How Does Jackson's Use of Manifolds Enhance Understanding of Electrodynamics?

[kidsasd987]

This is from Jackson, "Electrodynamics"
a field is a fuction mapping phi: M -> T, x -> phi(x) from a base manifold M into a target manifold T.

field X: R3 * R1 -> R3
x(r,t) ->X(x)I think this is eucledian R4 to R3 so I wonder why Jackson explained this with the concept of manifolds?
Is it necessary?

 

[Orodruin]

Impossible to say unless you specify what X is.

 

[kidsasd987]

Impossible to say unless you specify what X is.

http://www.thp.uni-koeln.de/alexal/pdf/electrodynamics.pdf

its on section 1.2.1 and field X is any arbitary EM fields (E,D,B,H)

 

[Cruz Martinez]

This is from Jackson, "Electrodynamics"
a field is a fuction mappingphi: M -> T, x -> phi(x) from a base manifold M into a target manifold T.

field X: R3 * R1 -> R3
x(r,t) ->X(x)I think this is eucledian R4 to R3 so I wonder why Jackson explained this with the concept of manifolds?
Is it necessary?

It is necessary if you want to consider spaces which have topologies other than that of R^n. For example a field on an n-dimensional sphere S^n.
If you restrict yourself to defining a field as a function from R^m to R^n, then you could not define a field on S^n as a global function because S^n is not homeomorphic to R^m for any n, m >= 0. In the example you mention here, the domain space is just R^4 with the usual topology and diferentiable structure, but this space is of course a manifold so it fits in the definition the book gives.

 

[The Bill]

Consider also that in practical applications, the boundaries of a particular region might not be made up of chunks of planes isometric to R². Often the boundary of a cavity or waveguide, or the surfaces of antennas and reflectors are best described as two dimensional manifolds rather than as subsets of R³ or R^3,1.
Also, thinking in terms of manifolds makes some nice simplifications and generalizations easier later on, like using the generalized Stokes theorem for all integration over bounded regions, curves, or surfaces.