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 \(x=\tan \dfrac{r}{s},\) \(y=\cot \dfrac{s}{r}.\)

Solución:

Observamos el siguiente diagrama

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}\left( \dfrac{1}{s}\sec ^{2}\left( \dfrac{r }{s}\right) \right) +\dfrac{\partial f}{\partial y}\left( \dfrac{s}{r^{2}} \csc ^{2}\left( \dfrac{s}{r}\right) \right) \\ & = &\dfrac{1}{s}\sec ^{2}\left( \dfrac{r}{s}\right) \dfrac{\partial f}{ \partial x}+\dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \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}}\sec ^{2}\left( \dfrac{r}{s}\right) \right) +\dfrac{\partial f}{\partial y}\left( -\dfrac{1}{ r}\csc ^{2}\left( \dfrac{s}{r}\right) \right) \\ & = &-\dfrac{r}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \dfrac{\partial f}{ \partial x}-\dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r}\right) \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}\sec ^{2}\left( \dfrac{r}{ s}\right) \dfrac{\partial f}{\partial x}+\dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial f}{\partial y}\right) \\ & = & \dfrac{\partial }{\partial r}\left( \dfrac{1}{s}\sec ^{2}\left( \dfrac{r}{ s}\right) \right) \dfrac{\partial f}{\partial x}+\dfrac{1}{s}\sec ^{2}\left( \dfrac{r}{s}\right) \dfrac{\partial }{\partial r}\left( \dfrac{\partial f}{ \partial x}\right) +\dfrac{\partial }{\partial r}\left( \dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \right) \dfrac{\partial f}{\partial y}+\dfrac{ s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial }{\partial r} \left( \dfrac{\partial f}{\partial y}\right) \\ & = & \dfrac{2}{s}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s} \right) \left( \dfrac{1}{s}\right) \dfrac{\partial f}{\partial x}+\dfrac{1}{s }\sec ^{2}\left( \dfrac{r}{s}\right) \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) + \\ & & + \left( -\dfrac{2s}{r^{3}}\csc ^{2}\left( \dfrac{s}{r}\right) -\dfrac{s}{r^{2}}\left( 2\right) \csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \left( - \dfrac{s}{r^{2}}\right) \right) \dfrac{\partial f}{\partial y}+ \dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \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{2}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{ s}\right) \dfrac{\partial f}{\partial x}+\dfrac{1}{s}\sec ^{2}\left( \dfrac{r }{s}\right) \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) + \\ & & + \left( -\dfrac{2s}{r^{3}}\csc ^{2}\left( \dfrac{s}{r} \right) +\dfrac{2s^{2}}{r^{4}}\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \right) \dfrac{\partial f}{\partial y}+ \dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \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{2}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{ s}\right) \dfrac{\partial f}{\partial x}+\dfrac{1}{s}\sec ^{2}\left( \dfrac{r }{s}\right) \left( \dfrac{\partial ^{2}f}{\partial x^{2}}\left( \dfrac{1}{s} \sec ^{2}\left( \dfrac{r}{s}\right) \right) +\dfrac{\partial ^{2}f}{\partial y\partial x}\left( \dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \right) \right) + \\ & & + \left( -\dfrac{2s}{r^{3}}\csc ^{2}\left( \dfrac{s}{r} \right) +\dfrac{2s^{2}}{r^{4}}\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \right) \dfrac{\partial f}{\partial y}+ \dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \left( \dfrac{ \partial ^{2}f}{\partial x\partial y}\left( \dfrac{1}{s}\sec ^{2}\left( \dfrac{r}{s}\right) \right) +\dfrac{\partial ^{2}f}{\partial y^{2}}\left( \dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \right) \right) \\ & = & \dfrac{2}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{ s}\right) \dfrac{\partial f}{\partial x}+\dfrac{1}{s^{2}}\sec ^{4}\left( \dfrac{r}{s}\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{1}{r^{2}} \sec ^{2}\left( \dfrac{r}{s}\right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+\left( -\dfrac{2s}{r^{3}}\csc ^{2}\left( \dfrac{s}{r}\right) +\dfrac{2s^{2}}{r^{4}}\csc ^{2}\left( \dfrac{s }{r}\right) \cot \left( \dfrac{s}{r}\right) \right) \dfrac{\partial f}{ \partial y} \\ & & +\dfrac{1}{r^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial x\partial y}+\dfrac{s^{2} }{r^{4}}\csc ^{4}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = & \dfrac{2}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \dfrac{r}{s} \dfrac{\partial f}{\partial x}+\dfrac{1}{s^{2}}\sec ^{4}\left( \dfrac{r}{s} \right) \dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{2}{r^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{ \partial ^{2}f}{\partial y\partial x}+\left( -\dfrac{2s}{r^{3}}\csc ^{2}\left( \dfrac{s}{r}\right) +\dfrac{2s^{2}}{r^{4}}\csc ^{2}\left( \dfrac{s }{r}\right) \cot \dfrac{s}{r}\right) \dfrac{\partial f}{\partial y}+ \\ & &+ \dfrac{s^{2}}{r^{4}}\csc ^{4}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{ \partial y^{2}} \end{eqnarray*}

