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
@2189.7:97P12:8-?.1:1ND4.:-!&&&$#!)*"-:82:87CWC-82-?7G70 C7N1C, 5C1j79?9CAAB*C?C+GLGD=+>/D7=>AB=G2,(2KK?B*2DB>,G>/ +>(=A,*>,DB,CC+ +=*72,B*G .QQ278U0>4-:W4-.I2:.9.5-/+829-?.1:D1,,.QQ.1:.:#!&& [8:52(O6C!gi+g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a.7,-::,-:.N1/8Q6i?9-:U72Q78?1Q7?25Q1,78.NN7C7:?.-/.87:?.?.7Q.:?47.:?C.:Q.9* 911C8.-:?7Q%.:8757:87:?N1C,?4-?5/-0?477QQ7:?.-/C1/7.:?47?471C01NA1C?.9.?080:-,.9QN1C?Z18.,7:Q.1:-/ N/1ZQ1:W7:7C-/N.V78Q,11?4Q2CN-97Q6 B3CD:27?!Q2CN-97WC-8.7:?157C-?1C'Y7A.%D.A.?-WC-8.7:?157C-?1C'.:?C.:Q.9W7:7C-/.b78>?1G7QN1C,2/-Q'N/2.8%Q1/.8 .:?7C-9?.1:QZ.?487N1C,-U/7U12:8-C.7Q'Q2CN-9787N1C,-?.1:?471C0'?Z18.,7:Q.1:-/N/1ZQ1:N.V78Q,11?4Q2CN-97 /U/JKLA32!T''& N:6BR*B?*R$6BRkX!*B$*R"!49$D"a+d& $ ->3BR*B?*R$3BRkX &*B &4R 9 ! !49$D"$""a+d& ?4-?-C7?7C,78-Q?47,7?C.9?7:Q1C-:8?4792CA-?2C7?7:Q1CC7Q579?.A7/06@-2QQ.-:92CA-?2C7.Q87N.:78 -Q.9.7:97" S1/6(#R16( T9?6#!&
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
E-Q781:?478.NN7C7:?.-/9-/92/2Q.: a+d&$1:74-Q?47N1//1Z.:WQ1%?7C,78NC-,7 ,1A7,7:? 732-?.1:Q% &*B &4R 9 !49$ D"X6H RB*Hd3RB"X6RB$H*Hd3RB" &*B &4R 9 !49$ D"X[6B RH*Hd3B R - . / " ' &" &4R 9 !49$ D"X[3RH*HX[3H R*H$ Z47C76RB$H-:86H RB-C7?47D4C.Q?1NN7/Q0,U1/Q1N?47N.CQ?-:8Q791:8G.:8QC7Q579?.A7/06i:-88.?.1:$1:7 4-Q?47C7/-?.1:U7?Z77:,7?C.9?7:Q1C-:8D4C.Q?1NN7/Q0,U1/ 6BR$HX& # &6RH &4B 9 d&6BH &4R 9 [&6BR ! &4H 9"!49$ D"# =:+ 8.,7:Q.1:-/Q,11?4Q2CN-977,U78878.:+ d&8.,7:Q.1:-/+29/.8Q5-97.Q:-?2C-//0- a.7,-::,-:.N1/8 Z.?4?47 ,7?C.9C75C7Q7:?78U0?47 ,7?C.9?7:Q1C-:8?4791A-C.-:?87C.A-?.A7* 8.NN7C7:?.-?.1:87:1?78U0 % ?87N.:78-Q$Q-0#kX!BR::H*B?*R?*H4"'!+"".Q-?7:Q1CN.7/8Z.?41C87C' 1:?47Q2CN-97$ % ?!BR::H> &!BR::H &4? 9 !49$ D"d6B ?G!GR::Hd6R ?G!BG::H[6G ?H!BR::G6 ;47N2:8-,7:?-/Q1N8.NN7C7:?.-/9-/92/2Q1:-Q2CN-979-:U7C7N7CC78?1?47 ,1:1WC-54QU0 c2UC1A.:7?-/(&) -:8@21(#) 6 ;Z1G.:8Q1N8.NN7C7:?.-/157C-?1CQ1:?47Q2CN-97-C7?1U7Q?28.78?4-?-C7?7C,78-QQ2CN-97 WC-8.7:?157C-?1C-:8Y7A.%D.A.?-WC-8.7:?157C-?1CC7Q579?.A7/06;47Z41/791:?7:?1N?475C7Q7:?5-57C 9-:U78.A.878.:?1?Z15-C?Q6;47N.CQ?5-C?.Q1:?47Q2CN-97WC-8.7:??7:Q1C?4-?.Q1C.W.:-?78NC1,?47 8.NN7C7:?.-?.1:1N-?7:Q1CN.7/887N.:781:?47Q2CN-976=Q-55/.9-?.1:Q$N12CC7/-?78-Q579?Q.:N/2.8-:8 Q1/.8,794-:.9Q-C7C7N7CC78?4-?.:9/287o#6&.:?C.:Q.9W7:7C-/.b78>?1G7QN1C,2/-Q.:a' Z.?4?4C77 G.:8Q1N-55/.9-?.1:Q$ o#6# 5C.,-C0 5C157C?.7Q1N87N1C,-?.1: WC-8.7:??7:Q1CN1C?4.: 7:12W4 91:?.:212Q,78.2,Q$o#6'Q?C-.:?7:Q1C1:-:-CU.?C-C087N1C,-U/7Q2CN-976;47Q791:85-C?.Q1:?47 Y7A.%D.A.?-WC-8.7:?157C-?1C?4-?.QU-Q78 1: Y7A.%D.A.?-91::79?.1: 51QQ7QQ78 U0-:0 a.7,-:: ,-:.N1/86=Q?1.?Q-55/.9-?.1:QC7N7C?1 o'6&Q1,75C.,-C0.87:?.?.7Q.:A1C?.9.?080:-,.9Q1N?Z1 8.,7:Q.1:-/N/1ZQ1:N.V78Q,11?4Q2CN-97Q-:8o'6#Q1,7.87:?.?.7Q1N-NN.:7Q2CN-97?7:Q1CQ6 @7:7C-//0$?47Q2CN-97WC-8.7:?157C-?1C.Q,1C7N-,./.