I How to use geometrical symmetries -- general advice? (Vector potential)

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Identifying useful symmetries in vector potential can be challenging, particularly in the context of an infinitely long hollow cylinder carrying current. The vector potential, denoted as ##\vec{A}##, is shown to be independent of ##z## and ##\varphi## due to the cylindrical symmetry of the current and charge densities. This symmetry implies that the magnetic flux density ##\vec{B}## must also be cylindrically symmetric, leading to the conclusion that ##\vec{A}## can be expressed solely as a function of the radial coordinate ##\rho##. While it is not possible to mathematically prove the independence of certain coordinates universally, symmetry arguments provide a strong basis for understanding the behavior of vector fields in this context. Understanding these symmetries is crucial for correctly applying them in calculations involving vector potentials.
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The following is an example from my script. I always have trouble identifying useful symmetries. Can someone explain to me why (for example) the vector potential doesn't have a ##z## dependence? I understand that there is no ##\varphi## dependency.
I don't understand why the field of ##\vec{A}## has to be parallel to the ##z## axis. What about ##d^{3} \overrightarrow{r^{\prime}}##??? Is there a way to show mathematically that the vector potential is independent of the two variables ##z\varphi##?
In general, I have problems identifying symmetries and using them correctly.

Magnetic flux density of an infinitely long hollow cylinder

The hollow cylinder is homogeneously traversed by the current ##I##. Calculate the magnetic flux ##\vec{B}(\vec{r})## as the curl of the vector potential ##\vec{A}(\vec{r})##.

$$
\begin{aligned}
\vec{A}(\vec{r}) & =\frac{\mu_{0}}{4 \pi} \int \limits_{V} \frac{\vec{j}\left(\overrightarrow{r^{\prime}}\right)}{\left|\vec{r}-\overrightarrow{r^{\prime}}\right|} d^{3} \overrightarrow{r^{\prime}} \\
\vec{j}(\vec{r}) & =j \vec{e}_{z}
\end{aligned}
$$

Use cylindrical coordinates ##\vec{r}=(\rho, \varphi, z)##. Due to the symmetry, we have:
$$
\vec{A}(\vec{r})=A(\rho, \varphi, z) \vec{e}_{z}=A(\rho) \vec{e}_{z}
$$
 
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If you have an infinitely long cylindrical object then the field can't depend on either ##z## or ##\varphi## because the charge and current densities are symmetrical under rotation around the ##z## axis and translation along it. If the sources are symmetric like that, how can the fields be otherwise? If they were (e.g.) weaker at some ##z=z_0##, why there?

So the ##\vec B## field is cylindrically symmetric. What ways can a vector field be cylindrically symmetric? What does that tell you about ##\vec A##?
 
There is no way to generally prove that the vector potential does not depend on particular coordinates since it is always possible to perform a gauge transformation ##\vec A \to \vec A + \nabla \phi## with the same resulting field for any scalar function ##\phi##.

Any symmetry argument must be based on this particular expression for the vector potential.
 
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