12—Tensors 39512.26 Compute the transformation laws for the other components of the electromagnetic field tensor.
12.27 The divergence of a tensor field is defined as above for a vector field. T is a tensor viewed as a vector
valued function of a vector variable
divT= lim
V→ 01
V
∮
T
(
dA~)
It is a vector. Note: As defined here it requires the addition of vectors at two different points of the manifold,
so it must be assumed that you can move a vector over parallel to itself and add it to a vector at another point.
Had I not made the point in the text about not doing this, perhaps you wouldn’t have noticed the assumption
involved, but now do it anyway. ComputedivT in rectangular coordinates.
12.28 ComputedivTin cylindrical coordinates using both the coordinate basis and the usual unit vector(ˆr,θ,ˆzˆ)
basis.
12.29 Show thatgijgj=δi.
12.30 If you know what it means for vectors at two different points to be parallel, give a definition for what it
means for two tensors to be parallel.
12.31 Fill in the missing steps in the derivation following Eq. ( 30 ), and show that the alternating tensor is unique
up to a factor.