here Tis denotes Christoffel symbol of the second kind. The contra-variant derivative relates generally to the co-variant one through V'2gVt. On the other hand, one defines the so termed Levi-Civita connection operator V=9V V。一重≡(gVg)。-(重189)会go-Va,(重18g3)=V重(go-9)8g where V_a denotes Levi-Civita connection defined on the surface/Riemannian manifold(see Durovin et al., 1992). It should be noted that the change of the order of co- and contra- variant derivatives must be related to Riemannian-Christoffel tensor that can be represented by Gaussian curvature and metric tensor for two dimensional riemannian manifolds as revealed by the relation =Vvp+KG(-99),+Kc(9p99-983) where R sp: 2 bibs -bspb"q denotes the component of Riemannian-Christoffel tensor It is worthy of note that Levi-Civita connection operator is just valid/ effective for surface tensor fields that just have components on the tangent plane, generally denoted by g(T2) where r E n denotes the order. However, the surface gradient operator can be applied to arbitrary tensor fields defined on the surface due to its definition is based on differential calculus of tensor normed spac 2.2 nematics Generally, the velocity of a fluid partial is determined as follows ar2 0∑ ()=a(s,1a()=vg Subsequently, the acceleration is 全(,t) (r, t)+Vvsv91+(vivi)n=agI+ann The material derivative of any surface tensor field is represented by ds9师 a(,t)=t(,)+Va(,1),V∈(T2) based on the velocity representation(1). The deformation gradient tensor denoted by F is introduced by the relation ∑(x+65,1,t)-x(x(:t),t)÷F·(∑(+6)-()),F:=(,tg()8G4() All kinds of deformation descriptions can be represented by the deformation gradient tensor with its properties as indicated in Xie et al.(2013 Similar to the familiar Helmholtz velocity decomposition, one has the relation v(x+6x,t)-v(x,t)=(V⑧V)·0∑ VV+V)g· i iVi-ViVi96g +(Vsbg)·∑,6∑=6x5gs(x)∈T∑where Γi ls denotes Christoffel symbol of the second kind. The contra-variant derivative relates generally to the co-variant one through ∇l , g lt∇t . On the other hand, one defines the so termed Levi-Civita connection operator ∇ ≡ g l∇ ∂ ∂xl ∇ ◦ −Φ ≡ (g l∇ ∂ ∂xl ) ◦ −(Φi ·jgi ⊗ g j ) , g l ◦ −∇ ∂ ∂xl (Φi ·jgi ⊗ g j ) = ∇lΦ i ·j (g l ◦ −gi ) ⊗ g j where ∇ ∂ ∂xl denotes Levi-Civita connection defined on the surface/Riemannian manifold (see Durovin et al., 1992). It should be noted that the change of the order of co- and contra￾variant derivatives must be related to Riemannian-Christoffel tensor that can be represented by Gaussian curvature and metric tensor for two dimensional Riemannian manifolds as revealed by the relation ∇p∇qΦ i ·j = ∇q∇pΦ i ·j + R i·· q ·sp ·Φ s ·j + R ·s·q j·p ·Φ i ·s = ∇q∇pΦ i ·j + KG(δ i p δ q s − gspg iq)Φs ·j + KG(gjpg sq − δ s p δ q j )Φi ·s where R i·· q ·sp · , b i p b q s − bspb iq denotes the component of Riemannian-Christoffel tensor. It is worthy of note that Levi-Civita connection operator is just valid/effective for surface tensor fields that just have components on the tangent plane, generally denoted by T r (T Σ) where r ∈ N denotes the order. However, the surface gradient operator can be applied to arbitrary tensor fields defined on the surface due to its definition is based on differential calculus of tensor normed space. 2.2 Kinematics Generally, the velocity of a fluid partial is determined as follows V , ∂Σ ∂t (ξ, t) = ∂xi ∂t (ξ, t) ∂Σ ∂xi (x) =: V i gi (1) Subsequently, the acceleration is a , ∂V ∂t (ξ, t) = ( ∂V l ∂t (x, t) + V s∇sV l ) gl + ( b ijViVj ) n =: a l gl + ann (2) The material derivative of any surface tensor field is represented by dΦ dt ≡ Φ˙ , ∂Φ ∂t (ξ, t) = ∂Φ ∂t (x, t) + V s ∂Φ ∂xs (x, t), ∀ Φ ∈ T r (T Σ) (3) based on the velocity representation (1). The deformation gradient tensor denoted by F is introduced by the relation Σ(x(ξ + δξ, t), t) − Σ(x(ξ, t), t) .= F · ( ◦ Σ(ξ + δξ) − ◦ Σ(ξ) ) , F := ∂xi ∂ξA (ξ, t)gi (x) ⊗ GA(ξ) All kinds of deformation descriptions can be represented by the deformation gradient tensor with its properties as indicated in Xie et al. (2013). Similar to the familiar Helmholtz velocity decomposition, one has the relation V (x + δx, t) − V (x, t) .= (V ⊗ ∇) · δ t Σ = [ 1 2 (∇jVi + ∇iVj )g i ⊗ g j ] · δ t Σ + [ 1 2 (∇jVi − ∇iVj )g i ⊗ g j ] · δ t Σ +(V s bstg t ) · δ t Σ, δ t Σ = δxs gs (x) ∈ T Σ (4) 3