Regla de la Cadena de Funciones de Varias Variables Angel Carrillo Hoyo, Elena de Oteyza de Oteyza\(^2\), Carlos Hernández Garciadiego\(^1\), Emma Lam Osnaya\(^2\) \(^1\) Instituto de Matemáticas, UNAM; \(^2\) Facultad de Ciencias, UNAM

Solución:

Observamos el siguiente diagrama

La función \(f\) depende de \(x\) y \(y;\) tanto \(x\) como \(y\) dependen de \(r\) y \(s\).

Calculamos

\begin{eqnarray*} \dfrac{\partial z}{\partial r} & = &\dfrac{\partial f}{\partial x}\dfrac{ \partial x}{\partial r}+\dfrac{\partial f}{\partial y}\dfrac{\partial y}{ \partial r} \\ & = &\dfrac{\partial f}{\partial x}2rs^{3}+\dfrac{\partial f}{\partial y}s^{4} \\ & = &2rs^{3}\dfrac{\partial f}{\partial x}+s^{4}\dfrac{\partial f}{\partial y} \end{eqnarray*}

y

\begin{eqnarray*} \dfrac{\partial z}{\partial s} & = &\dfrac{\partial f}{\partial x}\dfrac{ \partial x}{\partial s}+\dfrac{\partial f}{\partial y}\dfrac{\partial y}{ \partial s} \\ & = &\dfrac{\partial f}{\partial x}3r^{2}s^{2}+\dfrac{\partial f}{\partial y} 4rs^{3} \\ & = &3r^{2}s^{2}\dfrac{\partial f}{\partial x}+4rs^{3}\dfrac{\partial f}{ \partial y} \end{eqnarray*}

Ahora consideramos el siguiente diagrama

donde observamos que tanto \(\dfrac{\partial f}{\partial x}\) como \(\dfrac{ \partial f}{\partial y}\) dependen de \(x\) y \(y\) que a su vez dependen de \(r\) y \(s.\)

Calculamos

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial r^{2}} & = &\dfrac{\partial }{\partial r} \left( \dfrac{\partial z}{\partial r}\right) \\ & = &\dfrac{\partial }{\partial r}\left( 2rs^{3}\dfrac{\partial f}{\partial x} +s^{4}\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial r}\left( 2rs^{3}\right) \dfrac{\partial f}{ \partial x}+2rs^{3}\dfrac{\partial }{\partial r}\left( \dfrac{\partial f}{ \partial x}\right) +\dfrac{\partial }{\partial r}\left( s^{4}\right) \dfrac{ \partial f}{\partial y}+s^{4}\dfrac{\partial }{\partial r}\left( \dfrac{ \partial f}{\partial y}\right) \\ & = &2s^{3}\dfrac{\partial f}{\partial x}+2rs^{3}\left( \dfrac{\partial }{ \partial x}\left( \dfrac{\partial f}{\partial x}\right) \dfrac{\partial x}{ \partial r}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial x }\right) \dfrac{\partial y}{\partial r}\right) +0+s^{4}\left( \dfrac{ \partial }{\partial x}\left( \dfrac{\partial f}{\partial y}\right) \dfrac{ \partial x}{\partial r}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f }{\partial y}\right) \dfrac{\partial y}{\partial r}\right) \\ & = &2s^{3}\dfrac{\partial f}{\partial x}+2rs^{3}\left( \dfrac{\partial ^{2}f}{ \partial x^{2}}2rs^{3}+\dfrac{\partial ^{2}f}{\partial y\partial x} s^{4}\right) +s^{4}\left( \dfrac{\partial ^{2}f}{\partial x\partial y} 2rs^{3}+\dfrac{\partial ^{2}f}{\partial y^{2}}s^{4}\right) \\ & = &2s^{3}\dfrac{\partial f}{\partial x}+4r^{2}s^{6}\dfrac{\partial ^{2}f}{ \partial x^{2}}+2rs^{7}\dfrac{\partial ^{2}f}{\partial y\partial x}+2rs^{7} \dfrac{\partial ^{2}f}{\partial x\partial y}+s^{8}\dfrac{\partial ^{2}f}{ \partial y^{2}} \\ & = &2s^{3}\dfrac{\partial f}{\partial x}+4r^{2}s^{6}\dfrac{\partial ^{2}f}{ \partial x^{2}}+4rs^{7}\dfrac{\partial ^{2}f}{\partial y\partial x}+s^{8} \dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

