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
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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
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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
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