Mathematical Principles of Theoretical Physics

130 CHAPTER 3. MATHEMATICAL FOUNDATIONS

Letu∈W^1 ,p(T∗M),u= (u 1 ,···,un). Then,

divu=Dkuk=gklDl(uk) =Dl(glkuk)

=(by( 3. 2. 24 ))

=

1

√

−g

∂

∂xl

(

√

(3.2.25) −gglkuk).

The formula (3.2.24) and (3.2.25) give expression of divuas:

(3.2.26) divu=









1

√

−g

∂

∂xk

(

√

−guk) foru∈W^1 ,p(TM),

1

√

−g

∂

∂xk

(

√

−ggklul) foru∈W^1 ,p(T∗M).

3.Laplace operators.The Laplace operatorDkDkinRnis in the familiar form

DkDk=

∂

∂xk

∂

∂xk

.

However, the Laplace operatorsDkDkdefined on a Riemannian manifold are usually very
complex.
By (3.2.20) and (3.2.22), we have

W^2 ,p(TrkM)

∇k

→W^1 ,p(Trk+ 1 M)div→Lp(TrkM),

W^2 ,p(TrkM)∇

k

→W^1 ,p(Trk+^1 M)→divLp(TrkM).

Hence, the Laplace operator div·∇is the mapping:

(3.2.27) div·∇:W^2 ,p(TrkM)→Lp(TrkM).

By (3.2.21) and (3.2.23), div·∇is written as

div·∇=DkDk=gklDkDl.

4.Expression ofdiv·∇onM⊗pR^1 .A scalar fielduonMcan be regarded asu:M→
M⊗pR^1. In this case,∇uis written as

∇u=

(

∂u

∂x^1

,···,

∂u

∂xn

)

.

By (3.2.26), we get that

(3.2.28) div·∇u=

1

√

−g

∂

∂xk

(

√

−ggkl

∂u

∂xl

)

.

5.Expression ofdiv·∇on TM.A vector fieldu:M→TMcan be written as

u= (u^1 ,···,un),