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Area Mathematics - General concepts and linear algebra / Geometry

IEV ref102-04-18

en
length, <of a curve>
least upper bound, if it exists, of the lengths of any polygonal lines determined by successive points of the curve between the two points corresponding to the limiting values of the parameter interval

Note 1 to entry: For a curve from A to B defined by the position vector r=f(u) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahkhacqGH9aqpca WHMbGaaiikaiaadwhacaGGPaaaaa@3E10@ as a function of the parameter u in the given interval [a, b] where ab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadggacqGHKjYOca WGIbaaaa@3C4F@ , the length of the curve is the line integral A B | dr| = a b | df du |du MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapehabaWaaqWaae aajugibiGacsgakiaahkhaaiaawEa7caGLiWoaaSqaaiaabgeaaeaa caqGcbaaniabgUIiYdGccqGH9aqpnmaapehakeaadaabdaqaamaala aabaqcLbsaciGGKbGccaWHMbaabaqcLbsaciGGKbGccaWG1baaaaGa ay5bSlaawIa7aKqzGeGaciizaOGaamyDaaqaaiaadggaaeaacaWGIb aajugGciabgUIiYdaaaa@4FB4@ .

Note 2 to entry: In the usual geometrical space, the length of a curve is a quantity of the dimension length.


fr
longueur, <d'une courbe> f
borne supérieure, si elle existe, des longueurs de toutes les courbes polygonales déterminées par des point successifs de la courbe entre les deux points correspondant aux valeurs limites de l'intervalle du paramètre

Note 1 à l'article: Pour une courbe de A à B définie par le rayon vecteur r=f(u) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahkhacqGH9aqpca WHMbGaaiikaiaadwhacaGGPaaaaa@3E10@ en fonction du paramètre u dans l'intervalle [a, b] où ab MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaadggacqGHKjYOca WGIbaaaa@3C4F@ , la longueur de la courbe est l'intégrale curviligne A B | dr| = a b | df du |du MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapehabaWaaqWaae aajugibiGacsgakiaahkhaaiaawEa7caGLiWoaaSqaaiaabgeaaeaa caqGcbaaniabgUIiYdGccqGH9aqpnmaapehakeaadaabdaqaamaala aabaqcLbsaciGGKbGccaWHMbaabaqcLbsaciGGKbGccaWG1baaaaGa ay5bSlaawIa7aKqzGeGaciizaOGaamyDaaqaaiaadggaaeaacaWGIb aajugGciabgUIiYdaaaa@4FB4@ .

Note 2 à l'article: Dans l'espace géométrique usuel, la longueur d'une courbe est une grandeur ayant la dimension d'une longueur.


de
Länge (einer Kurve), f

es
longitud (de una curva)

ja
長さ, <曲線の>

pl
długość (krzywej)

pt
comprimento (de uma curva)

sr
дужина, <криве> ж јд

sv
längd (av en kurva)

zh
长度, <曲线的>

Publication date: 2008-08
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