Trabajo Fin de Grado
El teorema de la función implícita
Autor/es | Castaño Muñoz, José Carlos |
Director | Freniche Ibáñez, Francisco José |
Departamento | Universidad de Sevilla. Departamento de Análisis Matemático |
Fecha de publicación | 2018 |
Fecha de depósito | 2018-07-23 |
Titulación | Universidad de Sevilla. Grado en Matemáticas |
Resumen | In this work we study several proofs of the classical implicit function theorem,
which gives sufficient conditions for a vector equation to define a function. The first proof is the most elementary. It is obtained by ... In this work we study several proofs of the classical implicit function theorem, which gives sufficient conditions for a vector equation to define a function. The first proof is the most elementary. It is obtained by induction on the number of dependent variables. The second proof is also based on calculus, by looking directly at the linear approximation of the mapping. Finally, Banach fixed point theorem is used in the more abstract functional analytic third proof. In the second part we show an application of the implicit function theorem. We give four different definitions of a smooth surface and prove all of them are equivalent using the range theorem, an important consequence of the implicit function theorem. We finish the work showing an alternative proof of the implicit function theorem using a classical existence theorem of solutions in ordinary differential equations. |
Cita | Castaño Muñoz, J.C. (2018). El teorema de la función implícita. (Trabajo Fin de Grado Inédito). Universidad de Sevilla, Sevilla. |
Ficheros | Tamaño | Formato | Ver | Descripción |
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Castaño Muñoz José Carlos TFG.pdf | 948.2Kb | [PDF] | Ver/ | |