-C?1,794-:.9.-:Q-:8Y7A.%D.A.?-91::79?.1: .Q?1,-?47,-?.9.-:Q6e1Z7A7C$-//1N?47-55/.9-?.1:Q-Q.:8.9-?78.:?475C7Q7:?5-57C-C79/1Q7/0 /.:G78?1?47,794-:.9Q1N91:?.:212Q,78.2,QZ41Q7W71,7?C.9-/91:N.W2C-?.1:Q-C77.?47CU2/GQ1C Q2CN-97Q6=:8-//1N?47C7/-?78C7Q2/?Q-991,5-:0.:WZ.?487829?.1:Q-C7.:8757:87:??11?47CQ?28.7Q6 # >2CN-97WC-8.7:?157C-?1C @7:7C-//0$?47Q2CN-97WC-8.7:?157C-?1C9 " @*? & &4? 9 .Q87N.:78-Q$Q-0#4"#!a'"$ 9 " A[#@ *? & ! &4? 9"A[!!B:R*B?*Rd!B:'*B?"d!':R"?*Rd!':'"?""> *? A[ & &4? 9 !!B:R*B?*Rd!B:'*B?"d!':R"?*Rd!':'"?""X ( % ?!B:R!*? A[*B"?*Rd!B:R3?B!*? A[""?*Rd!B:R3R ?!*? A[*B"?")d ( % ?!B:'!*? A[*B"?"d!B:'3?B!*? A[""?"[!B:'3G ?!*? A[*B"?*G)d ( % ?!':R!*? A[""?*R[!':R3G ?!*? A[*G"?*Rd!':R3R ?!*? A[""?")d ( % ?!':'!*? A[""?"[!':'3G ?!*? A[*G"?"[!':'3G ?!*? A[""?*G)$ ( qrs%ÑÍÞA°bôFÖsTl ()*
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
Z47C7A[C75C7Q7:?Q-:0-A-./-U/7-/W7UC-?7:Q1C157C-?1C$ % ?87:1?7Q?4791A-C.-:?87C.A-?.A7* 8.NN7C7:?-?.1:1N?47?7:Q1C91,51:7:??4-?.Qj2Q?7NN79?.A7?1?47.:8.97QZ.?4C7Q579??1?47?-:W7:? 5/-:7$.676B$R.:?47-U1A7C75C7Q7:?-?.1:Q$ % ?!B:R>&!B:R &4? 9 !49$ D"d6B ?G!G:R[6G ?R!B:G$ % ?!B:'>&!B:' &4? 9 !49$ D"d6B ?G!G:'$ % ?!':R> &!':R &4? 9 !49$ D"[6G ?R!':G$ % ?!':'>&!':' &4? 9 !49$ D"$ Z47C76B ?G87:1?7QD4C.Q?1NN7/Q0,U1/1N?47Q791:8G.:86;4791:?C-A-C.-:?87C.A-?.A7C7/-?7QW7:7C-//0?1 ?4791%A-C.-:?1:7?4C12W4 ? % >6?D % D6;4794-:W71N?471C87C1N91%-:891:?C-A-C.-:?87C.A-?.A7Q,2Q? U7C7/-?78?1a.7,-::.-:%D4C.Q?1NN7/?7:Q1C$?4-?.Q$ % K ^ % !B:RX % ^ % K!B:Rd"B::^ :GK:!G:Rd":G:^ R:K:!B:G$ Z47C7"B::^:GK: >3B K3^ G[3GK3B^87:1?7Q?4791,51:7:?1Na.7,-::.-:%D4C.Q?1NN7/?7:Q1C(#) 6i:-88.?.1:$.:?47 9-Q71N?Z18.,7:Q.1:-/a.7,-::.-:,-:.N1/8Q$a.7,-::.-:%D4C.Q?1NN7/?7:Q1C9-:U7C75C7Q7:?78U0 @-2QQ.-:92CA-?2C7-:8,7?C.9?7:Q1C-QC7A7-/78U0?47C7/-?.1:"B::^:GK: X.-0#X!B:R'*B?*R?"4"'!a'"$1:74-Q #!49d%49$ D"X!B:R'!49d%49$ D"!*B?*R?""!49d%49$ D"4"'!a'" Z.?4?478.NN7C7:?.-?.1:Q1N?47?7:Q1C91,51:7:?-:8U-Q.QA79?1CQ !B:R'!49d%49$ D"X!B:R'!49$ D"d&!B:R' &4G 9 !49$ D"%4G 9d>B:R'!%49"4a$ *B!49d%49$ D"X*B!49$ D"d&*B &4G 9 !49$ D"%4G 9d/B!%49"4a'$ *R!49d%49$ D"X*R!49$ D"d&*R &4G 9 !49$ D"%4G 9d/R!%49"4a'$ "!49d%49$ D"X"!49$ D"d&" &4G 9 !49$ D"%4G 9d/'!%49"4a'# =991,5-:0.:W?47,2/?.%/.:7-C.?01N?47C75C7Q7:?-?.1:1N-:0Q.,5/7?7:Q1CZ.?4?47NC-,7,1A7,7:? 732-?.1:Q$?47-U1A7,7:?.1:78C75C7Q7:?-?.1:9-:U7-??-.:786i:?47A.7Z1N8.NN7C7:?.-?.1:$?47N2// 8.,7:Q.1:-/WC-8.7:?1N-?7:Q1CN./7887N.:781:-81,-.:9-:U7?-G7:-Q.?Q87C.A-?.A7(') 6>.,./-C/0$?47 Q2CN-97WC-8.7:?1N-?7:Q1CN.7/887N.:781:-Q2CN-97.Q.?Q87C.A-?.A7-/Q16 D1:Q7327:?/0$?475-C?.-/87C.A-?.A71N?47?7:Q1CZ.?4C7Q579??11:71N?4791,51:7:?1N?47 Q2CN-97911C8.:-?7Q9-:U787?7C,.:78% &# &4? 9 !49$ D">/.,)&! #!49d)0?$ D"[#!49$ D" ) X*?: 9 " ! ?#"X #? " ! "9 :*?$ 827?1 #!49d)0?$ D"[#!49$ D"X !)*?": 9 " ! ?#"d/!)"$ #? " ! "9 :!)*?"d/!)" - . / $ Z47C70?87:1?7Q?479-:1:.9-/U-Q.QA79?1C.:?475-C-,7?C.9Q5-976 (*! L & !#$%&" (#
