|
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field. == History of Development == Differential geometry arose and developed〔 http://www.encyclopediaofmath.org/index.php/Differential_geometry be referred to 〕 as a result of and in connection to the mathematical analysis of curves and surfaces. Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in Calculus, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships and symmetries in nature, which the standard methods of analysis could not address. When curves, surfaces enclosed by curves, and points on curves were found to be quantitatively, and generally, related by mathematical forms the formal study of the nature of curves and surfaces became a field of study in its own right, with Monge's paper in 1795, and especially, with Gauss's publication of his article, titled 'Disquisitiones Generales Circa Superficies Curvas', in ''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores''〔 'Disquisitiones Generales Circa Superficies Curvas' (literal translation from Latin: General Investigations of Curved Surfaces), ''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' (literally, Recent Perspectives, Gottingen's Royal Society of Science). Volume VI, pp. 99–146. A translation of the work, by A.M.Hiltebeitel and J.C.Morehead, titled, "General Investigations of Curved Surfaces" was published 1965 by Raven Press, New York. A digitised version of the same is available at http://quod.lib.umich.edu/u/umhistmath/abr1255.0001.001 for free download, for non-commercial, personal use. In case of further information, the library could be contacted. Also, the Wikipedia article on Gauss's works in the year 1827 at could be looked at. 〕 in 1827. Initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「differential geometry」の詳細全文を読む スポンサード リンク
|