Why use primed coordinates for this

[aaaa202]

Griffiths notation kind of bothers me. Can anyone explain why he uses primed coordinates in the attached picture. Wouldn't dl, da, dτ do just as well?
Cheers :)

 

[vanhees71]

I hate books that are so sloppy in their notation :-(. What he wants to write is

[tex]\Phi(\vec{x})=\int \mathrm{d}^3 \vec{x}' \frac{\rho(\vec{x}')}{4 \pi |\vec{x}-\vec{x}'|},[/tex]

which gives the electrostatic potential of a (time independent!) charge distribution [itex]\rho[/itex] (in Heaviside-Lorentz units). Note that there are two points involved: First there's the point [itex]\vec{x}[/itex] at which the potential is calculated and the point [itex]\vec{x}'[/itex] which is at the location of a charge [itex]\rho(\vec{x}') \mathrm{d}^3 \vec{x}'[/itex]. Then you "sum" (integrate) over all these charge elements.

 

[zezima1]

But why dl'? Is that because you might misinterpret dl unprimed as dl along the vector r (can't find that damn script letter)?

 

[Hassan2]

But why dl'? Is that because you might misinterpret dl unprimed as dl along the vector r (can't find that damn script letter)?

You integrate over primed coordinates so the integration element ( dl') is primed too. In vanhees71's notation, d³x' is an infinitesimal volume at point x'. For the 1D case, dl' is an infinitesimal length at point r'.

 

[PJC01]

There are a few reasons why primed coordinates may be used in certain situations. One reason is to distinguish between different coordinate systems or frames of reference. For example, in the attached picture, the primed coordinates may represent a moving frame of reference, while the unprimed coordinates represent a stationary frame of reference. This can be useful in situations where there are multiple frames of reference involved, such as in the study of relativity.

Another reason for using primed coordinates is to simplify calculations. In some cases, using primed coordinates may make the equations simpler and easier to work with. This can be especially helpful in more complex systems or when dealing with higher dimensions.

As for Griffiths' notation, it is simply a matter of personal preference and may not be the most intuitive for everyone. However, using primed coordinates in this notation allows for a clear and concise representation of the equations, making it easier to understand and work with.

In the end, whether to use primed coordinates or not depends on the specific problem at hand and the preferences of the scientist. In some cases, dl, da, and dτ may work just as well, but in others, primed coordinates may be necessary for a more accurate and efficient solution.