2.1 The integral form of Gauss's Law:
$\oint_{S}\;\overrightarrow{B}\;\circ\;\widehat{n}\;da = 0$ ⇒ Gauss's Law for Magnetic Field
The left side of this equation is a mathematical description of the flux of a vector field through a closed surface. In this case, Gauss's law refers to magnetic flux - the number of magnetic field lines passing through a closed surface 'S'. The right side is identically zero.
To understand the meaning of each symbol, here's an expanded view:
2.1 The integral form of Gauss's Law
Gauss's law in integral form: The total magnetic flux passing through any closed surface is Zero.
In another words: If you have a real or imaginary closed surface of any size or shape, the total magnetic flux through that surface must be zero.
Note: This does not mean that zero magnetic field lines penetrates the surface - it means that for magnetic field line that enters the volume enclosed by the surface, there must be a magnetic field line leaving that volume. Thus the inward (negative) magnetic flux must be exactly balanced by the outward (positive) magnetic flux.
2.2 The differential form of Gauss's Law:
$\nabla\;\circ\overrightarrow{B} = 0$ ⇒ Gauss's Law for Magnetic Field
The left side of this equation is a mathematical description of the divergence of the magnetic field - the tendency of the magnetic field to "flow" more strongly away from a point than toward it, while the right side is simply zero.To understand the meaning of each symbol, here's an expanded view:
2.2 The differential form of Gauss's Law
We will discuss about "Faraday's Law" later on.Gauss's Law in differential form: The divergence of the magnetic field at any point is zero.In other words: Why this is true is by analogy with the electric field, for which divergence at any location is proportional to the electric charge density at that locations. Since it is not possible to isolate magnetic poles, you can't have a north pole without a south pole, and the "magnetic charge density" must be zero everywhere. This means that the divergence of the magnetic field must also be zero.