第5期 谢锡麟:一般光滑曲面上的二类微分算子 691 2.1 Intrinsic generalized Stokes formulas in R3 In the first instance, the so termed semi orthogonal curvilin near coo nates with respect to a surface is constructed as shown in Fig. 1. i-line curve F-curve 2 Fig. 1 Sketch of the semi-orthogonal curvilinear coordinates with respect to a surface 2 Generally, the smooth surface in R takes the following form E(x):R29,3x=x E(x2)=X2|(x2)∈R, where n ixel2-i are surface parameters, in other words Gaussian coordinates. Without lost of the generality, it is assumed that E(xs) is an one-to-one/injective mapping on the definition domain 9r Subsequently, the mapping defined on the neighborhood of the surface can be constructed X(x,y):R3293 ,)△x(xx)+如(x)∈R3 where n(xx) is the unit normal vector of the surface. And the definition domain is 9,=9,X(-,A) in which A is a suitable positive number In order to calculate the determinant of the Jacobian matrix DX (s,5)ER3, the partial derivatives of X(xy, 5) with respect to its all coordinates are firstly calculated (xx)+(-b)g,(xx)=(-b)g,(xx) g3(s,5 )=g3(xx) where gi (xx))2-1 is the local covariant basis vectors of the surface. Secondly, the following calculation g1×g2=(-)(-)g,×g=(6-0一6+的妈)gp×g2 g2-yg1×g2+2(b远一6b)g1×g2 (1一b+detb])g1×g2=(1-H+Kc)g1×g Consequently, it is deduced that (1-H+gK)[g1,g2,n](x)=:(1-H+y2KG)√gs
#"! ;41284?83?9:2LKI(?84=) i:?47N.CQ?.:Q?-:97$?47Q1?7C,78Q7,.%1C?41W1:-/92CA./.:7-C911C8.:-?7QZ.?4C7Q579??1-Q2CN-97 .Q91:Q?C29?78-QQ41Z:.:P.W6&6 P.W6& >G7?941N?47Q7,.%1C?41W1:-/92CA./.:7-C911C8.:-?7QZ.?4C7Q579??1-Q2CN-979 @7:7C-//0$?47Q,11?4Q2CN-97.:a'?-G7Q?47N1//1Z.:WN1C,% "!49"%a#2UQ7327:?/0$?47,-55.:W87N.:781:?47:7.W4U1C41181N?47Q2CN-979-:U791:Q?C29?78 1!49$:"%a'"!49"d:"!49"4a'$ Z47C7"!49".Q?472:.?:1C,-/A79?1C1N?47Q2CN-976=:8?4787N.:.?.1:81,-.:.Q!4X!49H
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
Z47C7 槡69 kX87?(6BR)!49"2!.Q?7C,78-Q?47-C7-7/7,7:?1N?47Q2CN-976 P.:-//0$.?9-:U791:9/2878?4-??47,-55.:W1!49$:"-9?2-/.b7Q-Q,11?48.NN71,1C54.Q,U7?Z77: ?4787N.:.?.1:81,-.:!49 H
第5期 谢锡麟:一般光滑曲面上的二类微分算子 It should be pointed out that the surface gradients have nothing to do with the directional derivative with respect to the normal direction, in other words the quantity originally defined on the surface does not need to be extended if the surface gradient rather than the full dimensional gradient is adopted. Proposition 2(Intrinsic generalized Stokes formulas of the second kind) Φ十Hno-Φ)d 扣a-(xn业=(-+1一m), where the interaction direction Xn is perpendicular to the tangent vector but lies on the tangent plane as shown in Fig. 2 Fig 2 Sketch of the generalized Stokes formulas of the second kind Proof The proof of the second identity is carried out as follows. And the first one can be verified the same way5] Firstly, the integrant of the curve integral is expanded through the canonical basis d-(r×n)=(9iQi)。