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-Poincaré 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.
