Electromagnetic wave

A transverse wave composed of an oscillating electrical field and a magnetic field that oscillates perpendicular to the electric field.

Electromagnetic waves was predicted by the classical laws of electricity and magnetism, known as Maxwell's equations:

Name	Partial Differential Equations	Integral Equations
Gauss's law of Conservation:	$\nabla \cdot \mathbf {D} =\rho$	$\oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =\int _{V}\rho \,\mathrm {d} V$
Gauss' law Of Magnetism:	$\nabla \cdot \mathbf {B} =0$	$\oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0$
Faraday's Law of Induction:	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	$\oint _{C}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\int _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$
Ampère's Law of Circulation	$\nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}$	$\oint _{C}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {A} +\int _{S}{\frac {\partial \mathbf {D} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$

where B denotes the magnetic field, E denotes the electric field, H denotes the auxiliary magnetic field, J denotes the free current density, and $\rho$ denotes the free electric charge density.

In the language of Exterior Calculus, Maxwell's equations can be rewritten much more compactly as:

$\mathrm {d} {\mathbf {F}}=0$

$\mathrm {d} *{\mathbf {F}}={\mathbf {J}}$

where d is exterior derivative operator, and * is the Hodge star operator.