Mathematical Principles of Theoretical Physics

142 CHAPTER 3. MATHEMATICAL FOUNDATIONS

is closed. Hence it follows that there is aφ∈H^1 (M)such that

φk=

∂ φ

∂xk

for 1≤k≤n.

Assertion (2) is proved.

Now we prove Assertion (3). Taking the divergence on both sides of (3.3.26), we obtain
that

(3.3.32) DiDiφj=Diuij.

By the Laplace-Beltrami operator in (3.2.31),

(3.3.33) − ̃∆φj=DiDiφj+Rkjφk,

where ̃∆=δd+dδ. By the Hodge theory, forω=φidxiwe have

(3.3.34)

dω= 0 ⇔ φi=∂ φ/∂xi,

δ ω=∆φ ⇔ φi=∂ φ/∂xi.

Here∇is the gradient operator, and∆the Laplace operator as in (3.2.28). It follows from
(3.3.34) that
̃∆ω= (δd+dδ)ω=dδ ω ⇔φi=∂ φ/∂xi,

and

dδ ω=

∂

∂xi

(∆φ)dxi.

Hence we deduce from (3.3.33) that

(3.3.35) DiDjφj=−

∂

∂xj

(∆φ)−Rkj

∂ φ

∂xk

⇔ φi=∂ φ/∂xi.

Inserting (3.3.32) in (3.3.35) we obtain that the equations

∂

∂xj

(∆φ)+Rkj

∂ φ

∂xk

=−Diuij

have a solutionφif and only ifφjin (3.3.26) is a gradient field ofφ, i.e.φj=∂ φ/∂xj.
Assertion (3) is proven, and the proof of the theorem is complete.

Remark 3.21.The conclusions of Theorem3.20are also valid for second-order contra-
variant symmetric tensorsu={uij}, and the decomposition is given as follows:

uij=vij+gikgjlDkDlφ

Divij= 0 , vij=vji, φ∈H^2 (M).