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Sumario Transformaciones compuestas + Regla de la cadena
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  • Course
  • Análisis Matemático II
  • Institution
  • National University Of Luján (UNLu )

En este resumen vas a encontrar explicaciones, propiedades, teoremas, ejemplos de ejercicios resueltos y respuesta a ejercicios de libro "Lecciones de Análisis II" de Alfredo Novelli. Temas: - Transformaciones compuestas - Regla de la cadena

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Document information

  • March 16, 2021
  • 7
  • 2020/2021
  • Summary

Subjects

  • regla de la cadena
  • análisis matemático
  • transformaciones compuestas

Written for

  • National University of Luján (UNLu )
  • Análisis Matemático II
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Transformaciones compuestas. Regla de la cadena

Sea la transformación definida en un abierto de : : → (donde los valores de están
contenidos en un abierto de )
Y sea : → definida en el abierto tal que a cada vector perteneciente al abierto (de )
le corresponde según un vector del abierto , ( ) ∈ (de ) y a éste le corresponde según , un
vector ( ( )) de .




( ) ( ( ))




Si y son transformaciones diferenciables, entonces ( ( )) es diferenciable y resulta:
( ( )) ´ = ´( ( )). ´( )




= (", #)
= ( , , ) donde ! = (", #)$
%& %&
%' %
= (", #)
Supongamos y queremos calcular y , entonces


primeramente debemos tener bien claro que la composición se realiza de la siguiente forma:



(", #) ( , , )
( )



depende de " y de #, las variables , , son variables intermedias, además:


* * * * * * * * * * * * *
=+ , , ,.+ , , , = . + . + .
*" * * * *" *" *" * *" * *" * *"


* * * * * * * * * * * * *
=+ , , ,.+ , , , = . + . + .
*# * * * *# *# *# * *# * *# * *#




%& %&
en función de # y de 0 siendo = 1( , , ) = + + (
,
% %/
Ejemplo 1: Hallar y

= 0 + #, = 0 − #, = 0. #

, Gráficamente podemos llegar a las mismas fórmulas si tenemos en cuenta la variable según la cual
queremos hallar la derivada parcial:


3
%& %& %&
%4 %5 %6



7 8


%4 %4 %5 %5 %6 %6
%/ % %/ % %/ %


9 9 9


* * * * * * *
= . + . + .
*0 * *0 * *0 * *0


* * * * * * *
= . + . + .
*# * *# * *# * *#


* * * * * * * * * * * * * *
= . + . + . = = . + . + . =
*0 * *0 * *0 * *0 *# * *# * *# * *#
= 1 . 1 + 1 . 1 +2 . # = = 1 . 1 + 1 . (−1) + 2 . 0 =
= 2 + 2 0. #. # = 2 + 20. # ( = 2 0. #. 0 = 2#. 0 (




= (0, #)$
= ( , ) donde <
%6 %6
= (0, #) %/ %
Supongamos ahora una y queremos calcular y , entonces

8
%6 %6
%4 %5


7
%4 %4 %5 %5
%/ % %/ %

9 9

* * * * *
= . + .
*0 * *0 * *0
* * * * *
= . + .
*# * *# * *#

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