This book presents the classical theory of curves in the plane
and three-dimensional space, and the classical theory of surfaces
in three-dimensional space. It pays particular attention to the
historical development of the theory and the preliminary approaches
that support contemporary geometrical notions. It includes a
chapter that lists a very wide scope of plane curves and their
properties. The book approaches the threshold of algebraic
topology, providing an integrated presentation fully accessible to
undergraduate-level students.
At the end of the 17th century, Newton and Leibniz developed
differential calculus, thus making available the very wide range of
differentiable functions, not just those constructed from
polynomials. During the 18th century, Euler applied these ideas to
establish what is still today the classical theory of most general
curves and surfaces, largely used in engineering. Enter this
fascinating world through amazing theorems and a wide supply of
surprising examples. Reach the doors of algebraic topology by
discovering just how an integer (= the Euler-Poincare
characteristics) associated with a surface gives you a lot of
interesting information on the shape of the surface. And penetrate
the intriguing world of Riemannian geometry, the geometry that
underlies the theory of relativity.
The book is of interest to all those who teach classical
differential geometry up to quite an advanced level. The chapter on
Riemannian geometry is of great interest to those who have to
intuitively introduce students to the highly technical nature of
this branch of mathematics, in particular when preparing students
for courses on relativity."
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