La derivada mixta es

\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}\sec ^{2}\left( \dfrac{r}{ s}\right) \dfrac{\partial f}{\partial x}+\dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial f}{\partial y}\right) \\ & = & \dfrac{\partial }{\partial s}\left( \dfrac{1}{s}\sec ^{2}\left( \dfrac{r}{ s}\right) \right) \dfrac{\partial f}{\partial x}+\dfrac{1}{s}\sec ^{2}\left( \dfrac{r}{s}\right) \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{ \partial x}\right) +\dfrac{\partial }{\partial s}\left( \dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \right) \dfrac{\partial f}{\partial y}+\dfrac{ s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial }{\partial s} \left( \dfrac{\partial f}{\partial y}\right) \\ & = & \left( -\dfrac{1}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) +\dfrac{2}{s} \sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s}\right) \left( - \dfrac{r}{s^{2}}\right) \right) \dfrac{\partial f}{\partial x}+\dfrac{1}{s} \sec ^{2}\left( \dfrac{r}{s}\right) \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) + \\ & &+ \left( \dfrac{1}{r^{2}} \csc ^{2}\left( \dfrac{s}{r}\right) +\dfrac{s}{r^{2}}\left( -2\right) \csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \left( \dfrac{ 1}{r}\right) \right) \dfrac{\partial f}{\partial y}+ \dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \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) \\ & = & \left( -\dfrac{1}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) -\dfrac{2r}{ s^{3}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x}+\dfrac{1}{s}\sec ^{2}\left( \dfrac{r}{ s}\right) \left( \dfrac{\partial ^{2}f}{\partial x^{2}}\left( -\dfrac{r}{ s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \right) +\dfrac{\partial ^{2}f}{ \partial y\partial x}\left( -\dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r}\right) \right) \right) + \\ & & + \left( \dfrac{1}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) - \dfrac{2s}{r^{3}}\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r} \right) \right) \dfrac{\partial f}{\partial y}+ \dfrac{s}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \left( \dfrac{ \partial ^{2}f}{\partial x\partial y}\left( -\dfrac{r}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \right) +\dfrac{\partial ^{2}f}{\partial y^{2}}\left( - \dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r}\right) \right) \right) \\ & = & \left( -\dfrac{1}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) -\dfrac{2r}{ s^{3}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x}-\dfrac{r}{s^{3}}\sec ^{4}\left( \dfrac{r}{s}\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-\dfrac{1}{rs}\sec ^{2}\left( \dfrac{r}{s}\right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{ \partial ^{2}f}{\partial y\partial x}+ \\ & & +\left( \dfrac{1}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) -\dfrac{2s}{r^{3}}\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \right) \dfrac{\partial f}{\partial y} -\dfrac{1}{rs}\csc ^{2}\left( \dfrac{s}{r}\right) \sec ^{2}\left( \dfrac{r }{s}\right) \dfrac{\partial ^{2}f}{\partial x\partial y}-\dfrac{s}{r^{3}} \csc ^{4}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = & \left( -\dfrac{1}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) -\dfrac{2r}{ s^{3}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x}-\dfrac{r}{s^{3}}\sec ^{4}\left( \dfrac{r}{s}\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-\dfrac{2}{rs}\sec ^{2}\left( \dfrac{r}{s}\right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{ \partial ^{2}f}{\partial y\partial x}+ \\ & & + \left( \dfrac{1}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) -\dfrac{2s}{r^{3}}\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \right) \dfrac{\partial f}{\partial y}- \dfrac{s^{2}}{r^{3}}\csc ^{4}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f }{\partial y^{2}} \end{eqnarray*}

