作者: VEIL 時(shí)間: 2025-3-21 21:48 作者: Innocence 時(shí)間: 2025-3-22 02:51
https://doi.org/10.1007/978-3-642-98007-7In this chapter we discuss two mathematical formulations of the intuitive notion of a curve. The precise relation between them turns out to be quite subtle, so we shall begin by giving some examples of curves of each type and practical ways of passing between them.作者: 大漩渦 時(shí)間: 2025-3-22 08:01 作者: 迷住 時(shí)間: 2025-3-22 12:43
Curves in the Plane and in Space,In this chapter we discuss two mathematical formulations of the intuitive notion of a curve. The precise relation between them turns out to be quite subtle, so we shall begin by giving some examples of curves of each type and practical ways of passing between them.作者: 輕快帶來(lái)危險(xiǎn) 時(shí)間: 2025-3-22 16:52
,Gauss’s Theorema Egregium,One of Gauss’s most important discoveries about surfaces is that the gaussian curvature is unchanged when the surface is bent without stretching. Gauss called this result ‘egregium’, and the Latin word for ‘remarkable’ has remained attached to his theorem ever since.作者: 輕快帶來(lái)危險(xiǎn) 時(shí)間: 2025-3-22 19:08 作者: 打算 時(shí)間: 2025-3-23 00:45 作者: Indebted 時(shí)間: 2025-3-23 02:37
Lexikon der Wirtschaftsinformatikrface patch, is all that is needed for most of the book, it does not describe adequately most of the objects that we would want to call surfaces. For example, a sphere is not a surface patch, but it can be described by gluing two surface patches together suitably. The idea behind this gluing procedu作者: SEED 時(shí)間: 2025-3-23 09:07
https://doi.org/10.1007/978-3-662-08371-0rface. Of course, this will usually be different from the distance between these points as measured by an inhabitant of the ambient three dimensional space, since the straight line segment which furnishes the shortest path between the points in .. will generally not be contained in the surface. The 作者: CAGE 時(shí)間: 2025-3-23 13:02
Lexikon der Wirtschaftsinformatikt (see Theorem 10.4) that a surface patch is determined up to a rigid motion of .. by its first and second fundamental forms, just as a unit-speed plane curve is determined up to a rigid motion by its signed curvature.作者: 頭盔 時(shí)間: 2025-3-23 15:07
https://doi.org/10.1007/978-3-211-75607-2me information as the two principal curvatures, they turn out to have greater geometrical significance. The gaussian curvature, in particular, has the remarkable property, established in Chapter 10, that it is unchanged when the surface is bent without stretching, a property that is not shared by th作者: 預(yù)知 時(shí)間: 2025-3-23 21:29
Sabine Krist Doz. Mag. pharm. DDr.s in a surface is always a geodesic. We shall actually begin by giving a quite different definition of geodesics, since this definition is easier to work with. We give various methods of finding the geodesics on surfaces, before finally making contact with the idea of shortest paths towards the end 作者: mosque 時(shí)間: 2025-3-24 01:13
Lexikon der soziologischen Werkeigher dimension, that of finding a surface of minimal area with a fixed curve as its boundary. This is called .. The solutions to Plateau’s problem turn out to be surfaces whose mean curvature vanishes everywhere. The study of these so-called minimal surfaces was initiated by Euler and Lagrange in t作者: labile 時(shí)間: 2025-3-24 04:16 作者: Anthology 時(shí)間: 2025-3-24 09:52 作者: 推測(cè) 時(shí)間: 2025-3-24 12:12
Regine Witkowski,Otto Prokop,Eva Ullriche ‘global’ shape of the curve. In this chapter, we discuss some global results about curves. The most famous, and perhaps the oldest, of these is the ‘isoperimetric inequality’, which relates the length of certain ‘closed’ curves to the area they contain.作者: 大暴雨 時(shí)間: 2025-3-24 18:40 作者: 證明無(wú)罪 時(shí)間: 2025-3-24 19:03 作者: chisel 時(shí)間: 2025-3-25 01:50
How Much Does a Curve Curve?,curve is not contained in a straight line (so that straight lines have zero curvature), and the torsion measures the extent to which a curve is not contained in a plane (so that plane curves have zero torsion). It turns out that the curvature and torsion together determine the shape of a curve.作者: CANE 時(shí)間: 2025-3-25 04:55
