What does axially symmetric mean mathematically?

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What does "axially symmetric" mean mathematically?

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