-(cx)= T,(iQi2°-in) Secondly, the curve integral is transferred to the surface integral according to the Stokes formula in the prototype. The deduction of the surface integrant is as follows (en④。)(i:in-in) (n④s)(i8in-in) (6b-a0)n3(n)(i②x,-i) a-na一mn的]18,) Φ。-v-(v·n)(Φ。-n)-(n:(wΦ)) Thirdly, the full dimensional gradient is represented by the surface gradient RHS n(④。-n) The proof is completed 2. 1. 1 Some integral identities for soft matter studies Yin 6 reported some kinds of novel integral identities that are taken as meaningful for soft matter
i?Q412/8U751.:?7812??4-??47Q2CN-97WC-8.7:?Q4-A7:1?4.:W?181Z.?4?478.C79?.1:-/87C.A-?.A7 Z.?4C7Q579??1?47:1C,-/8.C79?.1:$.:1?47CZ1C8Q?4732-:?.?01C.W.:-//087N.:781:?47Q2CN-97817Q :1?:778?1U77V?7:878.N?47Q2CN-97WC-8.7:?C-?47C?4-:?47N2//8.,7:Q.1:-/WC-8.7:?.Q-815?786 '2:O:?818:4# !i:?C.:Q.9W7:7C-/.b78>?1G7QN1C,2/-Q1N?47Q791:8G.:8" :&9 ! !_""A[#8?S%9 9 " ! [#]8"A[#"8"$ :&9 #A[! !_""8?S%9 #A[ 9 " ! ]8#A[""8"$ Z47C7?47.:?7C-9?.1:8.C79?.1:!H".Q57C57:8.92/-C?1?47?-:W7:?A79?1CU2?/.7Q1:?47?-:W7:?5/-:7 -QQ41Z:.:P.W6#6 P.W6# >G7?941N?47W7:7C-/.b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d" & ! &4'"[ 9 " d" & (! &4'":")!#A[""[ ": 9 " d" & (! &4'"?#)A["X#A[9 " d8#A["# ;475C11N.Q91,5/7?786 #6&6& >1,7.:?7WC-/.87:?.?.7QN1CQ1N?,-??7CQ?28.7Q ].:()) C751C?78Q1,7G.:8Q1N:1A7/.:?7WC-/.87:?.?.7Q?4-?-C7?-G7:-Q,7-:.:WN2/N1CQ1N?,-??7C ( qrs%ÑÍÞA°bôFÖsTl (*&
694 复旦学报(自然科学版) 第52卷 studies. As a kind of applications, the intrinsic generalized Stokes formulas are utilized to deduce these identities as indicated in this subsection. Firstly, the the quantity termed as conjugate fundamental tensor[s] is introduced K|K=:△gg, where K Abig g is the curvature tensor, A denotes the adjugate matrix of [b;]. Certainly, it should be pointed out that this quantity can only make sense in the case that K is nonsingular, i. e. det[ z0. On A, the following fundamental relations can be concluded =b6-b,b=△8-△, V△=0 The first two relationships can be directly verified. The last one is due to the Codazzi equation as indicated by Yin et al. All of these relations play the essential role in the following deductions. tion 3 (n X where :=l gi ars ,i:=Kk-1 Proof Firstly, it is worthy of mention that Kg=det[b ]=:KI and V=KK. v As the application of the intrinsic generalized Stokes formula of the first kind, one has r·K。-Φdl=|(n×v).( wIt h Φ)=(n×g) an=(n×g)·(k axt big thanks to nXg)·a=(nxg)·(Tb1gg+b1bm8g+b2bg②m e3livbgjte(b, bi)n=0 e other nd. one (n×V)。@=[n×(Kk-1·)]-=nx(4ga)1- It's the end of the proof. Proposition 4 (r×n)·L°-dl 」-c+」。2K,(m,-)d where=KK
