The total magnetic flux through a closed surface is zero. Proof
The total magnetic flux through a closed surface is
\[\Phi = \int \int_S \mathbf{B} \cdot \mathbf{n} dS\]
.We can use the Divergence Theorem to equate this to a volume integral.
\[ \int \int_S \mathbf{B} \cdot \mathbf{n} dS = \int \int \int_V \mathbf{\nabla} \cdot \mathbf{B} dV\]
.\[V\]
in this integral is the volume enclosed by \[S\]
.
But \[\mathbf{\nabla} \cdot \mathbf{B} =0\]
from Maxwell's equations so \[\Phi = \int \int_S \mathbf{B} \cdot \mathbf{n} dS =0\]
.