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

Si \(z=f\left( x,y\right) \) es una función real de clase \(C^{2}\) en \(\mathbb{R}^{2}.\) Calcula \(\dfrac{\partial ^{2}z}{\partial r^{2}},\) \( \dfrac{\partial ^{2}z}{\partial s\partial r},\) \(\dfrac{\partial ^{2}z}{ \partial s^{2}}\ \) si \(x=\dfrac{r}{s},\) \(y=rs.\)

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}\dfrac{1}{s}+\dfrac{\partial f}{\partial y}s \\ & = &\dfrac{1}{s}\dfrac{\partial f}{\partial x}+s\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}\left( -\dfrac{r}{s^{2}}\right) +\dfrac{ \partial f}{\partial y}r \\ & = &-\dfrac{r}{s^{2}}\dfrac{\partial f}{\partial x}+r\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( \dfrac{1}{s}\dfrac{\partial f}{ \partial x}+s\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial r}\left( \dfrac{1}{s}\right) \dfrac{\partial f }{\partial x}+\dfrac{1}{s}\dfrac{\partial }{\partial r}\left( \dfrac{ \partial f}{\partial x}\right) +\dfrac{\partial }{\partial r}\left( s\right) \dfrac{\partial f}{\partial y}+s\dfrac{\partial }{\partial r}\left( \dfrac{ \partial f}{\partial y}\right) \\ & = &0\dfrac{\partial f}{\partial x}+\dfrac{1}{s}\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\dfrac{\partial f}{\partial y}+s\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) \\ & = &\dfrac{1}{s}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}\dfrac{\partial x }{\partial r}+\dfrac{\partial ^{2}f}{\partial y\partial x}\dfrac{\partial y}{ \partial r}\right) +s\left( \dfrac{\partial ^{2}f}{\partial x\partial y} \dfrac{\partial x}{\partial r}+\dfrac{\partial ^{2}f}{\partial y^{2}}\dfrac{ \partial y}{\partial r}\right) \\ & = &\dfrac{1}{s}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}\dfrac{1}{s}+ \dfrac{\partial ^{2}f}{\partial y\partial x}s\right) +s\left( \dfrac{ \partial ^{2}f}{\partial x\partial y}\dfrac{1}{s}+\dfrac{\partial ^{2}f}{ \partial y^{2}}s\right) \\ & = &\dfrac{1}{s^{2}}\dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{\partial ^{2}f}{\partial y\partial x}+\dfrac{\partial ^{2}f}{\partial x\partial y} +s^{2}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &\dfrac{1}{s^{2}}\dfrac{\partial ^{2}f}{\partial x^{2}}+2\dfrac{\partial ^{2}f}{\partial y\partial x}+s^{2}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

Calculamos la derivada mixta

\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( \dfrac{1}{s}\dfrac{\partial f}{ \partial x}+s\dfrac{\partial f}{\partial y}\right) \\ &=&\dfrac{\partial }{\partial s}\left( \dfrac{1}{s}\right) \dfrac{\partial f }{\partial x}+\dfrac{1}{s}\dfrac{\partial }{\partial s}\left( \dfrac{ \partial f}{\partial x}\right) +\dfrac{\partial }{\partial s}\left( s\right) \dfrac{\partial f}{\partial y}+s\dfrac{\partial }{\partial s}\left( \dfrac{ \partial f}{\partial y}\right) \\ & = & -\dfrac{1}{s^{2}}\dfrac{\partial f}{\partial x}+\dfrac{1}{s}\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) +\dfrac{\partial f }{\partial y}+s\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) \\ & = & -\dfrac{1}{s^{2}}\dfrac{\partial f}{\partial x}+\dfrac{1}{s}\left( \dfrac{ \partial ^{2}f}{\partial x^{2}}\dfrac{\partial x}{\partial s}+\dfrac{% \partial ^{2}f}{\partial y\partial x}\dfrac{\partial y}{\partial s}\right) + \dfrac{\partial f}{\partial y}+s\left( \dfrac{\partial ^{2}f}{\partial x\partial y}\dfrac{\partial x}{\partial s}+\dfrac{\partial ^{2}f}{\partial y^{2}}\dfrac{\partial y}{\partial s}\right) \\ & = & -\dfrac{1}{s^{2}}\dfrac{\partial f}{\partial x}+\dfrac{1}{s}\left( \dfrac{ \partial ^{2}f}{\partial x^{2}}\left( -\dfrac{r}{s^{2}}\right) +\dfrac{ \partial ^{2}f}{\partial y\partial x}r\right) +\dfrac{\partial f}{\partial y} +s\left( \dfrac{\partial ^{2}f}{\partial x\partial y}\left( -\dfrac{r}{s^{2}} \right) +\dfrac{\partial ^{2}f}{\partial y^{2}}r\right) \\ & = & -\dfrac{1}{s^{2}}\dfrac{\partial f}{\partial x}-\dfrac{r}{s^{3}}\dfrac{ \partial ^{2}f}{\partial x^{2}}+\dfrac{r}{s}\dfrac{\partial ^{2}f}{\partial y\partial x}+\dfrac{\partial f}{\partial y}-\dfrac{r}{s}\dfrac{\partial ^{2}f }{\partial x\partial y}+rs\dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = & -\dfrac{1}{s^{2}}\dfrac{\partial f}{\partial x}-\dfrac{r}{s^{3}}\dfrac{ \partial ^{2}f}{\partial x^{2}}+\dfrac{\partial f}{\partial y}+rs\dfrac{ \partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

