Furthermore. one can do the following deduction F NE+Flex(a F)+VX(V =-(V XV). ms+(Vxa).ng+(vxv(v)) 0u3+(V×a)·ny where the identities F=(VoV). F and d Fl/dt=8F(see Xie et al., 2013)are utilized As an appendant of the above proof, one has the conclusion Corollary 3(Lagrange Theorem on Vorticity) In the case of the acceleration field is ir rotational, a patch of continuous medium that is initially irrotational will keep irrotational at any time, conversely a patch of continuous medium with initially nonzero vorticity will possess vorticity at any time although the value can be changed Secondly, the following identity can be readily set up Lemma 4 V×(V×b)=V(V.b)+Kcb-△b,Vb∈T∑,△bV·(V×b) Proof V×(×b=Vx(oVa)×(g)=(r7m)×[)n e3qE3pi Vq(VPb)9k=( p 9-89)Vg(VPb)9k=Vi(Vb)9k-Vp(VPb)92 =V(Vb)9gk-△b Furthermore. one has Vi(V6)9k=V(Vib)+R. si-6'19k=V(Vb)+KG(6:8s-9sig)6'gk v(V·b)+Kcb where the change of the order of covariant or contra-variant derivatives should be related to Riemannian-Christoffel tensor. It is the end of the proof. It should be pointed out that this kind of identities is still keeping valid for any tensor field As an application, the governing equation of momentum conservation on the tangent plane (8)can be rewritten pag=VI-pV×u+2KGV+ fur. 1g,Ⅱ=-p+210 where V xw=E3 aF(a, t)g. Subsequently, the following coupling relations can be attained just by doing the dot and cross products by e on both sides of (14)respectively Corollary 4(Coupling Relations between Directional Derivatives of II and w pa·e+(Vxu)·e-2Kcv·e-f HDu's-lea, e, ns]+(VIL, e, ns]+(2uKGV, e, ns)+If>, e, nsl for alle st le=l. The coupling relations are valid at any point in the flow field Thirdly, the intrinsic decomposition is still valid for any surface tensor, i.e. there existsFurthermore, one can do the following deduction ω˙ 3 = − θ |F| [ ◦ ∇ × (V · F) ] · NΣ + 1 |F| [ ◦ ∇ × (a · F) + ◦ ∇ × ( V · (V ⊗ Σ ∇) · F )] · NΣ = −θ(∇ × V ) · nΣ + (∇ × a) · nΣ + ( ∇ × ∇ ( |V | 2 2 )) · nΣ = −θω3 + (∇ × a) · nΣ where the identities F˙ = (V ⊗ Σ ∇) · F and d|F|/dt = θ|F| (see Xie et al., 2013) are utilized. As an appendant of the above proof, one has the conclusion Corollary 3 (Lagrange Theorem on Vorticity) In the case of the acceleration field is ir￾rotational, a patch of continuous medium that is initially irrotational will keep irrotational at any time, conversely a patch of continuous medium with initially nonzero vorticity will possess vorticity at any time although the value can be changed. Secondly, the following identity can be readily set up Lemma 4 ∇ × (∇ × b) = ∇(∇ · b) + KGb − ∆b, ∀ b ∈ T Σ, ∆b , ∇ · (∇ × b) Proof: ∇ × (∇ × b) = ∇ × [(g p∇ ∂ ∂xp ) × ( big i ) ] = ( g q∇ ∂ ∂xq ) × [ (∇pbi)ϵ pi3n ] = ϵ 3kqϵ3pi∇q(∇p b i )gk = (δ k p δ q i − δ q p δ k i )∇q(∇p b i )gk = ∇i(∇k b i )gk − ∇p(∇p b i )gi = ∇i(∇k b i )gk − ∆b Furthermore, one has ∇i(∇k b i )gk = [∇k (∇ib i ) + R i··k ·si· b s ]gk = ∇(∇ · b) + KG(δ i i δ k s − gsig ik)b s gk = ∇(∇ · b) + KGb where the change of the order of covariant or contra-variant derivatives should be related to Riemannian-Christoffel tensor. It is the end of the proof. It should be pointed out that this kind of identities is still keeping valid for any tensor field. As an application, the governing equation of momentum conservation on the tangent plane (8) can be rewritten as ρ alg l = ∇Π − µ∇ × ω + 2µKGV + fsur,lg l , Π := −p + 2µθ (14) where ∇ × ω = ϵ k3l ∂ω3 ∂xk (x, t)gl . Subsequently, the following coupling relations can be attained just by doing the dot and cross products by e on both sides of (14) respectively. Corollary 4 (Coupling Relations between Directional Derivatives of Π and ω )    ∂Π ∂e = ρa · e + µ(∇ × ω) · e − 2µKGV · e − fΣ · e µ ∂ω3 ∂e = −[ρa, e, nΣ] + [∇Π, e, nΣ] + [2µKGV , e, nΣ] + [fΣ, e, nΣ] for all e ∈ T Σ s.t |e| = 1. The coupling relations are valid at any point in the flow field. Thirdly, the intrinsic decomposition is still valid for any surface tensor, i.e. there exists 7