|
Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry
|
Buy Now
Curves and Surfaces (Paperback, 2012 ed.)
Loot Price: R1,599
Discovery Miles 15 990
|
|
|
Curves and Surfaces (Paperback, 2012 ed.)
Series: UNITEXT
Expected to ship within 12 - 17 working days
|
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 in the
complementary material) of triangulations of surfaces. As an
application, we shall prove the Poincare-Hopf theorem on zeroes of
vector fields. Finally, the last chapter will be devoted to several
important results on the global theory of surfaces, like for
instance the characterization of surfaces with constant Gaussian
curvature, and the orientability of compact surfaces in R3.
General
Is the information for this product incomplete, wrong or inappropriate?
Let us know about it.
Does this product have an incorrect or missing image?
Send us a new image.
Is this product missing categories?
Add more categories.
Review This Product
No reviews yet - be the first to create one!
|
You might also like..
|
Welcome to Loot.co.za!
Sign in / Register |Wishlists & Gift Vouchers |Help | Advanced search
