Partial derivative of a function defined in a rectangular area












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I noticed in a book that it considers a function $u(x,y)$ in the rectangular area : $xin (0, 1)$ ,$yin (0, 1]$, and takes there the $partial_y$. As I am used to only to take partial derivatives in open sets, I would appreciate a clarification of the value of $partial_y(x, 1)$. Is it $$partial_y{u(x, 1)}=lim_{tto 1^{-}}frac{u(x, t) -u(x, 1)}{t-1}$$ or something else? Thanks in advance.










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    I noticed in a book that it considers a function $u(x,y)$ in the rectangular area : $xin (0, 1)$ ,$yin (0, 1]$, and takes there the $partial_y$. As I am used to only to take partial derivatives in open sets, I would appreciate a clarification of the value of $partial_y(x, 1)$. Is it $$partial_y{u(x, 1)}=lim_{tto 1^{-}}frac{u(x, t) -u(x, 1)}{t-1}$$ or something else? Thanks in advance.










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      I noticed in a book that it considers a function $u(x,y)$ in the rectangular area : $xin (0, 1)$ ,$yin (0, 1]$, and takes there the $partial_y$. As I am used to only to take partial derivatives in open sets, I would appreciate a clarification of the value of $partial_y(x, 1)$. Is it $$partial_y{u(x, 1)}=lim_{tto 1^{-}}frac{u(x, t) -u(x, 1)}{t-1}$$ or something else? Thanks in advance.










      share|cite|improve this question















      I noticed in a book that it considers a function $u(x,y)$ in the rectangular area : $xin (0, 1)$ ,$yin (0, 1]$, and takes there the $partial_y$. As I am used to only to take partial derivatives in open sets, I would appreciate a clarification of the value of $partial_y(x, 1)$. Is it $$partial_y{u(x, 1)}=lim_{tto 1^{-}}frac{u(x, t) -u(x, 1)}{t-1}$$ or something else? Thanks in advance.







      calculus multivariable-calculus derivatives






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      dmtri

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          Yes, your definition of partial derivative with respect to $y$ at $(x,1)$



          $$ partial_y{u(x, 1)}=lim_{tto 1^{-}}frac{u(x, t) -u(x, 1)}{t-1}$$



          is correct. Of course it is one sided because we do not have $y>1$ as part of the domain.






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            Yes, your definition of partial derivative with respect to $y$ at $(x,1)$



            $$ partial_y{u(x, 1)}=lim_{tto 1^{-}}frac{u(x, t) -u(x, 1)}{t-1}$$



            is correct. Of course it is one sided because we do not have $y>1$ as part of the domain.






            share|cite|improve this answer


























              2














              Yes, your definition of partial derivative with respect to $y$ at $(x,1)$



              $$ partial_y{u(x, 1)}=lim_{tto 1^{-}}frac{u(x, t) -u(x, 1)}{t-1}$$



              is correct. Of course it is one sided because we do not have $y>1$ as part of the domain.






              share|cite|improve this answer
























                2












                2








                2






                Yes, your definition of partial derivative with respect to $y$ at $(x,1)$



                $$ partial_y{u(x, 1)}=lim_{tto 1^{-}}frac{u(x, t) -u(x, 1)}{t-1}$$



                is correct. Of course it is one sided because we do not have $y>1$ as part of the domain.






                share|cite|improve this answer












                Yes, your definition of partial derivative with respect to $y$ at $(x,1)$



                $$ partial_y{u(x, 1)}=lim_{tto 1^{-}}frac{u(x, t) -u(x, 1)}{t-1}$$



                is correct. Of course it is one sided because we do not have $y>1$ as part of the domain.







                share|cite|improve this answer












                share|cite|improve this answer



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                answered yesterday









                Mohammad Riazi-Kermani

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