Teorema fundamental da teoria local das curvas e suas aplicações

Abstract

The present work is an introductory study of differential geometry, with the objective of demonstrating the Fundamental Theorem of the Local Theory of Curves in a clear, objective way and citing applications regarding the preliminary concepts for demonstrating the theorem. To achieve the objectives of the work, a bibliographical review was used and, as main tools, definitions of some properties of differentiable parameterized curves, such as tangent vector, arc length, curvature, torsion, Frenet trihedron, in addition to classic calculation results differential and integral. Furthermore, concepts from Ordinary Differential Equations, more precisely, the existence and uniqueness theorem, were also used. With this study, it was possible to develop interest in scientific research, improve knowledge in the area of differential geometry and develop the proposed objectives. Therefore, the study of Differential Geometry allows us to understand the properties of parameterized curves through fundamental concepts and use them to cite applications. Furthermore, this work can contribute to the dissemination of knowledge in Mathematics and serve as research material on Differential Geometry of differentiable parameterized curves.