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Line 30: − where '''B''' denotes the [[magnetic field]], '''E''' denotes the [[electric field]], '''H''' denotes the auxiliary magnetic field, '''J''' denotes the free [[current density]], and <math>\rho</math> denotes the free [[electric charge density]].+
Rework.
What follows is a survey of the various forms that these equations take, beginning with the most utilitarian and progressing to the most elegant. Which form you prefer depends on your scientific outlook, and perhaps your taste in T-shirts. The various <math>\nabla \cdot \mathbf{E}</math> and <math>\nabla \times \mathbf{E}</math> symbols appearing in some of the equations are the [[divergence]] and [[curl]] operators, respectively.
What follows is a survey of the various forms that these equations take, beginning with the most utilitarian and progressing to the most elegant. Which form you prefer depends on your scientific outlook, and perhaps your taste in T-shirts. The various <math>\nabla \cdot \mathbf{E}</math> and <math>\nabla \times \mathbf{E}</math> symbols appearing in some of the equations are the [[divergence]] and [[curl]] operators, respectively.
They are usually formulated as four equations (but later we will see some particularly elegant versions with only two), and the equations are usually expressed in ''differential form'', that is, as [[Partial Differential Equations]] involving the divergence and curl operators. They can also be expressed with integrals. They are often expressed in terms of four vector fields: E, D, B, and H, though the simpler form uses only E and B.
{| class="wikitable" border="1" cellpadding="8" cellspacing="0"
{| class="wikitable" border="1" cellpadding="8" cellspacing="0"
! Name
! Name
! [[Partial Differential Equations]]
! differential form
! [[Integral Equations]]
! integral form
|-
|-
| Gauss's Law of Conservation:
| Coulomb's law of electrostatics, or Gauss's Law:
| <math>\nabla \cdot \mathbf{D} = \rho</math>
| <math>\nabla \cdot \mathbf{D} = \rho</math>
| <math>\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V</math>
| <math>\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = \int_V \rho\, \mathrm{d}V</math>
|-
|-
| Gauss' Law Of Magnetism:
| Absence of magnetic monopoles:
| <math>\nabla \cdot \mathbf{B} = 0</math>
| <math>\nabla \cdot \mathbf{B} = 0</math>
| <math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>
| <math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>
| <math>\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math>
| <math>\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math>
|-
|-
| Ampère's Law of Circulation<br />
| Ampère's Law, or the Biot-Savart Law, plus displacement current:
| <math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math>
| <math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math>
| <math>\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} +
| <math>\oint_C \mathbf{H} \cdot \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} +
|}
|}
In these, '''E''' denotes the [[electric field]], '''B''' denotes the [[magnetic field]], '''D''' denotes the ''electric displacement field'', and '''H''' denotes the ''magnetic field strength'' or ''auxiliary field''. '''J''' denotes the free [[current density]], and <math>\rho</math> denotes the free [[electric charge density]].
In the language of [[Exterior Calculus]], Maxwell's equations can be rewritten much more compactly as:
In the language of [[Exterior Calculus]], Maxwell's equations can be rewritten much more compactly as: