Surfaces in Three-Dimensional Euclidean Space

This chapter deals with the local differential geometry of surfaces in three dimensional Euclidean space. This means the study of the geometric shape of surfaces in the neighbourhood of an arbitrary one of their points. The most important concepts arising in this task are those of the normal, principal, Gaussian and mean curvature. Additional topics are curves on surfaces, tangent planes and normal vectors of surfaces, the first and second fundamental coefficients, Meusnier's theorem, the principal directions, Euler's formula, the local shape of surfaces, Dupin's indicatrix, lines of curvature and asymptotic lines, triple orthogonal systems, the Gauss and Weingarten equations.