第52卷第5期 复旦学报(自然科学版) VoL. 52 No 5 2013年10月 Journal of Fudan University(Natural Science) Oct.2013 Article ID:0427-7104(2013)05-0688-24 On two Kinds of Differential Operators on General Smooth Surfaces lE Xi-lin Department of Mechanics Engineering Science, Fudan University, Shanghai 200433, China) Abstract: Two kinds of differential operators that can be generally defined on an arbitrary smooth surface in a finite dimensional Euclid space are studied, one is termed as surface gradient and the other one as Levi-Civita gradient. The surface gradient operator is originated from the differentiability of a tensor field defined on the surface. Some integral and differential identities have been theoretically studied that play the important role in the studies on continuous mediums whose geometrical configurations can be taken as surfaces and on interactions between fluids nd deformable boundaries. The definition of Levi-Civita gradient operator is based on Levi-Civita connections generally defined on Riemann manifolds, It can be used to set up some differential identities in the intrinsic/ coordiantes-independent form that play the essential role in the theory of vorticity dynamics for two dimensional flows on general fixed smooth surfaces. Keywords: surface gradient operator; Levi-Civita gradient operator ic generalized Stokes formulas; fluid-solid interactions with deformable boundaries; surface deformation theory; two dimensional flows on fixed smooth surface CLC Number:(331 Document code: A 1 Introduction Generally, an m-dimensional surface in m+1 Euclid space can be represented as (x,t)∈R In the case that Is is a nonsingular point, ig (xx,t):aE titutes the so-called covariant basis of the tangent space T, 2 and there exists uniquely one direction n(xx, t) that is particular to the tangent space, i.e. (n,gi )(xs,tr+1=0(i=1,2,",m) Two kinds of the fundamental affine tensor could be defined GA8i8'Og', g: =(gi,g, )(xx, t)R K仝bgQg,b that are termed as the metric tensor and the curvature tensor respectively. Gaussian curvature is defined as KG :=det [bi l/ detLg]=det[b]=: det B and mean curvature as H:=b:=: tr K. In the whole paper, Einstein summation convention is adopted accompanying he indices are represented by lower, upper case letters or Greek alphabets in the related studies. Foundation item: Projects support by National Nature Science Foundation of China(11172069)and undergraduate key reform project Curriculum system of theories and applications of modern continuum mechanics issued by Shanghai Municipal Education Commission in 2011 Biography:XIEXi-lin(1974-),male,Associateprofessor,E-mail:xiexilin@fudan.edu.cn
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第5期 谢锡麟:一般光滑曲面上的二类微分算子 Based on the differential calculus in Rmtl, one has the following sotermed frame move equations ar Ggk Tbin n an ge. axi ts,t)=-Nig+bi n where Tik and r are the Christoffel symbols of the first and second kinds respectively. In addition,one has the relation between metric tensor and Christoffel symbol r;=1(84+984-)(x2, An m dimensional smooth surface embedded in m+1 dimensional Euclid space is naturally a Riemann manifold with the metric represented by the metric tensor and the covariant derivative differentiation denoted by Vi defined as, say g: =gi g gE(TE)is a tensor field with order 3 on the surface Vc仝ax2(x,)++p一D The fundamentals of differential calculus on a surface can be referred to the monographs by Dubrovin et altl and guoli Two kinds of differential operators on the surface are to be studied that are termed as surface gradient operator and Levi-Civita gradient operator respectively The whole content of the present paper can be divided into two parts. The first part is on the surface gradient tensor that is originated from the differentiation of a tensor field defined on the surface. As applications, four related aspects in fluid and solid mechanics are referred that include $2. 1 intrinsic generalized Stokes formulas in R3 with three kinds of applications, 82.2 primary properties of deformation gradient tensor for thin enough continuous mediums, 82. 3 strain tensor on an arbitrary deformable surface. The second part is on the Levi-Civita gradient operator that is based on Levi-Civita connection possessed by any riemann manifold. As to its applications refer to 83. 1 some primary identities in vorticity dynamics of two dimensional flows on fixed smooth surfaces and $3. 2 some identities of affine surface tensors Generally, the surface gradient operator is more familiar to mechanicians and Levi-Civita connection is to mathematicians. However, all of the applications as indicated in the present paper are closely linked to the mechanics of continuous mediums whose geometrical configurations are either bulks surfaces. And all of the related results accompanying with deductions are independent to other studies. 2 Surface gradient operator Generally, the surface gradient operator vega is defined as, say E92(R) gi g'te g-ar(. 8, @8+6.38. n+. ng'+b. an@on) )⑧g+φ.bk(g-n)Qg+更.b(g VΦ∴.(go-g)n+Φ.sb(go-n)Qn-Φ.36i(ga-g)Qg,]+ Φ:,(go-n)⑧g-Φ.bi(g。-g,)⑧g+φ.b1(g-n)Qn 國:3(go-n)⑧n-Φ2.3b(g-g,)②n-.sb(g-n)⑧g
