What is dyadic Green tensor?

[Hanh Le]

Hi everyone,

Could you please explain to me what is dyadic Green tensor in electromagnetism? Thank you

 

[pancho_hideboo]

Electric Magnetic Fields are 3-dimensional vectors.

For example, current density vector is J=(Jx, Jy, Jz).
Electric field vector generated from this J is expressed as
E=(Ex, Ey, Ez)=[Gxx, Gxy, Gxz; Gyx, Gyy, Gyz; Gzx, Gzy, Gzz]*J

This matrix form green fuction, [Gxx, Gxy, Gxz; Gyx, Gyy, Gyz; Gzx, Gzy, Gzz] is called as tensor green fuction or dyadic green function.

Strictly speaking, this is not dyadic green function style.
Dyad means sets of base vector such as xx, xy, yx, .., zz.
So dyadic green function is expressed as
G=xx*Gxx+xy*Gxy+xz*Gxz+yx*Gyx+yy*Gyy+yz*Gyz+zx*Gzx+zy*Gzy+zz*Gzz
without using matrix style.
See equation (2) in https://deepblue.lib.umich.edu/bitstream/2027.42/8428/5/bab2445.0001.001.pdf
This is dyadic green function.
However this is equivalent to matrix style.

Also see ''Dyadic Green's functions in Electromagnetic theory by Tai C. T.