TY - JOUR
T1 - Explicit reconstruction of a displacement field on a surface by means of its linearized change of metric and change of curvature tensors
AU - Ciarlet, Philippe G.
AU - Gratie, Liliana
AU - Serpilli, Michele
PY - 2008/10
Y1 - 2008/10
N2 - Let ω be a simply-connected open subset in R2 and let θ : ω → R3 be a smooth immersion. If two symmetric matrix fields (γα β) and (ρα β) of order two satisfy appropriate compatibility relations in ω, then (γα β) and (ρα β) are the linearized change of metric and change of curvature tensor fields corresponding to a displacement vector field η of the surface θ (ω). We show here that, when the fields (γα β) and (ρα β) are smooth, the displacement vector η (y) at any point θ (y), y ∈ ω, of the surface θ (ω) can be explicitly computed by means of a "Cesàro-Volterra path integral formula on a surface", i.e., a path integral inside ω with endpoint y, and whose integrand is an explicit function of the functions γα β and ρα β and their covariant derivatives. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008). © 2008 Académie des sciences.
AB - Let ω be a simply-connected open subset in R2 and let θ : ω → R3 be a smooth immersion. If two symmetric matrix fields (γα β) and (ρα β) of order two satisfy appropriate compatibility relations in ω, then (γα β) and (ρα β) are the linearized change of metric and change of curvature tensor fields corresponding to a displacement vector field η of the surface θ (ω). We show here that, when the fields (γα β) and (ρα β) are smooth, the displacement vector η (y) at any point θ (y), y ∈ ω, of the surface θ (ω) can be explicitly computed by means of a "Cesàro-Volterra path integral formula on a surface", i.e., a path integral inside ω with endpoint y, and whose integrand is an explicit function of the functions γα β and ρα β and their covariant derivatives. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008). © 2008 Académie des sciences.
UR - http://www.scopus.com/inward/record.url?scp=53849087114&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-53849087114&origin=recordpage
U2 - 10.1016/j.crma.2008.09.001
DO - 10.1016/j.crma.2008.09.001
M3 - RGC 21 - Publication in refereed journal
VL - 346
SP - 1113
EP - 1117
JO - Comptes Rendus Mathematique
JF - Comptes Rendus Mathematique
SN - 1631-073X
IS - 19-20
ER -