Q?28.7Q6=Q-G.:81N-55/.9-?.1:Q$?47.:?C.:Q.9W7:7C-/.b78>?1G7QN1C,2/-Q-C72?./.b78?1878297?47Q7 .87:?.?.7Q-Q.:8.9-?78.:?4.QQ2UQ79?.1:6 P.CQ?/0$?4732-:?.?0?7C,78-Q91:j2W-?7N2:8-,7:?-/?7:Q1C()) .Q.:?C182978% 0-0-[&Xk%B R*B?*R$ Z47C7->3B R*B?*R.Q?4792CA-?2C7?7:Q1C$(%B R)87:1?7Q?47-8j2W-?7,-?C.V1N(3B R)6D7C?-.:/0$.?Q412/8U7 51.:?7812??4-??4.Q32-:?.?09-:1:/0,-G7Q7:Q7.:?479-Q7?4-?-.Q:1:Q.:W2/-C$.67687?(3B R)1!6 T:%B R$?47N1//1Z.:WN2:8-,7:?-/C7/-?.1:Q9-:U791:9/2878% %B RX3G G.B R[3B R$ 3B RX%G G.B R[%B R$ ='RB%? BX[='?B 3R B$#'RB%? BX[#'?B 3R B$ % B%B RX!# ;47N.CQ??Z1C7/-?.1:Q4.5Q9-:U78.C79?/0A7C.N.786;47/-Q?1:7.Q827?1?47D18-bb.732-?.1:-Q .:8.9-?78U0].:7?-/($) 6=//1N?47Q7C7/-?.1:Q5/-0?477QQ7:?.-/C1/7.:?47N1//1Z.:W87829?.1:Q6 '2:O:?818:4) :&9 !:-A[#8?S%9 "_ " ! "9 A[#8"$ Z47C79 " kX`%BR *B & &4R 9 $34kX.?1G7QN1C,2/-1N?47N.CQ?G.:8$1:74-Q :&9 !:-A[#8?X %9 "H " ! " 9 :!-A[#"8" Z.?4 ,H " ! " 9 :!-A[#"X!"H*?": &- &4? 9 A[#d-A[&# ! &4? 9"X!"H*?": -A[&# ! &4? 9"X#'?H 3R H*RA[&# &4? 9 $ ?4-:GQ?1 !"H*?":&- &4? 9 X!"H*?":! % ?3BR*B?*Rd3BR3B ?"?*Rd3BR3R ?*B?""X #'?B % ?3BR*Rd#'?B!3BR3R ?""X%# T:?471?47C4-:8$1:74-Q "H " ! " 9 A[#X "H 0-0-[&: " ( ! ") 9 A[#X "H %? B*B & ( ! &4? 9")A[#X "H!%? B* ( B")A[&# &4? 9 X#'BH%? B*HA[&# &4? 9 X#'BR%? B*RA[&# &4? 9 X [#'RB%? B*RA[&# &4? 9 X#'?B 3R B*RA[&# &4? 9 # i?^Q?477:81N?475C11N6 '2:O:?818:4$ :&9 ! !_"":34A[#8?X %9 9 " A[#8"d %9 #.< !"A[#"8"$ Z47C734X0-0-[&6 (*' L & !#$%&" (#
第5期 谢锡麟:一般光滑曲面上的二类微分算子 Proof On the left hand side. one has r×n)·(|K|K1)dl (r×n)·(K|K1.-Φ)dl LV·(K|K1。Φ)+Hn·(|K|K1a-Φ)]do 氵.(K|-1-@)d To deal with v·(|K|K1。-Φ)=g akk-1。-)= ars (|K|K1)。-Φ+|K|K-1。 one deduces the second term on the right hand side as g·(k|K-1-c a)=g·(kk-)],-9 az(|KK-1·g1)-o业= K·(ga and the first term on the right hand side (△gg) g·[(v△gg+△ban②g+△b1gQn)。-Φ] (V1△g'+△bn)。一=(b-b)bn一=(的b一b)n。-= The last identity is due to the relationship Sg=det[b]=do In studies by Yin with his collaborators[7. 8] on some integral identities, the following one plays the ssential role v·(一)b=(Xm)·(Q一)业V∈(1),yp∈R Its validity can be confirmed as soon as the intrinsic generalized Stokes formula of the second kind is namely m)·(,-)=[·(-)+hm:(e,-)J=「v,(,-)d By other ways, one can do the following calculation, let 0= g. g i without lost of the generality ( do=[1-a g an(geg;°-)(x,)d=中(r×n)·(o。-)d. The first identity is due to
