Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 131


and its gradient (i.e. covariant derivatives) read as


(3.2.29)


∇u={Dkui},

Dkui=
∂ui
∂xk

+Γiklul,

and the divergence of∇uis


div·∇ui=DkDkui=gklDl(Dkui) =gkl

[



∂xl

(Dkui)+Γil jDkuj−ΓkljDjui

]


.


Then, by (3.2.29) we obtain that


div·∇u=gkl

[



∂xl

(


∂ui
∂xk

+Γik juj

)


+Γil j

(


∂uj
∂xk

+Γksjus

)


−Γklj

(


∂ui
∂xj

+Γijsus

)]


(3.2.30).


6.Laplace-Beltrami operators.The Laplace-Beltrami operators are defined by

∆=dδ+δd,

wheredis the differential operator, andδthe Hodge operator, which are defined on the spaces
of all differential forms. For a scalar field,∆is the same as div·∇, i.e.


∆u=div·∇u, asu∈W^2 ,p(M⊗pR^1 ).

For vector fields and covector fields, we have


(3.2.31)


∆ui=−div(∇ui)−gijRjkuk,
∆ui=−div(∇uj)−gk jRijuk,

whereRijis the Ricci curvature tensor defined by (2.3.31), and in (3.2.30) we give the formula
of div(∇uk). The expression of div(∇ui)is written as


div(∇ui) =gkl

{



∂xl

[


∂ui
∂xk

−Γikjuj

]


−Γlkj

[


∂ui
∂xj

−Γrijur

]


−Γlij

[


∂uj
∂xk

−Γrjkur

]}


(3.2.32).


Physically, it suffices to introduce the Laplace-Beltrami operators∆only for vector and
covector fields, i.e. the formula in (3.2.31).


Remark 3.10.The Navier-Stokes equations governing a fluid flow on a sphere(e.g. the
surface of a planet) are expressed in the spherical coordinate system, and the viscosity dif-
ferential operators are the Laplace-Beltrami operators given by (3.2.31); see Chapter 7 for
details.


3.2.4 Gauss formula


IfM=Rn, it is known that the Gauss formula is


Rn

u·∇fdx=−


Rn

fdivudx ∀u∈H^1 (Rn×Rn),f∈H^1 (Rn),
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