Artículo Original
Publicado 1999-08-18
Cómo citar
Pinzón, S., & Paredes, M. (1999). La derivada de Carathéodory en R2. Revista Integración, Temas De matemáticas, 17(2), 65–98. Recuperado a partir de https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/838
Resumen
En este trabajo hacemos una completa revisión de la noción de derivada, para lo cual presentamos las definiciones de derivada de Gateaux, Fréchet y Carathéodory. Se muestran las relaciones entre ellas y finalmente se da una topología sobre R2 para la cual los tres conceptos de derivada son equivalentes.
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Referencias
[1] Acosta E.,Differentiability in Topological Groups, Reporte interno No. 39, Uni-versidad Nacional de Colombia, Bogotá, 1994.
[2] Acosta E. , Delgado C., “Frécht vs Carathéodory”,American Mathematical Monthly, Vol. 101, No. 2, 4, April (1994).
[3] Apostol T. M.,Análisis Matemático,Editorial Reverté, Barcelona (1986).
[4] Bartle R. G.,The Elements of Real Analysis, second edition, Wiley International Edition, 1964.
[5] Bruckner A. M., Leonard J. L. “Derivatives”,American Mathematical Monthly,Vol. 73 No. 4, April (Slaught Memorial Papers, 1966).
[6] Carathéodory C.,Theory of Functions of a Complex Variable, Chelsea, NewYork, 1954.
[7] Dubrovskii, “Sur certaines equations integrales non linéaire”,Uc. Zap. Moskov,Gos. Univ. 30 1939, 49–60.
[8]Fréchet M.,La Notion de differentielle dans l’analyse g ́en ́erale,C. R. Acad. Sci.(Paris), No. 180 (1925).
[9]Kreyszig E.,Introductory Functional Analysis With Aplications, John Wiley &Sons, New York, (1978).
[10]Kuhn S. , “The Derivative a la Carath ́eodory”,American Mathematical Monthly,Vol. 98, No. 1, January, (1991).
[11]Nashed M. Z., “Differentiability and Related Properties of Nonlinear Operators:Some Aspects of the Role of Differentials in Nonlinear Functional Analysis.”
[12]Nashed M. Z. , “Some Remarks on Variations and differentials”.American. Mat-hematical Monthly, Vol. 73 (1966) (Slaught Memorial Papers).
[13]Nicolescu L. J., “On Some Properties of the direct second order differentials inGˆateaux Sense”,Review Mathematics Pures Appicated, No. 8 (1963).
[14]Spivak M.,Calculus On Manifolds,W. A. Benjamin (1965).
[15] Tadeu Guerreiro G. R. de,Sobre as Várias Noc ̃oes de Diferenciabilidade, Disser-ta ̧cao apresentada para obtencao de Grau de Mestre em Matemática, Institutode Matemática Pura e Aplicada, Rio de Janeiro (1982).
[16] Rudin W.,Real and Complex Analysis, Third Edition, McGraw–Hill Book Com-pany (1987).
[17]Vainberg M. M.,Variational Methods for the Study of Nonlinear Operators,Holden–Day, San Francisco (1964).
[18]Willard S.,General Topology, Addison–Wesley Publishing Company, (1970)
[2] Acosta E. , Delgado C., “Frécht vs Carathéodory”,American Mathematical Monthly, Vol. 101, No. 2, 4, April (1994).
[3] Apostol T. M.,Análisis Matemático,Editorial Reverté, Barcelona (1986).
[4] Bartle R. G.,The Elements of Real Analysis, second edition, Wiley International Edition, 1964.
[5] Bruckner A. M., Leonard J. L. “Derivatives”,American Mathematical Monthly,Vol. 73 No. 4, April (Slaught Memorial Papers, 1966).
[6] Carathéodory C.,Theory of Functions of a Complex Variable, Chelsea, NewYork, 1954.
[7] Dubrovskii, “Sur certaines equations integrales non linéaire”,Uc. Zap. Moskov,Gos. Univ. 30 1939, 49–60.
[8]Fréchet M.,La Notion de differentielle dans l’analyse g ́en ́erale,C. R. Acad. Sci.(Paris), No. 180 (1925).
[9]Kreyszig E.,Introductory Functional Analysis With Aplications, John Wiley &Sons, New York, (1978).
[10]Kuhn S. , “The Derivative a la Carath ́eodory”,American Mathematical Monthly,Vol. 98, No. 1, January, (1991).
[11]Nashed M. Z., “Differentiability and Related Properties of Nonlinear Operators:Some Aspects of the Role of Differentials in Nonlinear Functional Analysis.”
[12]Nashed M. Z. , “Some Remarks on Variations and differentials”.American. Mat-hematical Monthly, Vol. 73 (1966) (Slaught Memorial Papers).
[13]Nicolescu L. J., “On Some Properties of the direct second order differentials inGˆateaux Sense”,Review Mathematics Pures Appicated, No. 8 (1963).
[14]Spivak M.,Calculus On Manifolds,W. A. Benjamin (1965).
[15] Tadeu Guerreiro G. R. de,Sobre as Várias Noc ̃oes de Diferenciabilidade, Disser-ta ̧cao apresentada para obtencao de Grau de Mestre em Matemática, Institutode Matemática Pura e Aplicada, Rio de Janeiro (1982).
[16] Rudin W.,Real and Complex Analysis, Third Edition, McGraw–Hill Book Com-pany (1987).
[17]Vainberg M. M.,Variational Methods for the Study of Nonlinear Operators,Holden–Day, San Francisco (1964).
[18]Willard S.,General Topology, Addison–Wesley Publishing Company, (1970)