Intuitive Guide to Maxwell's Equations • Hacker News Board

Re: Intuitive Guide to Maxwell's Equations

#11

Post by ABeeSea » Wed, Jul 01, 2020, 1:40 PM UTC

chronolitus wrote: post #3

These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…

Most mathematicians consider the equations for div and curl to be the “dirty equations” while the “perfect mathematical concept” is the exterior derivative of a 1-form.

Re: Intuitive Guide to Maxwell's Equations

#12

Post by OscarCunningham » Wed, Jul 01, 2020, 1:41 PM UTC

pa7x1 wrote: post #9

antman wrote: post #5

As a side note there was a historical debate, and the shape of these equations is the end result of the chosen system. In geometric algebra which did not prevail, these are one equation.

Geometric algebra is a bit awkward notationally. Physicist prefer to use an alternative notation based on exterior calculus that does provide a compact representation of Maxwell equations: d F=J dF=0 Geometric algebra produces two equations too, by the way, not one.

You can even write them in terms of the four-vector A rather than F (related by F = dA) to reduce them to ddA = J.

Re: Intuitive Guide to Maxwell's Equations

#13

Post by JadeNB » Wed, Jul 01, 2020, 1:41 PM UTC

layoutIfNeeded wrote: post #8

contravariant wrote: post #2

For a truly intuitive understanding I think you will have to understand relativistic differential geometry. Once you have that you just have 1 (4-dimensional) vector potential, the Laplacian of which is equal to the current. This single remaining equation corresponds to the Maxwell equations for the electric field, the equations for the magnetic field just correspond to the fact that the 'curl' of this vector potenti…

How do you define curl in 4d?

Div, grad, and curl are a manifestation of de Rham cohomology that make use of a lot of lucky coincidences. See https://en.wikipedia.org/wiki/De_Rham_cohomology for the formal definition (in which curl becomes the exterior derivative from 1-forms to 2-forms) and https://web.ma.utexas.edu/users/a.debray/lecture_notes/idea_... for a nice exposé.

Re: Intuitive Guide to Maxwell's Equations

#14

Post by alanbernstein » Wed, Jul 01, 2020, 1:43 PM UTC

Daub wrote: post #6

James Clark Maxwell is one of the Demi-gods of Colour science. He produced the first Colour photograph, and was the first to quantify Colour. Every time you define an RGB value, he is sitting on your shoulder.

Or, you're standing on his shoulder, you might say.

Re: Intuitive Guide to Maxwell's Equations

#15

Post by tarsinge » Wed, Jul 01, 2020, 1:46 PM UTC

Interesting, I always thought of fields as the interface to measure underlying particles interactions instead of the "base thing" itself. My reasoning was that since it was impossible to model effects from individual particles behavior, we were modeling the aggregate effect using fields, but underneath it was just "small objects" interactions.

Edit: reading the rest of the guide, I find it very enlightening and at the perfect level of complexity perfect for me (no formal maths since Uni 15 years ago).

Re: Intuitive Guide to Maxwell's Equations

#16

Post by reikonomusha » Wed, Jul 01, 2020, 2:24 PM UTC

Daub wrote: post #6

James Clark Maxwell is one of the Demi-gods of Colour science. He produced the first Colour photograph, and was the first to quantify Colour. Every time you define an RGB value, he is sitting on your shoulder.

Small nit, it’s “Clerk” not “Clark”.

Re: Intuitive Guide to Maxwell's Equations

#17

Post by contravariant » Wed, Jul 01, 2020, 2:25 PM UTC

layoutIfNeeded wrote: post #8

contravariant wrote: post #2

For a truly intuitive understanding I think you will have to understand relativistic differential geometry. Once you have that you just have 1 (4-dimensional) vector potential, the Laplacian of which is equal to the current. This single remaining equation corresponds to the Maxwell equations for the electric field, the equations for the magnetic field just correspond to the fact that the 'curl' of this vector potenti…

How do you define curl in 4d?

You use the outer derivative which generalizes the curl as well as a few similar constructs. Technically it's slightly different as it returns a bivector (which is a bit like a plane spanned by 2 vectors), but in 3D both vectors and bivectors form a 3D space and you can freely convert between the two. The difference between the divergence and the curl is basically whether you switch between vectors and bivectors before or after you take the outer derivative.

Re: Intuitive Guide to Maxwell's Equations

#18

Post by king07828 » Wed, Jul 01, 2020, 2:31 PM UTC

This is awesome. What a concept, identifying the variables in the equations and providing diagrams to show the geometric meaning of the equations. Really wish wikipedia (or some sort of companion website) would similarly present an explanation of the practical meaning of variables and operators in the heavy math pages so that people with an understanding of algebra could understand the formulaic descriptions in addition to the qualitative descriptions after landing on a random page with heavy math.

Re: Intuitive Guide to Maxwell's Equations

#19

Post by cs702 » Wed, Jul 01, 2020, 2:31 PM UTC

ghaff wrote: post #7

chronolitus wrote: post #3

These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…

Wikipedia is probably the most obvious modern exemplar of this. The editors of most mathematics articles clearly are much fonder of playing with the equations editor than they are of actually explaining things.

Probably true!

That said, Wikipedia's mathy pages have proved repeatedly useful to me as a reference, e.g., whenever I remember a mathematical concept only vaguely or intuitively and just need to find a detailed formalization to implement it in code. My browser is always open, so Wikipedia is often the "reference of least resistance."

Re: Intuitive Guide to Maxwell's Equations

#20

Post by ghaff » Wed, Jul 01, 2020, 2:47 PM UTC

cs702 wrote: post #19

ghaff wrote: post #7

Earlier quoted context omitted.

Wikipedia is probably the most obvious modern exemplar of this. The editors of most mathematics articles clearly are much fonder of playing with the equations editor than they are of actually explaining things.

Probably true! That said, Wikipedia's mathy pages have proved repeatedly useful to me as a reference, e.g., whenever I remember a mathematical concept only vaguely or intuitively and just need to find a detailed formalization to implement it in code. My browser is always open, so Wikipedia is often the "reference of least resistance."

I'm at least ambivalent about a general purpose encyclopedia having whole categories of articles that are mostly only accessible to specialists. In an ideal world, there would probably be a companion "Mathepedia" or something along those lines with a different target audience. But, of course, Wikipedia doesn't really have the concept of a specific reader persona like a conventionally edited reference does.