By Philippe G. Ciarlet
curvilinear coordinates. This therapy contains particularly a right away evidence of the three-d Korn inequality in curvilinear coordinates. The fourth and final bankruptcy, which seriously will depend on bankruptcy 2, starts off by way of a close description of the nonlinear and linear equations proposed through W.T. Koiter for modeling skinny elastic shells. those equations are “two-dimensional”, within the experience that they're expressed when it comes to curvilinear coordinates used for de?ning the center floor of the shell. The lifestyles, distinctiveness, and regularity of options to the linear Koiter equations is then tested, thank you this time to a primary “Korn inequality on a floor” and to an “in?nit- imal inflexible displacement lemma on a surface”. This bankruptcy additionally contains a short creation to different two-dimensional shell equations. apparently, notions that pertain to di?erential geometry in line with se,suchas covariant derivatives of tensor ?elds, also are brought in Chapters three and four, the place they seem such a lot clearly within the derivation of the fundamental boundary worth difficulties of three-d elasticity and shell conception. sometimes, parts of the cloth coated listed below are tailored from - cerpts from my publication “Mathematical Elasticity, quantity III: conception of Shells”, released in 2000by North-Holland, Amsterdam; during this recognize, i'm indebted to Arjen Sevenster for his sort permission to depend on such excerpts. Oth- clever, the majority of this paintings was once considerably supported by means of provides from the study delivers Council of Hong Kong specified Administrative sector, China [Project No. 9040869, CityU 100803 and undertaking No. 9040966, CityU 100604].
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The proof is thus complete. Remarks. (1) The assumptions ∂j Γpik − ∂k Γpij + Γik Γpj − Γij Γpk = 0 in Ω, made in part (ii) on the functions Γpij = Γpji are thus suﬃcient conditions for the equations ∂i F j = Γpij F p in Ω to have solutions. Conversely, a simple Sect. 6] Existence of an immersion with a prescribed metric tensor 29 computation shows that they are also necessary conditions, simply expressing that, if these equations have a solution, then necessarily ∂ik F j = ∂ki F j in Ω. , ∂ik g j = ∂ij g k in Ω.
As usual, given any immersion Θ ∈ C 3 (Ω; E3 ), let g i = ∂i Θ, let gij = g i · g j , and let the vectors g q be deﬁned by the relations g i · g q = δiq . It is then immediately veriﬁed that ∂ij Θ = ∂i g j = (∂i g j · g q )g q = 1 (∂j giq + ∂i gjq − ∂q gij )g q . 2 Sect. 8] An immersion as a function of its metric tensor 41 Applying this relation to the mappings Θn thus gives ∂ij Θn = 1 n n n (∂j giq + ∂i gjq − ∂q gij )(g q )n , n ≥ 0, 2 where the vectors (g q )n are deﬁned by means of the relations ∂i Θn · (g q )n = δiq .
By (i), there thus exist c ∈ E3 and Q ∈ O3+ such that Φ(x) = Θ(x) = c + Q ox for almost all x = Θ(x) ∈ U , or equivalently, such that Ξ(x) := ∇Θ(x)∇Θ(x)−1 = Q for almost all x ∈ U. Since the point x0 ∈ Ω is arbitrary, this relation shows that Ξ ∈ L1loc (Ω). By a classical result from distribution theory (cf. 6]), Sect. 8] Uniqueness of immersions with the same metric tensor 37 we conclude from the assumed connectedness of Ω that Ξ(x) = Q for almost all x ∈ Ω, and consequently that Θ(x) = c + QΘ(x) for almost all x ∈ Ω.
An Introduction to Differential Geometry with Applications to Elasticity by Philippe G. Ciarlet