It may be inevitable in this subject that the writing at times becomes rather dense. The sections devoted to explaining the tensor notations more prevalent in physics textbooks are even equipped with problems that deal with physics: the moment of inertia tensor and the Minkowski metric. They range from the almost trivial computations to a carefully outlined exercise deriving the transformation rules for Christoffel symbols. I think the exercises in this book are wonderfully balanced. It becomes particularly helpful when less intuitive concepts (e.g. This type of exposition pervades most chapters of the book. Later, after curvature and torsion have been defined, they return to flesh out a couple of these examples. For instance, rather than simply defining space curves analogously to plane curves and moving on to torsion, Banchoff and Lovett present a long series of examples ranging from simple lines in three-space through the twisted cubic. One very satisfying characteristic of this book is the geometric intuition that the authors nurture through their careful exposition and examples. Geared toward advanced undergraduates, specifically those that are particularly calculus-savvy, this text carefully develops the rudiments of differential geometry by first examining plane curves, then space curves, moving on to regular surfaces, the first and second fundamental forms, and finally curves on regular surfaces. This is well illustrated by the non-linear Euler–Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding.Differential Geometry of Curves and Surfaces, by Thomas Banchoff and Stephen Lovett, presents a thorough introduction to the study of curves and surfaces. On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. These Lie groups can be used to describe surfaces of constant Gaussian curvature they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. The mathematics of smooth surfaces Carl Friedrich Gauss in 1828
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