What is the name of the law that states the total magnetic flux through a closed surface is zero?

The law stating that the total magnetic flux through a closed surface is zero is Gauss's law for magnetism.

In integral form, the law says that the net magnetic flux through any closed surface equals zero: as many magnetic field lines enter the surface as leave it. In differential form, it is written as the divergence of the magnetic field being zero. The two forms are connected by the divergence theorem.

Gauss's law for magnetism is one of Maxwell's four equations of classical electrodynamics. Its physical interpretation is that isolated magnetic monopoles have not been observed: magnets have north and south poles together, so a cut magnet produces smaller dipoles rather than a lone pole.

The law is often confused with Faraday's law. Faraday's law concerns an electromotive force caused by changing magnetic flux through an open loop, whereas Gauss's law for magnetism concerns the net flux through a closed surface. Magnetic flux through an open surface can be nonzero; only the closed-surface total must cancel.