rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Curves and Surfaces
  name:
    name part: Abate, Marco.
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: Tovena, Francesca.
    role:
      role term: author.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: it
    date issued: 2012
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XIII, 396 p. online resource.
  abstract: The book provides an introduction to Differential Geometry of Curves and Surfaces. The theory of curves starts with a discussion of possible definitions of the concept of curve, proving in particular the classification of 1-dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in n-dimensional space are discussed in the complementary material): curvature, torsion, Frenet’s formulas and the fundamental theorem of the local theory of curves. Then, after a self-contained presentation of degree theory for continuous self-maps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves. The local theory of surfaces begins with a comparison of the concept of parametrized (i.e., immersed) surface with the concept of regular (i.e., embedded) surface. We then develop the basic differential geometry of surfaces in R3: definitions, examples, differentiable maps and functions, tangent vectors (presented both as vectors tangent to curves in the surface and as derivations on germs of differentiable functions; we shall consistently use both approaches in the whole book) and orientation. Next we study the several notions of curvature on a surface, stressing both the geometrical meaning of the objects introduced and the algebraic/analytical methods needed to study them via the Gauss map, up to the proof of Gauss’ Teorema Egregium. Then we introduce vector fields on a surface (flow, first integrals, integral curves) and geodesics (definition, basic properties, geodesic curvature, and, in the complementary material, a full proof of minimizing properties of geodesics and of the Hopf-Rinow theorem for surfaces). Then we shall present a proof of the celebrated Gauss-Bonnet theorem, both in its local and in its global form, using basic properties (fully proved …
  note: by Marco Abate, Francesca Tovena.
  subject:
    topic: Mathematics
  subject:
    topic: Computer graphics
  subject:
    topic: Computer mathematics
  subject:
    topic: Geometry
  subject:
    topic: Differential geometry
  subject:
    topic: Mathematics
  subject:
    topic: Mathematics, general
  subject:
    topic: Differential Geometry
  subject:
    topic: Geometry
  subject:
    topic: Computational Science and Engineering
  subject:
    topic: Computer Imaging, Vision, Pattern Recognition and Graphics
  classification: QA1-939
  classification: 510
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: UNITEXT
  identifier: 9788847019416
  identifier: http://dx.doi.org/10.1007/978-88-470-1941-6
  record info:
    record creation date: 120611
    record change date: 20151204174026.0
    record identifier: 978-88-470-1941-6