Global Properties of Curves,e ‘global’ shape of the curve. In this chapter, we discuss some global results about curves. The most famous, and perhaps the oldest, of these is the ‘isoperimetric inequality’, which relates the length of certain ‘closed’ curves to the area they contain.作者: fiscal 時(shí)間: 2025-3-25 09:06
Curvature of Surfaces,t (see Theorem 10.4) that a surface patch is determined up to a rigid motion of .. by its first and second fundamental forms, just as a unit-speed plane curve is determined up to a rigid motion by its signed curvature.作者: 借喻 時(shí)間: 2025-3-25 13:06 作者: Meander 時(shí)間: 2025-3-25 15:57 作者: 放縱 時(shí)間: 2025-3-25 20:50 作者: 仔細(xì)檢查 時(shí)間: 2025-3-26 01:04 作者: 觀點(diǎn) 時(shí)間: 2025-3-26 08:16 作者: 緩解 時(shí)間: 2025-3-26 10:20 作者: 死貓他燒焦 時(shí)間: 2025-3-26 13:51
Surfaces in Three Dimensions, proofs in a section at the end of the chapter; this section is not used anywhere else in the book and can safely be omitted if desired. In fact, surfaces (as opposed to surface patches) will be used in a serious way on only a few occasions in this book.作者: 團(tuán)結(jié) 時(shí)間: 2025-3-26 17:37 作者: Veneer 時(shí)間: 2025-3-26 21:09 作者: 連系 時(shí)間: 2025-3-27 04:57 作者: resilience 時(shí)間: 2025-3-27 08:50 作者: NIP 時(shí)間: 2025-3-27 12:54 作者: 間諜活動(dòng) 時(shí)間: 2025-3-27 14:46
Gaussian Curvature and the Gauss Map, remarkable property, established in Chapter 10, that it is unchanged when the surface is bent without stretching, a property that is not shared by the principal curvatures. In the present chapter, we discuss some more elementary properties of the gaussian and mean curvatures, and what a knowledge of them implies about the geometry of the surface.作者: BILK 時(shí)間: 2025-3-27 17:50
Minimal Surfaces,rn out to be surfaces whose mean curvature vanishes everywhere. The study of these so-called minimal surfaces was initiated by Euler and Lagrange in the mid-18th century, but new examples of minimal surfaces have been discovered quite recently.作者: 得意牛 時(shí)間: 2025-3-27 23:10
https://doi.org/10.1007/978-3-322-87461-0he surface does . change. The real importance of the Gauss—Bonnet theorem is as a prototype of analogous results which apply in higher dimensional situations, and which relate . properties to . ones. The study of such relations is one of the most important themes of 20th century Mathematics.作者: 奇怪 時(shí)間: 2025-3-28 04:05 作者: 頂點(diǎn) 時(shí)間: 2025-3-28 09:39 作者: 懲罰 時(shí)間: 2025-3-28 13:17 作者: filicide 時(shí)間: 2025-3-28 18:39
How Much Does a Curve Curve?,curve is not contained in a straight line (so that straight lines have zero curvature), and the torsion measures the extent to which a curve is not contained in a plane (so that plane curves have zero torsion). It turns out that the curvature and torsion together determine the shape of a curve.作者: 爭(zhēng)吵加 時(shí)間: 2025-3-28 20:26
Global Properties of Curves,e ‘global’ shape of the curve. In this chapter, we discuss some global results about curves. The most famous, and perhaps the oldest, of these is the ‘isoperimetric inequality’, which relates the length of certain ‘closed’ curves to the area they contain.作者: mendacity 時(shí)間: 2025-3-29 00:33
Surfaces in Three Dimensions,rface patch, is all that is needed for most of the book, it does not describe adequately most of the objects that we would want to call surfaces. For example, a sphere is not a surface patch, but it can be described by gluing two surface patches together suitably. The idea behind this gluing procedu作者: moribund 時(shí)間: 2025-3-29 04:37
The First Fundamental Form,rface. Of course, this will usually be different from the distance between these points as measured by an inhabitant of the ambient three dimensional space, since the straight line segment which furnishes the shortest path between the points in .. will generally not be contained in the surface. The 作者: ornithology 時(shí)間: 2025-3-29 10:02
Curvature of Surfaces,t (see Theorem 10.4) that a surface patch is determined up to a rigid motion of .. by its first and second fundamental forms, just as a unit-speed plane curve is determined up to a rigid motion by its signed curvature.
