2 ms·
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) v
by contravariant 6y ago
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 potential has 0 divergence (which is just a basic fact of geometry, and is also why magnetic monopoles are unlikely).
- layoutIfNeeded 6y agoHow do you define curl in 4d?
- JadeNB 6y agoDiv, 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 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_of_cohomology.pdf https://web.ma.utexas.edu/users/a.debray/lecture_notes/idea_... for a nice exposé.
- contravariant 6y agoYou 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.