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Maxwell's equations

The divergence theorem and Stokes's theorem are generalizations of integration that invoke the divergence and curl operations on vectors. A familiar vector field is the electromagnetic field and Maxwell's equations depend on these vector derivatives as well:

$$\begin{split}\nabla \cdot \vec{B} = 0 \;\text{\hspace{1in}}\;... ...ext{\hspace{1in}}\; & \nabla \cdot \vec{D} = \rho\ \end{split}$$ (16-9)

in MKS units and the total electric displacement $\vec{D}$ is related to the total polarization $\vec{P}$ and the electric field $\vec{E}$ through:

$$\vec{D} = \vec{P} + \epsilon_o \vec{E}$$ (16-10)

where $\epsilon_o$ is the dielectric permittivity of vacuum. The total magnetic induction $\vec{B}$ is related to the induced magnetic field $\vec{H}$ and the material magnetization through

$$\vec{B} = \mu_o ( \vec{H} + \vec{M} )$$ (16-11)

where $\mu_o$ is the magnetic permeability of vacuum.