y

\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( -\dfrac{r}{s^{2}}\dfrac{\partial f}{ \partial x}+r\dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( -\dfrac{r}{s^{2}}\right) \dfrac{ \partial f}{\partial x}-\dfrac{r}{s^{2}}\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{\partial s}\left( r\right) \dfrac{\partial f}{\partial y}+r\dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{2r}{s^{3}}\dfrac{\partial f}{\partial x}-\dfrac{r}{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) +0+r\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) \\ & = &\dfrac{2r}{s^{3}}\dfrac{\partial f}{\partial x}-\dfrac{r}{s^{2}}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}\dfrac{\partial x}{\partial s}+\dfrac{ \partial ^{2}f}{\partial y\partial x}\dfrac{\partial y}{\partial s}\right) +r\left( \dfrac{\partial ^{2}f}{\partial x\partial y}\dfrac{\partial x}{ \partial s}+\dfrac{\partial ^{2}f}{\partial y^{2}}\dfrac{\partial y}{ \partial s}\right) \\ & = &\dfrac{2r}{s^{3}}\dfrac{\partial f}{\partial x}-\dfrac{r}{s^{2}}\left( \dfrac{\partial ^{2}f}{\partial x^{2}}\left( -\dfrac{r}{s^{2}}\right) + \dfrac{\partial ^{2}f}{\partial y\partial x}r\right) +r\left( \dfrac{ \partial ^{2}f}{\partial x\partial y}\left( -\dfrac{r}{s^{2}}\right) +\dfrac{ \partial ^{2}f}{\partial y^{2}}r\right) \\ & = &\dfrac{2r}{s^{3}}\dfrac{\partial f}{\partial x}+\dfrac{r^{2}}{s^{4}} \dfrac{\partial ^{2}f}{\partial x^{2}}-\dfrac{r^{2}}{s^{2}}\dfrac{\partial ^{2}f}{\partial y\partial x}-\dfrac{r^{2}}{s^{2}}\dfrac{\partial ^{2}f}{ \partial x\partial y}+r^{2}\dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &\dfrac{2r}{s^{3}}\dfrac{\partial f}{\partial x}+\dfrac{r^{2}}{s^{4}} \dfrac{\partial ^{2}f}{\partial x^{2}}-\dfrac{2r^{2}}{s^{2}}\dfrac{\partial ^{2}f}{\partial y\partial x}+r^{2}\dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

Por lo tanto,

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial r^{2}} & = & \dfrac{1}{s^{2}}\dfrac{\partial ^{2}f}{\partial x^{2}}+2\dfrac{\partial ^{2}f}{\partial y\partial x}+s^{2} \dfrac{\partial ^{2}f}{\partial y^{2}} \\ \\ \dfrac{\partial ^{2}z}{\partial s\partial r} & = & -\dfrac{1}{s^{2}}\dfrac{ \partial f}{\partial x}-\dfrac{r}{s^{3}}\dfrac{\partial ^{2}f}{\partial x^{2} }+\dfrac{\partial f}{\partial y}+rs\dfrac{\partial ^{2}f}{\partial y^{2}} \\ \\ \dfrac{\partial ^{2}z}{\partial s^{2}} & = & \dfrac{2r}{s^{3}}\dfrac{\partial f }{\partial x}+\dfrac{r^{2}}{s^{4}}\dfrac{\partial ^{2}f}{\partial x^{2}}- \dfrac{2r^{2}}{s^{2}}\dfrac{\partial ^{2}f}{\partial y\partial x}+r^{2} \dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}