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

Como la función \(z\) es de clase \(C^{2},\) entonces \(\dfrac{\partial ^{2}f }{\partial y\partial x}=\dfrac{\partial ^{2}f}{\partial x\partial y},\) de donde

\begin{equation*} \dfrac{\partial ^{2}z}{\partial r^{2}}=12r^{2}\dfrac{\partial f}{\partial x} +16r^{6}\dfrac{\partial ^{2}f}{\partial x^{2}}+24r^{5}\dfrac{\partial ^{2}f}{ \partial y\partial x}+6r\dfrac{\partial f}{\partial y}+9r^{4}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{equation*}

Ahora 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( 4r^{3}\dfrac{\partial f}{\partial x} +3r^{2}\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( 4r^{3}\right) \dfrac{\partial f}{ \partial x}+4r^{3}\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{ \partial x}\right) +\dfrac{\partial }{\partial s}\left( 3r^{2}\right) +3r^{2} \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &4r^{3}\left( \dfrac{\partial }{\partial x}\dfrac{\partial f}{\partial x} \dfrac{\partial x}{\partial s}+\dfrac{\partial }{\partial y}\dfrac{\partial f }{\partial x}\dfrac{\partial y}{\partial s}\right) +3r^{2}\left( \dfrac{ \partial }{\partial x}\dfrac{\partial f}{\partial y}\dfrac{\partial x}{ \partial s}+\dfrac{\partial }{\partial y}\dfrac{\partial f}{\partial y} \dfrac{\partial y}{\partial s}\right) \\ & = &4r^{3}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}2s+\dfrac{\partial ^{2}f}{\partial y\partial x}3s^{2}\right) +3r^{2}\left( \dfrac{\partial ^{2}f }{\partial x\partial y}2s+\dfrac{\partial ^{2}f}{\partial y^{2}}3s^{2}\right) \\ & = &8r^{3}s\dfrac{\partial ^{2}f}{\partial x^{2}}+12r^{3}s^{2}\dfrac{\partial ^{2}f}{\partial y\partial x}+6r^{2}s\dfrac{\partial ^{2}f}{\partial x\partial y}+9r^{2}s^{2}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &8r^{3}s\dfrac{\partial ^{2}f}{\partial x^{2}}+\left( 12r^{3}s^{2}+6r^{2}s\right) \dfrac{\partial ^{2}f}{\partial y\partial x} +9r^{2}s^{2}\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( 2s\dfrac{\partial f}{\partial x} +3s^{2}\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( 2s\right) \dfrac{\partial f}{\partial x}+2s\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial x} \right) +\dfrac{\partial }{\partial s}\left( 3s^{2}\right) \dfrac{\partial f }{\partial y}+3s^{2}\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{ \partial y}\right) \\ & = &2\dfrac{\partial f}{\partial x}+2s\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) +6s\dfrac{\partial f}{\partial y} +3s^{2}\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) \\ & = &2\dfrac{\partial f}{\partial x}+2s\left( \dfrac{\partial ^{2}f}{\partial x^{2}}2s+\dfrac{\partial ^{2}f}{\partial y\partial x}3s^{2}\right) +6s\dfrac{ \partial f}{\partial y}+3s^{2}\left( \dfrac{\partial ^{2}f}{\partial x\partial y}2s+\dfrac{\partial ^{2}f}{\partial y^{2}}3s^{2}\right) \\ & = &2\dfrac{\partial f}{\partial x}+4s^{2}\dfrac{\partial ^{2}f}{\partial x^{2}}+6s^{3}\dfrac{\partial ^{2}f}{\partial y\partial x}+6s\dfrac{\partial f }{\partial y}+6s^{3}\dfrac{\partial ^{2}f}{\partial x\partial y}+9s^{4} \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &2\dfrac{\partial f}{\partial x}+4s^{2}\dfrac{\partial ^{2}f}{\partial x^{2}}+12s^{3}\dfrac{\partial ^{2}f}{\partial y\partial x}+6s\dfrac{\partial f}{\partial y}+9s^{4}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

Por lo tanto,

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