{"title":"关于Tonelli的一个定理的注记","authors":"W. Wilczyński","doi":"10.2478/tmmp-2022-0006","DOIUrl":null,"url":null,"abstract":"Abstract It is well known that if the surface f : [−1, 1] × [−1, 1] → ℝ has a finite area, then the total variations of both sections fx(y)= f (x, y)and f y(x) = f (x, y)of f are finite almost everywhere in [−1, 1]. In the note it is proved that these variations can be infinite on residual subsets of [−1, 1].","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"89 - 92"},"PeriodicalIF":0.0000,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Remark on a Theorem of Tonelli\",\"authors\":\"W. Wilczyński\",\"doi\":\"10.2478/tmmp-2022-0006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract It is well known that if the surface f : [−1, 1] × [−1, 1] → ℝ has a finite area, then the total variations of both sections fx(y)= f (x, y)and f y(x) = f (x, y)of f are finite almost everywhere in [−1, 1]. In the note it is proved that these variations can be infinite on residual subsets of [−1, 1].\",\"PeriodicalId\":38690,\"journal\":{\"name\":\"Tatra Mountains Mathematical Publications\",\"volume\":\"81 1\",\"pages\":\"89 - 92\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Tatra Mountains Mathematical Publications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2478/tmmp-2022-0006\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tatra Mountains Mathematical Publications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/tmmp-2022-0006","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
摘要
摘要众所周知，如果曲面f:[−1,1]x[−1,1]→∈有一个有限的面积，那么f的两个截面fx(y)= f (x, y)和f (x) = f (x, y)在[−1,1]内几乎处处都是有限的。在注释中证明了这些变化在[−1,1]的残差子集上是无限的。
Abstract It is well known that if the surface f : [−1, 1] × [−1, 1] → ℝ has a finite area, then the total variations of both sections fx(y)= f (x, y)and f y(x) = f (x, y)of f are finite almost everywhere in [−1, 1]. In the note it is proved that these variations can be infinite on residual subsets of [−1, 1].
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