Appendix: Exercises... 421
The second term is related to the derivative of the energy densityu. This is seen
as follows. Renaming the summation indicesm→mand usingHm=−Hm,
Fmk=−Fkmleads toHm∂mFk=Hm∂Fkmand consequently
Hm∂mFk=1
2
Hm(∂mFk+∂Fkm).Due to the homogeneous Maxwell equation (18.58), this expression is equal to
Hm∂mFk=−1
2
Hm∂kFm=1
2
Hm∂kFm.For the special case of a linear medium whereHm∼Fmapplies, one has
Hm∂mFk=1
4
∂k(HmFm),and this then leads to the expression (18.77) for the 4D stress tensor.
18.4 Flatlanders Invent the Third Dimension and Formulate their Maxwell
Equations(p.388)
The flatlanders of Exercise7.3noticed, they can introduce contra- and contra-variant
vectors
xi=(r 1 ,r 2 ,ct), xi=(−r 1 ,−r 2 ,ct).With the Einstein summation convention for the three Roman indices, the scalar
product of their 3-vectors is
xixi=−(r 12 +r 22 )+c^2 t^2 =−r^2 +c^2 t^2.They use the differential operator
∂i=(
∂
∂r 1,
∂
∂r 2,
∂
∂ct)
,
and form the 3-vectors
JI=(j 1 ,j 2 ,cρ), Φi=(A 1 ,A 2 ,φ/c)from their current and charge densities and their vector and scalar potentials.
How are the relations
B=∂ 1 A 2 −∂ 2 A 1 , Ei=−∂iφ−∂A/∂t, i= 1 , 2 ,