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Add DISPLAYTITLE: title the page by its section rather than its path

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			{{DISPLAYTITLE:On Physical Lines of Force — Modern derivations}}
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	''Start with [[On Physical Lines of Force/Modern derivations/From components to vectors\|From components to vectors]] · [[On Physical Lines of Force\|Back to the contents]]''		''Start with [[On Physical Lines of Force/Modern derivations/From components to vectors\|From components to vectors]] · [[On Physical Lines of Force\|Back to the contents]]''

BFP Rauch: 1 revision imported: Maxwell, On Physical Lines of Force — initial import

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1 revision imported: Maxwell, On Physical Lines of Force — initial import

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''[[On Physical Lines of Force]] → Modern derivations''

Every one of the four equations now called Maxwell's equations appears somewhere in this paper, along with the constitutive relations, the vector potential, the field energy densities, the stress tensor, the motional term of the Lorentz force, and the identification of light. This section takes each result in Maxwell's 1861 notation and carries it to the form a reader meets today, making the changes of convention explicit at every step.

== The master table ==

{| class="wikitable"
! modern equation !! Maxwell's number !! where in the paper !! derivation
|-
| <math>\nabla\cdot\mathbf{D}=\rho</math> || (113) || [[On Physical Lines of Force/Part III/Prop. XIV|Part III, Prop. XIV]] || [[On Physical Lines of Force/Modern derivations/Gauss's law for electricity|Gauss's law for electricity]]
|-
| <math>\nabla\cdot\mathbf{B}=0</math> || (56) || [[On Physical Lines of Force/Part II/Prop. VIII|Part II, Prop. VIII]] || [[On Physical Lines of Force/Modern derivations/No magnetic monopoles|No magnetic monopoles]]
|-
| <math>\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t</math> || (54) || [[On Physical Lines of Force/Part II/Prop. VIII|Part II, Prop. VIII]] || [[On Physical Lines of Force/Modern derivations/Faraday's law of induction|Faraday's law]]
|-
| <math>\nabla\times\mathbf{H}=\mathbf{J}+\partial\mathbf{D}/\partial t</math> || (9), corrected at (112) || [[On Physical Lines of Force/Part I/Prop. III|Part I, Prop. III]] and [[On Physical Lines of Force/Part III/Prop. XIV|Part III, Prop. XIV]] || [[On Physical Lines of Force/Modern derivations/Ampere's law and its correction|Ampère–Maxwell]]
|}

And the supporting results:

{| class="wikitable"
! result !! Maxwell's number !! where !! derivation
|-
| <math>\mathbf{B}=\nabla\times\mathbf{A}</math> || (55) || [[On Physical Lines of Force/Part II/Prop. VIII|Part II, Prop. VIII]] || [[On Physical Lines of Force/Modern derivations/The vector potential|The vector potential]]
|-
| <math>\mathbf{E}=-\partial\mathbf{A}/\partial t-\nabla\varphi</math> || (58) || [[On Physical Lines of Force/Part II/Prop. VIII|Part II, Prop. VIII]] || [[On Physical Lines of Force/Modern derivations/The vector potential|The vector potential]]
|-
| <math>\mathbf{E}=\mathbf{v}\times\mathbf{B}-\partial\mathbf{A}/\partial t-\nabla\varphi</math> || (77) || [[On Physical Lines of Force/Part II/Prop. XI|Part II, Prop. XI]] || [[On Physical Lines of Force/Modern derivations/The Lorentz force|The Lorentz force]]
|-
| <math>\mathbf{B}=\mu\mathbf{H}</math>, <math>\mathbf{D}=\varepsilon\mathbf{E}</math> || (105) and Part I || [[On Physical Lines of Force/Part III/Prop. XIII|Part III, Prop. XIII]] || [[On Physical Lines of Force/Modern derivations/The constitutive relations|Constitutive relations]]
|-
| the magnetic stress tensor || Prop. II, then (12)–(14) || [[On Physical Lines of Force/Part I/Prop. II|Part I, Props. II and III]] || [[On Physical Lines of Force/Modern derivations/The Maxwell stress tensor|The stress tensor]]
|-
| magnetic energy density <math>\mathbf{B}\cdot\mathbf{H}/8\pi</math> || (46) || [[On Physical Lines of Force/Part II/Prop. VI|Part II, Prop. VI]] || [[On Physical Lines of Force/Modern derivations/The electromagnetic wave equation|used in the wave equation]]
|-
| <math>c=1/\sqrt{\varepsilon_{0}\mu_{0}}</math> || unnumbered || [[On Physical Lines of Force/Part III/Prop. XVI|Part III, Prop. XVI]] || [[On Physical Lines of Force/Modern derivations/The unit ratio|The unit ratio and c]]
|-
| Faraday rotation, results (A)–(F) || — || [[On Physical Lines of Force/Part IV/The six results|Part IV]] || [[On Physical Lines of Force/Modern derivations/Faraday rotation today|Faraday rotation today]]
|}

