UID:
almafu_9961429316902883
Umfang:
1 online resource (204 pages)
Ausgabe:
1st ed. 2024.
ISBN:
3-031-39838-6
Serie:
Compact Textbooks in Mathematics,
Inhalt:
This open access book covers the main topics for a course on the differential geometry of curves and surfaces. Unlike the common approach in existing textbooks, there is a strong focus on variational problems, ranging from elastic curves to surfaces that minimize area, or the Willmore functional. Moreover, emphasis is given on topics that are useful for applications in science and computer graphics. Most often these applications are concerned with finding the shape of a curve or a surface that minimizes physically meaningful energy. Manifolds are not introduced as such, but the presented approach provides preparation and motivation for a follow-up course on manifolds, and topics like the Gauss-Bonnet theorem for compact surfaces are covered.
Anmerkung:
Intro -- Preface -- Acknowledgement -- Contents -- Part I Curves -- 1 Curves in Rn -- 1.1 What is a Curve in Rn? -- 1.2 Length and Arclength -- 1.3 Unit Tangent and Bending Energy -- 2 Variations of Curves -- 2.1 One-Parameter Families of Curves -- 2.2 Variation of Length and Bending Energy -- 2.3 Critical Points of Length and Bending Energy -- 2.4 Constrained Variation -- 2.5 Torsion-Free Elastic Curves and the Pendulum Equation -- 3 Curves in R2 -- 3.1 Plane Curves -- 3.2 Area of a Plane Curve -- 3.3 Planar Elastic Curves -- 3.4 Tangent Winding Number -- 3.5 Regular Homotopy -- 3.6 Whitney-Graustein Theorem -- 4 Parallel Normal Fields -- 4.1 Parallel Transport -- 4.2 Curvature Function of a Curve in Rn -- 4.3 Geometry in Terms of the Curvature Function -- 5 Curves in R3 -- 5.1 Total Torsion of Curves in R3 -- 5.2 Elastic Curves in R3 -- 5.3 Vortex Filament Flow -- 5.4 Total Squared Torsion -- 5.5 Elastic Framed Curves -- 5.6 Frenet Normals -- Part II Surfaces -- 6 Surfaces and Riemannian Geometry -- 6.1 Surfaces in Rn -- 6.2 Tangent Spaces and Derivatives -- 6.3 Riemannian Domains -- 6.4 Linear Algebra on Riemannian Domains -- 6.5 Isometric surfaces -- 7 Integration and Stokes' Theorem -- 7.1 Integration on Surfaces -- 7.2 Integration Over Curves -- 7.3 Stokes' Theorem -- 8 Curvature -- 8.1 Unit Normal of a Surface in R3 -- 8.2 Curvature of a Surface -- 8.3 Area of Maps Into the Plane or the Sphere -- 9 Levi-Civita Connection -- 9.1 Derivatives of Vector Fields -- 9.2 Equations of Gauss and Codazzi -- 9.3 Theorema Egregium -- 10 Total Gaussian Curvature -- 10.1 Curves on Surfaces -- 10.2 Theorem of Gauss and Bonnet -- 10.3 Parallel Transport on Surfaces -- 11 Closed Surfaces -- 11.1 History of Closed Surfaces -- 11.2 Defining Closed Surfaces -- 11.3 Boy's Theorem -- 11.4 The Genus of a Closed Surface -- 12 Variations of Surfaces.
,
12.1 Vector Calculus on Surfaces -- 12.2 One-Parameter Families of Surfaces -- 12.3 Variation of Curvature -- 12.4 Variation of Area -- 12.5 Variation of Volume -- 13 Willmore Surfaces -- 13.1 The Willmore Functional -- 13.2 Variation of the Willmore Functional -- 13.3 Willmore Functional Under Inversions -- A Some Technicalities -- A.1 Smooth Maps -- A.2 Function Toolbox -- B Timeline -- References -- Index.
Weitere Ausg.:
ISBN 3-031-39837-8
Sprache:
Englisch
Schlagwort(e):
Llibres electrònics
DOI:
10.1007/978-3-031-39838-4