JC11N T:?47/7N?4-:8Q.87$1:74-Q :&9 ! !_"":!Z-Z-[&"A[#8?S:&9 ! !_"":!Z-Z-[&A[#"8?S %9 9 " :!Z-Z-[&A[#"]8":!Z-Z-[& ( A[#")8"S %9 9 " :!Z-Z-[&A[#"8"# ;187-/Z.?4 9 " :!0-0-[& A[#"X*?: & &4? 9 !0-0-[& A[#"X *?: & &4? 9 !0-0-[&"A[#d0-0-[& A[&# ( &4? 9)$ 1:7878297Q?47Q791:8?7C,1:?47C.W4?4-:8Q.87-Q *?: 0-0-[& A[&# ! &4? 9"X(*?:!0-0-[&")A[&# &4? 9 X!0-0-[&:*?"A[&# &4? 9 X 0-0-[&: *? & ( ! &4? 9")A[#X9 " A[#$ -:8?47N.CQ??7C,1:?47C.W4?4-:8Q.87-Q *?: & &4? 9 ( !0-0-[&"A[#)X*?: & &4? 9 !%B ( R*B?*R"A[#)X *?: ! % ?%B R*B?*Rd%B R3?B"?*Rd%B R3R ( ?*B?""A[#)X ! % ?%? R*Rd%B R3R B""A[#X!. B R3G G[3B R"3R B"A[# X!3R R3G G[3B R3R B""A[#X #.?1G7QN1C,2/-1N?47Q791:8G.:8.Q -815?78$:-,7/0 :&9 ! !_"":!$A[#"8?X %9 9 " ( :!$A[#"d8":!$A[#")8"X %9 9 " :!$A[#"8"# E01?47CZ-0Q$1:79-:81?47N1//1Z.:W9-/92/-?.1:$/7?$X;BR *B?*R Z.?412?/1Q?1N?47W7:7C-/.?0$ %9 9 " :!$A[#"8"X %9 & 槡6 & &4G 9 !槡6;GR *RA[#"!49$ D"8"X %!4 & &4G 9 !槡6;GR *RA[#"!49$ D"8"X :&9 ! !_"":!$A[#"8?# ;47N.CQ?.87:?.?0.Q827?1 ( qrs%ÑÍÞA°bôFÖsTl (*#
复旦学报(自然科学版) 第52卷 .(3p)=gas(6"g8g)-@+e [Teyg⑧g,+6(ng;+bg8n)—+g,-如= (vg,+eybn)-①+6g,-ar eg+6"bn=(e+re+re)g,+eb,n= +1 s are)8, +e(r,g:b,m 1 (√g6y)g,+6 (√g6yg;) then RS=[4g),-a+g,-e]=a(8-0) The last identity is essentially due to the green formula. In detail, firstly one has the relation as the prototype n)·a ya∈T As the left hand side is considered it can be calculated as follows (√a)+(vga2)d= gr oxy [一√ga2(1)+ga2(t)]d=vge到ai(t)d= E3a'I; (t)da ∫[n,cg,2(Og]d=5[ma,业=小。(rxn),a Subsequently, the relation as the general type 9OVg6g;-Φ)(xx,t) (r×n)·(⊙。Φ)dl fied. The essentials of the deduction is to transfer some indices of the tensor with respect to the local bases to the ones with respect to the canonical bases, namely yg°-φ=√g6"g-(Φ.g8g)=vkge%i-(叭.i⑧F), whereΦ∈2(T) can be extended to2(R) with the constrainΦ(·,n)=①(n,·)=0∈R, then the transformation can be carried out e"g;=6"g:=6(g,g)g:=6(g,(g,i)k)g=6(g,p)[(i,g2)k2g=:6in Consequently, one can do the following deduction 「a(eg-d=Lna(e0,)y,= (n×r)·(eΦ.g,) i (nxr)·[g:,-(c③P)
9 " :!$?#"X*?: & &4? 9 !;BR *B?*R"A[#d$A[&# ( &4? 9)X *?:( % ?;BR *B?*Rd;BR!3?B"?*Rd3?R*B?"")A[#d;?G *GA[&# &4? 9 X ! % B;BR *Rd;BR3BR""A[#d;?G *GA[&# &4? 9 $ Z47C7 % B;BR *Rd;BR3BR"X!&;BR &4B 9 d6B BG;GRd6R BG;BG "*Rd;BR3BR" ! X &;GR &4G 9 d& 槡6 &槡6 &4G;GR "*Rd;GR!6H GR*Hd3GR""X & 槡6 & &4G 9 !槡6;GR"*Rd;GR&*R &4G 9 X& 槡6 & &4G 9 !槡6;GR *R"$ ?47: "8JX& 槡6 & &4G 9 !槡6;GR *R"A[#d槡6;GR *RA[&# ( &4G 9)X& 槡6 & &4G 9 !槡6;GR *RA[#"# ;47/-Q?.87:?.?0.Q7QQ7:?.-//0827?1?47@C77:N1C,2/-6i:87?-./$N.CQ?/01:74-Q?47C7/-?.1:-Q?47 5C1?1?057 %!4 & &4G 9 !槡62G"8"X %9 ! !_"":28? F24+"# =Q?47/7N?4-:8Q.87.Q91:Q.87C78$.?9-:U79-/92/-?78-QN1//1ZQ% %!4 & &4G 9 !槡62G"8"X %!4 & &4& 9 !槡62&"d & &4# 9 ( !槡62#")8"X %&!4 ([槡62# 0&d槡62& 0#):!8?X % 3 2 ([槡62#4 :&! D"d槡62&4 :#! D")8DX % 3 2槡6='BR2B4 :R! D"8DX % 3 2 #'BR2B4 : R! D"8DX % 3 2 ("$2B *B$4 :R! D"*R)8DX :&9 ("$2$ !)8DS:&9 ! !H"":28D# >2UQ7327:?/0$?47C7/-?.1:-Q?47W7:7C-/?057 %!4 & &4G 9 !槡6;GR *RA[#"!49$ D"8"S:&9 ! !_"":!$A[#"8? 9-:U7A7C.N.786;477QQ7:?.-/Q1N?4787829?.1:.Q?1?C-:QN7CQ1,7.:8.97Q1N?47?7:Q1CZ.?4C7Q579??1 ?47/19-/U-Q7Q?1?471:7QZ.?4C7Q579??1?479-:1:.9-/U-Q7Q$:-,7/0 槡6;GR *RA[#X槡6;GR *RA[!