![]() ![]() ![]() The defining contribution to the theory of surfaces was made by Gauss in two remarkable papers written in 18. Monge laid down the foundations of their theory in his classical memoir L'application de l'analyse à la géometrie which appeared in 1795. In 1760 he proved a formula for the curvature of a plane section of a surface and in 1771 he considered surfaces represented in a parametric form. Curvature of general surfaces was first studied by Euler. The development of calculus in the seventeenth century provided a more systematic way of computing them. The volumes of certain quadric surfaces of revolution were calculated by Archimedes. 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. 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. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. ![]() Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. ![]()
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