What does axially symmetric mean mathematically?

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"Axially symmetric" refers to a mathematical property where a system exhibits symmetry around a central axis. In the context of magnetic fields, if a magnetic field \(\vec B\) is axially symmetric, it is defined mathematically by the condition \(\frac{\partial \vec B}{\partial \phi} = 0\) in cylindrical coordinates, indicating that the field does not change with respect to the azimuthal angle \(\phi\). This concept is also referred to as "axisymmetric" and is crucial in fields such as electromagnetism and fluid dynamics.

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What does "axially symmetric" mean mathematically?

If we, for example, say that a magnetic field [tex]\vec B[/tex] is axially symmetric, does that mean that (in cylindrical coordinates) we have [tex]\frac{\partial \vec B}{\partial \phi} = 0[/tex], where [tex]\phi[/tex] is the azimuthal angle?
 
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Yes, exactly. (Sometimes called "axisymmetric.")
 

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