Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

3.3. ORTHOGONAL DECOMPOSITION FOR TENSOR FIELDS 137


1) The tensor field u can be orthogonally decomposed into

(3.3.9) u=∇Aφ+v with divAv= 0 ,

whereφ∈H^1 (Trk−^1 M)orφ∈H^1 (Trk− 1 M).

2) IfMis a compact Riemannian manifold, then u can be orthogonallydecomposed into

(3.3.10) u=∇Aφ+v+h,

whereφand v are as in (3.3.9), and h is a harmonic field, i.e.

divAh= 0 , ∇Ah= 0.

In particular, the subspace of all harmonic tensor fields in L^2 (TrkM)is of finite dimen-
sional:

(3.3.11)


H(TrkM) ={h∈L^2 (TrkM)|∇Ah= 0 ,divAh= 0 },and
dimH(TrkM)<∞.

Remark 3.18.The above orthogonal decomposition theorem implies thatL^2 (TrkM)can be
decomposed into


(3.3.12)


L^2 (TrkM) =G(TrkM)⊕L^2 D(TrkM) for general case,
L^2 (TrkM) =G(TrkM)⊕H(TrkM)⊕L^2 N(TrkM) forMcompact Riemainnian.

HereHis as in (3.3.11), and


G(TrkM) ={v∈L^2 (TrkM)|v=∇Aφ,φ∈H^1 (Trk− 1 M)},
L^2 D(TrkM) ={v∈L^2 (TrkM)|divAv= 0 },
L^2 N(TrkM) ={v∈L^2 D(TrkM)|∇Av 6 = 0 }.

They are orthogonal to each other:


L^2 D(TrkM)⊥G(TrkM), L^2 N(TrkM)⊥H(TrkM), G(TrkM)⊥H(TrkM).

Remark 3.19.The orthogonal decomposition (3.3.12) ofL^2 (TrkM)implies that if a tensor
fieldu∈L^2 (TrkM)satisfies that


〈u,v〉L 2 =


M

(u,v)


−gdx= 0 ∀divAv= 0 ,

thenumust be a gradient field, i.e.


u=∇Aφ for someφ∈H^1 (Trk−^1 M)orH^1 (Trk− 1 M).

Likewise, ifu∈L^2 (TrkM)satisfies that


〈u,v〉L 2 = 0 ∀v∈G(TrkM),

thenu∈L^2 D(TrkM). It is the reason why we define a divA-free field by (3.3.8).

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