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复旦学报(自然科学版) 第52卷 where .- represents any available algebra tensor operator, vi denotes the covariant derivative/ differentation of the tensor component that is just effective to the indices with respect to the tangent plane, i.e. i, j in the above representations W,△axg(x,)+0,一,四,△2(x2)+ (rs,t)-Ty a① where Ti denotes Christoffel symbol of the second kind. The contravariant derivative relates generally to the co-variant one through vag"'V. The change of the order of co- and contravariant derivatives must be related to riemannian- Christoffel tensor. that is VΦ,=wVp·,+R更,+R:更, where R- p. Abpbg-bypbin denotes the component of Riemannian-Christoffel tensor[2. In addition,in the case of two dimensional Riemannian manifolds, Riemannian-Christoffel tensor can be represented by Gaussian curvature and metric tensor as revealed by the relation R*=KG(8po-gsgig) It should be noted that the definition of the surface gradient operator is based on the differential calculus in the normed linear tensor space, namely, one has (△g,)·(vΦ) ①(xx+△rx,t)-Φ(xx,t) o(△xx) (ΦQv)·(△rg,) yΦ=φ.g⑧gn∈73(R3), one has Φ(x+△x,t)=更.(xx+△xx,t)(gQgn)(x+△x,t)∈3(R3) with the differentiations of the tensor component and basis vectors (x2+△2,)=5(x2,)+∞0 a:(x,t)△x+0.(△x)∈R g(xx+△x,t)=g(xx,t)+(x,t)△x+01(△x)∈R3, g(xs+ATs, t)=g(xs,t)+ ax(xx,t)△x+o(△x)∈R3, n(xx+△x,t)=n(xx,t)+(x,t)△x+o3(△xx)∈R Accompanying the multi-linearity of the representation of any simple tensor with the frame movement equations, the above mentioned representation can be attained. In the view of differentiation, the full dimensional gradient of a tensor filed defined on a domain can be taken as its derivative. Similarly, the surface gradient of a tensor field defined on a surface is its derivative also Consequently, the partial derivative of the tensor with respect to one of the component of the surface coordinates can be determined g·(VΦ)=(8v)·g1 dx+Mn,D)-d(,)(g)·(W)+0(A), (d⑧v)·(g1)+o() where i, denotes the canonical basis vector in the parametric space
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第5期 谢锡麟:一般光滑曲面上的二类微分算子 691 2.1 Intrinsic generalized Stokes formulas in R3 In the first instance, the so termed semi orthogonal curvilin near coo nates with respect to a surface is constructed as shown in Fig. 1. i-line curve F-curve 2 Fig. 1 Sketch of the semi-orthogonal curvilinear coordinates with respect to a surface 2 Generally, the smooth surface in R takes the following form E(x):R29,3x=x E(x2)=X2|(x2)∈R, where n ixel2-i are surface parameters, in other words Gaussian coordinates. Without lost of the generality, it is assumed that E(xs) is an one-to-one/injective mapping on the definition domain 9r Subsequently, the mapping defined on the neighborhood of the surface can be constructed X(x,y):R3293 ,)△x(xx)+如(x)∈R3 where n(xx) is the unit normal vector of the surface. And the definition domain is 9,=9,X(-,A) in which A is a suitable positive number In order to calculate the determinant of the Jacobian matrix DX (s,5)ER3, the partial derivatives of X(xy, 5) with respect to its all coordinates are firstly calculated (xx)+(-b)g,(xx)=(-b)g,(xx) g3(s,5 )=g3(xx) where gi (xx))2-1 is the local covariant basis vectors of the surface. Secondly, the following calculation g1×g2=(-)(-)g,×g=(6-0一6+的妈)gp×g2 g2-yg1×g2+2(b远一6b)g1×g2 (1一b+detb])g1×g2=(1-H+Kc)g1×g Consequently, it is deduced that (1-H+gK)[g1,g2,n](x)=:(1-H+y2KG)√gs
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复旦学报(自然科学版) 第52卷 where vgs:=degu](xs)>0 is termed as the area element of the surface. Finally, it can be concluded that the mapping X(xs, 5) actualizes a smooth diffeomorphism between the definition domain X(-1,A)and the range of arrival, provided a is small enough. Smooth diffeomorphism can also be termed as smooth curvilinear coordinatesL4. In the present case, it is evident that g3(Ts, 5) is perpendicular to (g; (s, 5))=1. Namely, the curvilinear coordinates (xx, 5) is semi- orthogonal. On the other hand, Iry, s) is still a kind of full dimensional curvilinear coordinates so that the following identity is keeping valid g(x,3)(x,)+g(x5) F(x32(x,9+n(2)是2)(a2:∈92x(-, due to the chain-rule in differential calculus. On the surface, one has V=i.k(X)=g(x)(x,0)+n(x)(x,0)=V+n(xx)(xx,0) through the continuously extension. Proposition 1 (Intrinsic generalized Stoke es formulas of the first kind) d=(n×)。-olr Proof It is well known that Stokes formula in the fundamental calculus takes the following form where all of the quantities are represented through the canonical basis, and X" denotes Cartesian coordinates in the full paper. Consequently, the vector field a should be extended differentially to a three dimensional open set in which the surface is embedded in order to fulfil the full dimensional curl operator. This kind of Stokes formula is termed as the prototype in the present paper. In order to prove the second identity listed in Proposition 1, firstly the integrant of the curve integral is expanded through the canonical basis, that is Φ-r=(ii1)°-(rin)=rpk(iCi,°-in), where i. and so on denote the canonical basis vectors econdly, the Stokes formula in the prototype is adopted to attain the surface gradient (aae,)(i Oi, -ia) Thirdly, the full dimensional gradient is represented through the surface gradient (V+max) in the process of the deduction the semi-orthogonal curvilinear coordinates with respect to the surface 2 adopted. The proof of the first identity can be obtained in the same way5]. Therefore, the proof is completed
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