#(:-*(?*-"X槡6;G+0+A[!!):%0)?0%"$ Z47C7#4"#!+""9-:U77V?7:878?1"#!a'"Z.?4?4791:Q?C-.:#!:$""X#!"$:"X!4a$?47:?47 ?C-:QN1C,-?.1:9-:U79-CC.7812? ;GR *RX;G(*(kX;!*G$*("*(X$!*G$!*($0+"a'0+"*(X$!*G$0+"(!0+$*("a'*()Xk;G+0+# D1:Q7327:?/0$1:79-:81?47N1//1Z.:W87829?.1:% %!4 & &4G 9 !槡6;GR *RA[#"8"X%!4 & &4G 9 !槡6;G+ !) ( :%"8")0+A[0)?0%X :&9 !"_!":!;G+ !) ( :%*G")0+A[0)?0%X :&9 !"_!":(;G+ *G?0+A[!!):%0)?0%")8?$ (*( L & !#$%&" (#
第5期 谢锡麟:一般光滑曲面上的二类微分算子 where the relation as the prototype is adopted. The relations as the prototype and general type have been adopted directly by Yin et all. J respe 2.1.2 A kind of ways to deduce governing equations for thin enough continuous mediums To study the representation of the natural law of momentum conservation for the continuous medium whose geometrical configuration can be taken as a surface, the so termed surface stress can be introduced t.,g,②g+t.3g Subsequently, the momentum conservation can be set up in the integral form (r×n)·tdl+|,fxdo where f denotes the distribution of the action imposed directly on the surface such as the weight fraction and electromagnetic force. The differential equation of momentum conservation can be directly attained through the intrinsic generalized Stokes formula of the second kind t+ with the component forms par=V,tbt.3+, ca,=V,/3+b;t';tfm On the other hand, the moment of momentum conservation can be represented as a×d=中.[(r×n)·n×Ed+|f:×+m=do with the differential form X∑V·(tX∑)+f×∑+ms=[(V·t)×∑+g·(t×g;)]+∫x×∑+msx, where my denotes the surface force couple. Substituting the governing equation of momentum conservation, one arrives at the governing equation of moment of momentum conservation 0=g·(txg)+m=-p"in+√g(-tsg2+.3g2)+mx,g:=det[g] Consequently, it can be concluded that the symmetry of the components of surface stress tensor on the tangent space, i.e. t;=t;, corresponds to the vanishing of the component of surface force couple in the surface normal direction. And the appearance of surface stress tensor in the surface normal direction i. e. t3+0, corresponds to the existence of components of surface force couple on the tangent space. The governing equations of the statical force equilibrium of elastic plates and shells put forward by Chienti are included in the above mentioned equation of momentum conservation. Comparatively, the deduction based on the intrinsic generalized Stokes formula of the second kind seems more compactly Both Chient and ArisLioJ have introduced the concept of membrane or surface stress tensor in their studies on solids or fluids whose geometrical configurations can be taken as surfaces. Subsequently, the stress force can be represented as the surface divergence of the stress tensor and the differential equations of nature laws can be readily deduced from the integral representations through the intrinsic generalized Stokes formula of the second kind. 2. 1. 3 A differential identity for vorticity dynamics Proposition 5 On any deformable smooth surface, the following identity is keeping valid