Calculamos

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial s\partial r} & = &\dfrac{\partial }{\partial s} \left( \dfrac{\partial z}{\partial r}\right) \\ & = &\dfrac{\partial }{\partial s}\left( 2rs^{3}\dfrac{\partial f}{\partial x} +s^{4}\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( 2rs^{3}\right) \dfrac{\partial f}{ \partial x}+2rs^{3}\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{ \partial x}\right) +\dfrac{\partial }{\partial s}\left( s^{4}\right) \dfrac{ \partial f}{\partial y}+s^{4}\dfrac{\partial }{\partial s}\left( \dfrac{ \partial f}{\partial y}\right) \\ & = &6rs^{2}\dfrac{\partial f}{\partial x}+2rs^{3}\left( \dfrac{\partial }{ \partial x}\left( \dfrac{\partial f}{\partial x}\right) \dfrac{\partial x}{ \partial s}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial x }\right) \dfrac{\partial y}{\partial s}\right) +4s^{3}\dfrac{\partial f}{ \partial y}+s^{4}\left( \dfrac{\partial }{\partial x}\left( \dfrac{\partial f }{\partial y}\right) \dfrac{\partial x}{\partial s}+\dfrac{\partial }{ \partial y}\left( \dfrac{\partial f}{\partial y}\right) \dfrac{\partial y}{ \partial s}\right) \\ & = &6rs^{2}\dfrac{\partial f}{\partial x}+2rs^{3}\left( \dfrac{\partial ^{2}f }{\partial x^{2}}3r^{2}s^{2}+\dfrac{\partial ^{2}f}{\partial y\partial x} 4rs^{3}\right) +4s^{3}\dfrac{\partial f}{\partial y}+s^{4}\left( \dfrac{ \partial ^{2}f}{\partial x\partial y}3r^{2}s^{2}+\dfrac{\partial ^{2}f}{ \partial y^{2}}4rs^{3}\right) \\ & = &6rs^{2}\dfrac{\partial f}{\partial x}+6r^{3}s^{5}\dfrac{\partial ^{2}f}{ \partial x^{2}}+8r^{2}s^{6}\dfrac{\partial ^{2}f}{\partial y\partial x} +4s^{3}\dfrac{\partial f}{\partial y}+3r^{2}s^{6}\dfrac{\partial ^{2}f}{ \partial x\partial y}+4rs^{7}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &6rs^{2}\dfrac{\partial f}{\partial x}+6r^{3}s^{5}\dfrac{\partial ^{2}f}{ \partial x^{2}}+\left( 8r^{2}s^{6}+3r^{2}s^{6}\right) \dfrac{\partial ^{2}f}{ \partial y\partial x}+4s^{3}\dfrac{\partial f}{\partial y}+4rs^{7}\dfrac{ \partial ^{2}f}{\partial y^{2}} \\ & = &6rs^{2}\dfrac{\partial f}{\partial x}+6r^{3}s^{5}\dfrac{\partial ^{2}f}{ \partial x^{2}}+11r^{2}s^{6}\dfrac{\partial ^{2}f}{\partial y\partial x} +4s^{3}\dfrac{\partial f}{\partial y}+4rs^{7}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

