EQUATIONS OF ELASTIC HYPERSURFACES

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80 SHELLS The Laplace-Beltrami operator ∆ S on S takes the form ( ) n∑ ∆ S ϕ = Div S ∇ S ϕ = −∇ ∗ S ∇ S ϕ = Dj 2 ϕ (3.63) = ∑ j
3. CALCULUS <strong>OF</strong> DIFFERENTIAL OPERATORS ON <strong>HYPERSURFACES</strong> 81 for k = 1, . . . , n; also we recall that n∑ ν j D j = 0 and M jk = ν j D k − ν k D j (cf. Lemma j=1 3.11.vi, 3.11.v)) and proceed as follows: 1 2 n∑ j,k=1 M 2 jkϕ= 1 2 n∑ [ν j D k − ν k D j ] 2 ϕ = 1 2 j,k=1 + ν k D j ν k D j ϕ − ν k D j ν j D k ϕ] = = n∑ Dkϕ 2 − k=1 n∑ [ν j D k ν j D k ϕ − ν j D k ν k D j ϕ j,k=1 n∑ [ν j D k ν j D k ϕ − ν j D k ν k D j ϕ] j,k=1 n∑ [ νj ν k D k D j ϕ + ( ) D k ν k νj D j ϕ ] = j,k=1 n∑ Dkϕ 2 = ∆ S ϕ . k=1 We remind that the surface gradient ∇ S maps scalar functions to the tangential vector fields ∇ S : C 1 (S ) → TV (S ) := C(S , TS ) (3.66) and the scalar product with the normal vector vanishes everywhere on the surface S : ν(X ) · ∇ S ϕ(X ) ≡ 0 for all ϕ ∈ C 1 (S ) . (3.67) The directing vectors { } n d j of Günter’s derivatives { } n D j=1 j are tangential and represent j=1 a full system (cf. (3.25)-(3.27)). But the derivative D j V is not an automorphisms of the tangential vector space D j : TV (S ) ↛ TV (S ), i.e., did not represent covariant derivative of vector fields (cf. § 2.7). To improve this we just eliminate, as customary, the normal component of the vector by applying the canonical orthogonal projection π S onto TS (cf. (1.11)) and obtain D S j V := π S D j V = D j V − 〈ν, D j V 〉ν = D j V + (W S V ) j ν = D j V + (D V ν j )ν , (3.68) where W S := [ n∑ D j ν k ]n×n , D V ϕ := V k D k ϕ . k=1 because n∑ n∑ [ ] 〈ν, D j V 〉= ν m D j V m = Dj (ν m V m ) − V m D j ν m m=1 =− m=1 n∑ V m D j ν m = −(W S V ) j = − m=1 m=1 n∑ V m D m ν j = −D V ν j . It is easy to check that the operators D S j are automorphisms of the tangential vector space D S j : TV (S ) → TV (S ) . (3.69)
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Grant GNSF/ST06/3-001 of the Georgi
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1. INTRODUCTION 3 1 INTRODUCTION Pa
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1. INTRODUCTION 5 defines the canon
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2. OUTLINE OF DIFFERENTIAL GEOMETRY
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5. BOUNDARY VALUE PROBLEMS: EXAMPLE
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BIBLIOGRAPHY 181 [CD1] O.Chkadua, R
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BIBLIOGRAPHY 183 [Hr1] L. Hörmande
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BIBLIOGRAPHY 185 [Si1] I. Simonenko
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EQUATIONS OF ELASTIC HYPERSURFACES

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