Divergence and curl of vectorfields

According to the Helmholtz-theorem a vectorfield is completely defined by the divergence and  curl of the vectorfield.

the divergence is a measure of the strength of the source of the vectorfield whereas the degree of rotation of the field is given by the curl.

The divergence is defined as  ∇·F  = lim Δv→0 ∫A⋅ ds/Δv i.e. the scalarproduct(dotproduct) of the nabla operator and the vector.

The ∇-operator is defined as the vector differential operator
∇=∂/∂x + ∂/∂y + ∂/∂z.

When this operates on a scalar V one obtains the gradient  ∇V of that scalar i.e. a vector that represents both the magnitude and the direction of the maximum space rate of increase of  of that scalar.

The curl is defined by

∇xF. = (dFz/dy – dFy/dz) i + (dFx/dz – dFz/dx)j + (dFy/dx – dFx/dy) k

The electromagnetic field is defined by the divergence and curl of the Electric field vector E and the magnetic field vector B:
∇· E= ρ

∇xE=∂B/∂t

∇· B=0; This can be interpretated as stating the fact that there are no magnetic charges.
∇xB=∂D/∂t

These are the famous Maxwellian equations which gives a full description of the electromagnetic theory.
Every electromagnetic law can be deduced from them.