An Introduction to Differential GeometryA solid introduction to the methods of differential geometry and tensor calculus, this volume is suitable for advanced undergraduate and graduate students of mathematics, physics, and engineering. Rather than a comprehensive account, it offers an introduction to the essential ideas and methods of differential geometry. Part 1 begins by employing vector methods to explore the classical theory of curves and surfaces. An introduction to the differential geometry of surfaces in the large provides students with ideas and techniques involved in global research. Part 2 introduces the concept of a tensor, first in algebra, then in calculus. It covers the basic theory of the absolute calculus and the fundamentals of Riemannian geometry. Worked examples and exercises appear throughout the text. |
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algebra angle arbitrary arc length asymptotic lines calculus called Cartesian Chapter coefficients coeflicients compact surface components condition connexion consider constant curvature contravariant vector coordinate neighbourhood coordinate system corresponding covariant tensors covariant vector curvature tensor defined denote differential equations differential geometry difierential direction Euclidean space example EXERCISE find first fixed follows formula function Gaussian curvature geodesic arc geodesic curvature given gives helicoid helix Hence identity integrable intrinsic isometric isomorphism linear lines of curvature mapping matrix metric tensor null obtained osculating plane parallel field parameter parametric curves position vector principal curvatures principal normal proof prove r-planes radius real numbers real-valued region relation respect Riemannian manifold Riemannian space satisfied scalar Show sphere sufiicient suflixes surface of revolution symmetric tangent plane tangent space tangent vector tensor field tensor of type theory topological torsion total curvature transformation uniquely unit vector values vector space verified zero