partial \left( \gamma \phi - \gamma \beta A_x \right)}{\partial y} + \dfrac{\partial A_y}{\partial \left( \gamma ct - \gamma \beta x \right)} \right)##,

which I know is supposed to equal:

##- \gamma \left( \dfrac{\partial \phi}{\partial y} + \dfrac{\partial A_y}{\partial (ct)} \right) - \gamma \beta \left( \dfrac{\partial A_y}{\partial x} - \dfrac{\partial A_x}{\partial y} \right)##

(##\beta## and ##\gamma=(1 - \beta^2)^{-1/2}## are constants).

I didn't know what to do with those ##\partial (\textrm{foo + bar})## objects, but I forged ahead with the kind of "algebra" shown above in my OP (as if they were ordinary differentials), and I seem to have succeeded. I just don't have experience treating "partial differentials" this way, so I don't know if what I did was valid.
 
  • #6
gleem said:
If a,c,f, and h are independent variables and y does not depend and a and c and x does not depend on f and h then x is not a function of y.

So, in fact there are only two new independent variables x and t not four. Have you tried the chain rule yet?
 
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  • #7
Yes, thank you, I think I understand now:

##\dfrac{\partial A_y}{\partial ct^\prime (x, ct)} = \dfrac{\partial A_y}{\partial ct} \dfrac{\partial}{\partial ct^\prime} (\gamma ct^\prime + \gamma \beta x^\prime) + \dfrac{\partial A_y}{\partial x} \dfrac{\partial}{\partial ct^\prime} (\gamma x^\prime + \gamma \beta ct^\prime) = \gamma \dfrac{\partial A_y}{\partial ct} + \gamma \beta \dfrac{\partial A_y}{\partial x}##

Been a while since I've worked with partials...
 

Related to Is it possible to manipulate partial derivatives in this manner?

1. What is the purpose of manipulating partials in a scientific experiment?

The purpose of manipulating partials is to control and observe the effects of specific variables on the overall outcome of an experiment. By manipulating partials, scientists can isolate and study the impact of individual factors on the results, allowing for a more accurate understanding of the relationships between variables.

2. How do scientists manipulate partials in an experiment?

Scientists manipulate partials by changing one variable while keeping all other variables constant. This can be done by adjusting the conditions of the experiment, altering the amount or concentration of a substance, or using control groups to compare the results of the manipulated variable to a baseline measurement.

3. What is the benefit of manipulating partials in an experiment?

The benefit of manipulating partials is that it allows scientists to determine the causal relationship between variables. By controlling all other factors and only changing one variable, scientists can confidently attribute any changes in the outcome to the manipulated partial, providing a more accurate understanding of the experiment's results.

4. Are there any limitations to manipulating partials in an experiment?

Yes, there are limitations to manipulating partials in an experiment. In some cases, it may be difficult or impossible to isolate and manipulate a single partial without affecting other variables. Additionally, some variables may have an indirect or interactive effect on the outcome, making it challenging to determine the exact impact of a partial on the results.

5. Can manipulating partials be used in all scientific experiments?

While manipulating partials is a common technique in many scientific experiments, it may not be suitable for all studies. Some experiments may require the manipulation of multiple variables or may not have easily identifiable or controllable partials. In these cases, alternative methods may be used to achieve the same goals as manipulating partials.

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