Explain Gauss's law for magnetism.

(N/A) Consider a closed surface $S$ placed in a magnetic field $\vec{B}$. We need to determine the magnetic flux associated with this surface.
Imagine the surface $S$ is divided into small area elements. For one such element $\Delta \vec{S}$, the magnetic field is $\vec{B}$. The magnetic flux $\Delta \phi_{B}$ through this element is defined as:
$\Delta \phi_{B} = \vec{B} \cdot \Delta \vec{S}$
The total magnetic flux $\phi_{B}$ through the closed surface is the sum of the fluxes through all such elements:
$\phi_{B} = \sum_{\text{all}} \Delta \phi_{B} = \sum_{\text{all}} \vec{B} \cdot \Delta \vec{S} = 0 \quad \dots (1)$
Since the number of magnetic field lines leaving the closed surface is equal to the number of lines entering it, the net magnetic flux through any closed surface is always zero.
Gauss's law for magnetism states:
"The net magnetic flux through any closed surface is zero."
In the limit where $\Delta S \rightarrow 0$, the summation becomes an integral:
$\phi_{B} = \oint_{S} \vec{B} \cdot d\vec{S} = 0$
This integral form is the mathematical representation of Gauss's law for magnetism.