Mathematical Principles of Theoretical Physics

122 CHAPTER 3. MATHEMATICAL FOUNDATIONS

It is known that

(3.1.64) FkGk=a scalar field, andD(FkGk) =∂(FkGk).

We infer then from (3.1.63) and (3.1.64) that

(3.1.65) DkFiGi+FiDkGi=∂kFiGi+Fi∂kGi.

Inserting (3.1.61) in (3.1.65) we deduce that

DkFi=∂kFi−ΓkijFj.

6.Derivative on TrkM.For two vector fieldsAkandBk, their tensor productA⊗B=
{AiBj}is a second order tensor. By (3.1.63) we have

Dk(A⊗B) =Dk(AiBj) =DkAiBj+AiDkBj

=(∂kAi+ΓiklAl)Bj+Ai(∂kBj+ΓkljBl)

=∂k(AiBj)+ΓiklAlBj+ΓkljAiBl.

ReplacingA⊗BbyTij, we obtain

DkTij=∂kTij+ΓiklTl j+ΓkljTil.

In the same fashion, for general(k,r)-tensors

T=Tji 11 ······jikr:M→TrkM,

its covariant derivative can be expressed as

(3.1.66) DkTji 11 ······ijkr=∂kTij^11 ······jikl+Γikl^1 Tjli 12 ······jrik+···+

+ΓiklkT

i 1 ···ik− 1 l

j 1 ···jr −Γ

l

k j 1 T

i 1 ···ik

j 2 ···jr− ··· −Γ

l

k jrT

i 1 ···ik

j 1 ···jk− 1 l.

The derivative (3.1.66) were given in (2.3.26).

Remark 3.5.We have encountered tensor products in (3.1.59) and (3.1.62). A further expla-
nation of this considered is now in order. LetAandBbe two matrices with ordersnandm
respectively. ThenA⊗Bis a matrix of orderN=nm, defined by

(3.1.67) A⊗B=







a 11 B ··· a 1 nB

..

.

..

.

an 1 B ··· annB





, aijB=







aijb 11 ··· aijb 1 m

..

.

..

.

aijbm 1 ··· aijbmm





.

In (3.1.60) and (3.1.62), the components ofDF={DkFji 11 ······ijkr}andΓ={Γkij}are arranged to
be in two vectorial forms.