== Suggested order ==

If you are reading this section on its own rather than following links from the Part pages:

# [[On Physical Lines of Force/Modern derivations/From components to vectors|From components to vectors]] — read this first; everything else assumes it
# [[On Physical Lines of Force/Modern derivations/No magnetic monopoles|No magnetic monopoles]]
# [[On Physical Lines of Force/Modern derivations/Ampere's law and its correction|Ampère's law and its correction]]
# [[On Physical Lines of Force/Modern derivations/Faraday's law of induction|Faraday's law of induction]]
# [[On Physical Lines of Force/Modern derivations/Gauss's law for electricity|Gauss's law for electricity]]
# [[On Physical Lines of Force/Modern derivations/The constitutive relations|The constitutive relations]]
# [[On Physical Lines of Force/Modern derivations/The displacement current|The displacement current]]
# [[On Physical Lines of Force/Modern derivations/The unit ratio|The unit ratio, and why it is a velocity]]
# [[On Physical Lines of Force/Modern derivations/The electromagnetic wave equation|The electromagnetic wave equation]]
# [[On Physical Lines of Force/Modern derivations/The vector potential|The vector potential]]
# [[On Physical Lines of Force/Modern derivations/The Lorentz force|The motional term and the Lorentz force]]
# [[On Physical Lines of Force/Modern derivations/The Maxwell stress tensor|The Maxwell stress tensor]]
# [[On Physical Lines of Force/Modern derivations/Faraday rotation today|Faraday rotation today]]

== Three things to keep in mind throughout ==

=== Maxwell's equations are not Maxwell's ===

The four equations in the box in every textbook were assembled into that shape by '''Oliver Heaviside''' in 1884–85, working independently of and in parallel with '''Josiah Willard Gibbs''', who developed the same vector calculus. Maxwell's own 1865 paper lists twenty equations in twenty variables. He had no del operator, no curl, no divergence as symbols — only components. What is called "Maxwell's equations" is Maxwell's physics in Heaviside's notation.

This is not a quibble. Reading the 1861 paper is much easier once you stop looking for equations that are not there and start looking for the ''triples'' that became them. See [[On Physical Lines of Force/Modern derivations/From components to vectors|From components to vectors]].

=== The 4π is a units convention, not physics ===

Maxwell works in a Gaussian-style system where the factor <math>4\pi</math> — the surface area of a unit sphere — appears explicitly in the field equations. SI absorbs it into <math>\varepsilon_{0}</math> and <math>\mu_{0}</math>. Neither is more correct; the 4π has to appear ''somewhere'', because the inverse-square law and the sphere are related, and the choice is where to put it.

{| class="wikitable"
!   !! Gaussian (Maxwell's style) !! SI
|-
| Ampère || <math>\nabla\times\mathbf{H}=\dfrac{4\pi}{c}\mathbf{J}+\dfrac{1}{c}\dfrac{\partial\mathbf{D}}{\partial t}</math> || <math>\nabla\times\mathbf{H}=\mathbf{J}+\dfrac{\partial\mathbf{D}}{\partial t}</math>
|-
| Gauss || <math>\nabla\cdot\mathbf{D}=4\pi\rho</math> || <math>\nabla\cdot\mathbf{D}=\rho</math>
|-
| Coulomb || <math>F=\dfrac{q_{1}q_{2}}{r^{2}}</math> || <math>F=\dfrac{1}{4\pi\varepsilon_{0}}\dfrac{q_{1}q_{2}}{r^{2}}</math>
|}

Maxwell's own system is not quite modern Gaussian either — the explicit factors of <math>c</math> in the table above come from a later tidying. Converting his results is mostly a matter of tracking the 4π, and it is where most transcription errors happen.

=== The equations here are reconstructions ===

No equation in this paper could be transcribed from a page scan for this edition. Equation ''numbers'' are attested where the pages say so; equation ''bodies'' in Maxwell's notation are reconstructions, correct in content but possibly differing from the printing in sign convention, placement of 4π, and whether a relation is written once or three times. The modern equations are, of course, just the modern equations. See [[On Physical Lines of Force/Sources and editorial method|Sources and editorial method]] for the full account.

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''Start with [[On Physical Lines of Force/Modern derivations/From components to vectors|From components to vectors]] · [[On Physical Lines of Force|Back to the contents]]''

On Physical Lines of Force/Modern derivations - Revision history

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

BFP Rauch: 1 revision imported: Maxwell, On Physical Lines of Force — initial import

opl>Import: Imported from the DokuWiki edition