
Maxwell's Equations first published 150 years ago - cft
http://www.theguardian.com/science/life-and-physics/2015/nov/22/maxwells-equations-150-years-of-light
======
chris_va
Also credit to this guy:
[https://en.wikipedia.org/wiki/Oliver_Heaviside](https://en.wikipedia.org/wiki/Oliver_Heaviside)

... Who took the original 20 equations and simplified them to the 4 we
normally see.

~~~
Retric
_That same year he patented, in England, the coaxial cable._

That's a fairly major footnote, on an interesting career. He seems to be one
of those people who quietly had a significant impact on the future.

~~~
juped
Heaviside was somewhat written out of history because he was considered a
crank during his lifetime. His reputation is basically rehabilitated these
days, but it's only physicists/mathematicians/engineers who have heard of him.

~~~
sklogic
His name was all over the physics textbooks, probably even more frequently
than Maxwell, so he is not entirely forgotten.

------
mhartl
Some HN readers might be interested to know that Maxwell's equations can be
written in a single expression (with one auxiliary condition).

The first step is to define the _Faraday tensor_ _F_ , which can be written as
a 4x4 antisymmetric matrix. Such a matrix has 16 elements, but the diagonal is
all zeroes (by antisymmetry: _x_ = - _x_ => _x_ = 0), and the 6 off-diagonal
elements on one side of the diagonal are simply the negatives of their
mirrored values. This leaves 16 - 4 - 6 = 6 independent elements—exactly
enough to contain the 6 elements of the electric and magnetic field vectors (3
each for _E_ and _B_ ), which in fact are exactly the elements of the Faraday
tensor.

In terms of _F_ , the Maxwell equations for the divergence of _B_ and the curl
of _E_ can be combined in the single equation

    
    
        dF = 0
    

where _d_ is the _exterior derivative_. As with ordinary potential theory,
where a vector field with zero curl can be written as the gradient of a scalar
function, a tensor whose exterior derivative is zero can be derived from
another tensor, called in this case the 4-potential _A_ (which contains both
the ordinary electric potential and the magnetic vector potential):

    
    
        F = dA      (1)
    

(This is the auxiliary condition alluded to above.) The second expression for
_F_ , corresponding to the divergence of _E_ and the curl of _B_ , combines
the exterior derivative and the _Hodge star operator_ ⭑ [1]:

    
    
        d⭑F = J     (2)
    

where _J_ is the 4-current density (which contains both the charge density and
the 3-dimensional current density) [2]. Substituting (1) for _F_ in (2) then
yields Maxwell's equations in a single expression:

    
    
        d⭑dA = J    Maxwell's equation(s)
    

This equation is perhaps a more compact and elegant choice for those T-shirts:
"God said: d⭑dA = J—and there was light!"

[1]: The ⭑ character may not display in some browsers. In this case, you can
use an asterisk instead.

[2]: Eq. (2) is written in "God's units", equivalent to SI with μ₀ = ε₀ = 1 =>
_c_ = 1.

------
McKayDavis
...and in two days it is the 100th anniversary of Einstein's publication of
The Field Equations of Gravitation. (The original paper on General
Relativity).

[1]
[http://einsteinpapers.press.princeton.edu/vol6-trans/129](http://einsteinpapers.press.princeton.edu/vol6-trans/129)

[2]
[https://en.wikipedia.org/wiki/History_of_general_relativity#...](https://en.wikipedia.org/wiki/History_of_general_relativity#The_development_of_the_Einstein_field_equations)

[3] [http://www-history.mcs.st-
and.ac.uk/HistTopics/General_relat...](http://www-history.mcs.st-
and.ac.uk/HistTopics/General_relativity.html#60)

------
cnvogel
Of course, like every article published by a newspaper or magazine, its only
purpose is to carry links to _other_ articles published by this newspaper, and
omit any reference to the original source.

So here's the original publication:
[http://rstl.royalsocietypublishing.org/content/155.toc](http://rstl.royalsocietypublishing.org/content/155.toc)

Philosophical Transactions of the Royal Society of London, For the year
ⅯⅮⅭⅭⅭⅬⅩⅤ, Vol. 155

[http://rstl.royalsocietypublishing.org/content/155/local/fro...](http://rstl.royalsocietypublishing.org/content/155/local/front-
matter.pdf) (Title page and Contents)
[http://rstl.royalsocietypublishing.org/content/155/459.full....](http://rstl.royalsocietypublishing.org/content/155/459.full.pdf)
(Article)

------
madengr
Maxwell's equations as most of us know it are the Heaviside-Gibbs
reforumulation. Good article though.

