The statement that the divergence of magnetic fields is zero is a fundamental property of magnetism described by Maxwell's equations. The divergence of a vector field represents how the field lines "spread out" or "converge" at a given point. In the case of magnetic fields, the divergence is indeed zero everywhere, including at the poles of a magnet.
To understand why the divergence of magnetic fields is zero, we can look at one of Maxwell's equations, specifically Gauss's law for magnetism:
∇ · B = 0
This equation states that the divergence (represented by the symbol ∇ ·) of the magnetic field B is zero. In other words, the total number of magnetic field lines that enter a closed surface is equal to the total number that exits. This implies that there are no "sources" or "sinks" of magnetic field lines (monopoles) like there are for electric fields (electric charges).
When considering a magnet, the magnetic field lines form closed loops that extend from one pole to the other. At each pole, the field lines appear to "diverge" outward, but they also "converge" into the other pole. This behavior cancels out the divergence, resulting in a net divergence of zero.
So while it may appear that the magnetic field diverges at the poles of a magnet, the overall divergence of the magnetic field is zero due to the closed loop nature of magnetic field lines.