This is mathematically equivalent to saying that the divergence of is zero . ( Such vector fields are called solenoidal vector fields . ) This property is called Gauss's law for magnetism and is equivalent to the statement that there are no isolated magnetic poles or magnetic monopoles.
32.
The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field ( the order of the equations is not casual, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following ).
33.
When it came to deriving the electromagnetic wave equation from displacement current in his 1865 paper A Dynamical Theory of the Electromagnetic Field, he got around the problem of the non-zero divergence associated with Gauss's law and dielectric displacement by eliminating the Gauss term and deriving the wave equation exclusively for the solenoidal magnetic field vector.
34.
Because an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, the Helmholtz decomposition states that a vector field ( satisfying appropriate smoothness and decay conditions ) can be decomposed as the sum of the form-\ operatorname { grad } \ Phi + \ operatorname { curl } \ mathbf { A } where is a scalar field, called scalar potential, and is a vector field called a vector potential.