~~~
jacobolus
And the one we should know is the Geometric Algebra formulation, where it ends
up as Maxwell’s Equation, singular.

Gibbs/Heaviside vectors are cumbersome and confusing, and the faster we can
dump them as a society the better.

~~~
noselfrighteous
Can you link to an explanation/derivation of the geometric formulation?

~~~
jacobolus
[https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...](https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field#Geometric_algebra_formulations)

Or open any of these and search for “Maxwell”:
[http://www.av8n.com/physics/maxwell-
ga.htm](http://www.av8n.com/physics/maxwell-ga.htm)
[http://arxiv.org/pdf/1101.3619.pdf](http://arxiv.org/pdf/1101.3619.pdf)
[http://ieeexplore.ieee.org/ielx7/5/6879517/06876131.pdf](http://ieeexplore.ieee.org/ielx7/5/6879517/06876131.pdf)
[http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf](http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf)
[http://faculty.luther.edu/~macdonal/GA&GC.pdf](http://faculty.luther.edu/~macdonal/GA&GC.pdf)

~~~
iheartmemcache
Yep. I'm real glad Hestenes has been championing VA/VC so hard. That first
text is a perfect supplementary reading that any undergraduate should read.
Who was the author?

I skipped the a few and went straight to the last PDF (the Survey) and coming
from a stronger foundational math background, I felt right at home. (I'd
imagine those who went down the physics route would be thrown off by the
notation though, so I understand the author's preface). He mentions physics
and CS applications, but I really think this has structural engineering
applications as well -- e.g. R^4 seems like a perfect domain to explore
bidirectional tensile and shear strength of materials (especially composite
materials/aggregates), then perform some FEA with all of the simple operations
within that come inherent to operating within the bounds of the 4-vector form.
I mean that's a real trivial example (and I obviously don't know enough about
structural engineering to make a good analogy) but it seems like one of those
cases where you move the problem set lifted into another domain, perform your
manipulation then finally lower back into your original (co-)domain.
(Frequently done in algebraic topology, most famously by Perelman on Poincare)

~~~
semi-extrinsic
I think there is a very good reason why structural engineers never see this
type of thing, and it's because most structural engineers don't like math.
They like clicking buttons in their CAD/FEA/BIM/<insert-latest-TLA> computer
system.

The people who write those systems on the other hand enjoy this math very
much, and tend to use Galerkin as a verb and will casually point out just how
wonderful the Lax-Milgram theorem is.

------
drmpeg
Here's my favorite article on the equations.

[http://www.setileague.org/articles/ham/maxwell.pdf](http://www.setileague.org/articles/ham/maxwell.pdf)

------
msravi
Where are all the gradient and curl operators in the picture of the original
manuscript on the page? Or was that notation developed later?

~~~
dboreham
Long after Maxwell was dead.
[https://en.wikipedia.org/wiki/Oliver_Heaviside](https://en.wikipedia.org/wiki/Oliver_Heaviside)

Supposedly they're only known as "Maxwell Equations" now because Einstein
began his 1905 paper naming them such. Prior to that they were known by other
names. Before even my time though.

Frank Yang's talk (and the associated paper, but that's only available behind
a paywall) might be of interest to those looking for more background:
[https://www.youtube.com/watch?v=SG343kojbnU](https://www.youtube.com/watch?v=SG343kojbnU)