Por último, calculamos

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial s^{2}} & = &\dfrac{\partial }{\partial s} \left( \dfrac{\partial z}{\partial s}\right) \\ & = &\dfrac{\partial }{\partial s}\left( 3r^{2}s^{2}\dfrac{\partial f}{ \partial x}+4rs^{3}\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( 3r^{2}s^{2}\right) \dfrac{\partial f}{ \partial x}+3r^{2}s^{2}\dfrac{\partial }{\partial s}\left( \dfrac{\partial f }{\partial x}\right) +\dfrac{\partial }{\partial s}\left( 4rs^{3}\right) \dfrac{\partial f}{\partial y}+4rs^{3}\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &6r^{2}s\dfrac{\partial f}{\partial x}+3r^{2}s^{2}\left( \dfrac{\partial }{ \partial x}\left( \dfrac{\partial f}{\partial x}\right) \dfrac{\partial x}{ \partial s}+\dfrac{\partial }{\partial y}\left( \dfrac{\partial f}{\partial x }\right) \dfrac{\partial y}{\partial s}\right) +12rs^{2}\dfrac{\partial f}{ \partial y}+4rs^{3}\left( \dfrac{\partial }{\partial x}\left( \dfrac{ \partial f}{\partial y}\right) \dfrac{\partial x}{\partial s}+\dfrac{ \partial }{\partial y}\left( \dfrac{\partial f}{\partial y}\right) \dfrac{ \partial y}{\partial s}\right) \\ & = &6r^{2}s\dfrac{\partial f}{\partial x}+3r^{2}s^{2}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}3r^{2}s^{2}+\dfrac{\partial ^{2}f}{\partial y\partial x }4rs^{3}\right) +12rs^{2}\dfrac{\partial f}{\partial y}+4rs^{3}\left( \dfrac{ \partial ^{2}f}{\partial x\partial y}3r^{2}s^{2}+\dfrac{\partial ^{2}f}{ \partial y^{2}}4rs^{3}\right) \\ & = &6r^{2}s\dfrac{\partial f}{\partial x}+9r^{4}s^{4}\dfrac{\partial ^{2}f}{ \partial x^{2}}+12r^{3}s^{5}\dfrac{\partial ^{2}f}{\partial y\partial x} +12rs^{2}\dfrac{\partial f}{\partial y}+12r^{3}s^{5}\dfrac{\partial ^{2}f}{ \partial x\partial y}+16r^{2}s^{6}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &6r^{2}s\dfrac{\partial f}{\partial x}+9r^{4}s^{4}\dfrac{\partial ^{2}f}{ \partial x^{2}}+24r^{3}s^{5}\dfrac{\partial ^{2}f}{\partial y\partial x} +12rs^{2}\dfrac{\partial f}{\partial y}+16r^{2}s^{6}\dfrac{\partial ^{2}f}{ \partial y^{2}} \end{eqnarray*}

Por lo tanto,

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial r^{2}} & = &2s^{3}\dfrac{\partial f}{\partial x }+4r^{2}s^{6}\dfrac{\partial ^{2}f}{\partial x^{2}}+4rs^{7}\dfrac{\partial ^{2}f}{\partial y\partial x}+s^{8}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s\partial r} & = &6rs^{2}\dfrac{\partial f}{ \partial x}+6r^{3}s^{5}\dfrac{\partial ^{2}f}{\partial x^{2}}+\left( 8r^{2}s^{6}+3r^{2}s^{6}\right) \dfrac{\partial ^{2}f}{\partial y\partial x} +4s^{3}\dfrac{\partial f}{\partial y}+4rs^{7}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s^{2}} & = &6r^{2}s\dfrac{\partial f}{\partial x}+9r^{4}s^{4}\dfrac{\partial ^{2}f}{\partial x^{2}}+24r^{3}s^{5}\dfrac{ \partial ^{2}f}{\partial y\partial x}+12rs^{2}\dfrac{\partial f}{\partial y} +16r^{2}s^{6}\dfrac{\partial ^{2}f}{\partial y^{2}}. \end{eqnarray*}