Z47C7?47C7/-?.1:-Q?475C1?1?057.Q-815?786 ;47C7/-?.1:Q-Q?475C1?1?057-:8W7:7C-/?0574-A7U77:-815?788.C79?/0U0 ].:7?-/($$M) C7Q579?.A7/06 #6&6# =G.:81NZ-0Q?1878297W1A7C:.:W732-?.1:QN1C?4.:7:12W491:?.:212Q,78.2,Q ;1Q?280?47C75C7Q7:?-?.1:1N?47:-?2C-//-Z1N,1,7:?2,91:Q7CA-?.1:N1C?4791:?.:212Q,78.2,Z41Q7 W71,7?C.9-/91:N.W2C-?.1:9-:U7?-G7:-Q-Q2CN-97$?47Q1?7C,78Q2CN-97Q?C7QQ9-:U7.:?C182978 'XDB:R*B?*RdDB:'*B?"# >2UQ7327:?/0$?47,1,7:?2,91:Q7CA-?.1:9-:U7Q7?25.:?47.:?7WC-/N1C, %9 D/28"S:&9 D! !_"":'8?]%9 D&98"$ Z47C7&9 87:1?7Q?478.Q?C.U2?.1:1N?47-9?.1:.,51Q788.C79?/01:?47Q2CN-97Q294-Q?47Z7.W4?$ NC-9?.1:-:87/79?C1,-W:7?.9N1C976;478.NN7C7:?.-/732-?.1:1N,1,7:?2,91:Q7CA-?.1:9-:U78.C79?/0 -??-.:78?4C12W4?47.:?C.:Q.9W7:7C-/.b78>?1G7QN1C,2/-1N?47Q791:8G.:8 /2X9 " :'d&9 Z.?4?4791,51:7:?N1C,Q% /2?X % GDG:?[3G?DG:'d/?$/2,X % GDG:'d3B RDR:Bd/,# T:?471?47C4-:8$?47,1,7:?1N,1,7:?2,91:Q7CA-?.1:9-:U7C75C7Q7:?78-Q %9 D/2_"8"S:&9 D(! !_"":')_"8?]%9 D&9 _"8"]%9 D598" Z.?4?478.NN7C7:?.-/N1C, /2H"X9 " :!'H""d&9H"d59X 9 " ! :'"H"d* ( ?:!'H*?")d&9H"d59$ Z47C75987:1?7Q?47Q2CN-97N1C979125/76 >2UQ?.?2?.:W?47W1A7C:.:W732-?.1:1N ,1,7:?2, 91:Q7CA-?.1:$1:7-CC.A7Q-??47W1A7C:.:W 732-?.1:1N,1,7:?1N,1,7:?2,91:Q7CA-?.1: %X*?:!'H*?"d59X[DBR #BR'"d槡6![D#:'*&dD&:'*#"d59$ 6kX87?(6BR)# D1:Q7327:?/0$.?9-:U791:9/2878?4-??47Q0,,7?C01N?4791,51:7:?Q1NQ2CN-97Q?C7QQ?7:Q1C1:?47 ?-:W7:?Q5-97$.676DBRXDRB$91CC7Q51:8Q?1?47A-:.Q4.:W1N?4791,51:7:?1NQ2CN-97N1C979125/7.:?47 Q2CN-97:1C,-/8.C79?.1:6=:8?47-557-C-:971NQ2CN-97Q?C7QQ?7:Q1C.:?47Q2CN-97:1C,-/8.C79?.1:$ .676DB:'1!$91CC7Q51:8Q?1?477V.Q?7:971N91,51:7:?Q1NQ2CN-97N1C979125/71:?47?-:W7:?Q5-976 ;47W1A7C:.:W732-?.1:Q1N?47Q?-?.9-/N1C97732./.UC.2,1N7/-Q?.95/-?7Q-:8Q47//Q52?N1CZ-C8U0 D4.7:(*) -C7.:9/2878.:?47-U1A7,7:?.1:78732-?.1:1N,1,7:?2,91:Q7CA-?.1:6D1,5-C-?.A7/0$?47 87829?.1:U-Q781:?47.:?C.:Q.9W7:7C-/.b78>?1G7QN1C,2/-1N?47Q791:8G.:8Q77,Q,1C791,5-9?/06 E1?4D4.7:(*)-:8=C.Q(&!)4-A7.:?C182978?4791:975?1N,7,UC-:71CQ2CN-97Q?C7QQ?7:Q1C.:?47.C Q?28.7Q1:Q1/.8Q1CN/2.8QZ41Q7W71,7?C.9-/91:N.W2C-?.1:Q9-:U7?-G7:-QQ2CN-97Q6>2UQ7327:?/0$?47 Q?C7QQN1C979-:U7C75C7Q7:?78-Q?47Q2CN-978.A7CW7:971N?47Q?C7QQ?7:Q1C-:8?478.NN7C7:?.-/ 732-?.1:Q1N:-?2C7/-ZQ9-:U7C7-8./08782978NC1,?47.:?7WC-/C75C7Q7:?-?.1:Q?4C12W4?47.:?C.:Q.9 W7:7C-/.b78>?1G7QN1C,2/-1N?47Q791:8G.:86 #6&6' =8.NN7C7:?.-/.87:?.?0N1CA1C?.9.?080:-,.9Q '2:O:?818:4* T:-:087N1C,-U/7Q,11?4Q2CN-97$?47N1//1Z.:W.87:?.?0.QG775.:WA-/.8% ( qrs%ÑÍÞA°bôFÖsTl (*