Ahora calculamos la derivada parcial de segundo orden

\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}}\sec ^{2}\left( \dfrac{r}{s}\right) \dfrac{\partial f}{\partial x}-\dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial f}{\partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( -\dfrac{r}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x}-\dfrac{r}{s^{2}} \sec ^{2}\left( \dfrac{r}{s}\right) \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{\partial }{\partial s}\left( - \dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r}\right) \right) \dfrac{\partial f}{ \partial y}-\dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &\left( \dfrac{2r}{s^{3}}\sec ^{2}\left( \dfrac{r}{s}\right) +\dfrac{2r^{2} }{s^{4}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x}-\dfrac{r}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \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{2}{r^{2}}\csc ^{2}\left( \dfrac{s}{r} \right) \cot \left( \dfrac{s}{r}\right) \dfrac{\partial f}{\partial y} -\dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r}\right) \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) \\ & = &\left( \dfrac{2r}{s^{3}}\sec ^{2}\left( \dfrac{r}{s}\right) +\dfrac{2r^{2} }{s^{4}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x}-\dfrac{r}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \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{2}{r^{2}}\csc ^{2}\left( \dfrac{s}{r} \right) \cot \left( \dfrac{s}{r}\right) \dfrac{\partial f}{\partial y}- \dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r}\right) \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) \\ & = &\left( \dfrac{2r}{s^{3}}\sec ^{2}\left( \dfrac{r}{s}\right) +\dfrac{2r^{2} }{s^{4}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x}-\dfrac{r}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \left( \dfrac{\partial ^{2}f}{\partial x^{2}}\left( - \dfrac{r}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \right) +\dfrac{\partial ^{2}f}{\partial y\partial x}\left( -\dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r} \right) \right) \right) + \\ & &+\dfrac{2}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \dfrac{\partial f}{\partial y} -\dfrac{1}{r}\csc ^{2}\left( \dfrac{s}{r}\right) \left( \dfrac{\partial ^{2}f}{\partial x\partial y}\left( -\dfrac{r}{s^{2}}\sec ^{2}\left( \dfrac{r }{s}\right) \right) +\dfrac{\partial ^{2}f}{\partial y^{2}}\left( -\dfrac{1}{ r}\csc ^{2}\left( \dfrac{s}{r}\right) \right) \right) \\ & = &\left( \dfrac{2r}{s^{3}}\sec ^{2}\left( \dfrac{r}{s}\right) +\dfrac{2r^{2} }{s^{4}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x}+\dfrac{r^{2}}{s^{4}}\sec ^{4}\left( \dfrac{r}{s}\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{1}{s^{2}} \sec ^{2}\left( \dfrac{r}{s}\right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+ \\ & &+\dfrac{2}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \dfrac{\partial f}{ \partial y}+ \dfrac{1}{s^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \sec ^{2}\left( \dfrac{r}{s}\right) \dfrac{\partial ^{2}f}{\partial x\partial y}+\dfrac{1}{ r^{2}}\csc ^{4}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &\left( \dfrac{2r}{s^{3}}\sec ^{2}\left( \dfrac{r}{s}\right) +\dfrac{2r^{2} }{s^{4}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x}+\dfrac{r^{2}}{s^{4}}\sec ^{4}\left( \dfrac{r}{s}\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{2}{s^{2}} \sec ^{2}\left( \dfrac{r}{s}\right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+ \\ & &+\dfrac{2}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \dfrac{\partial f}{ \partial y}+\dfrac{1}{r^{2}}\csc ^{4}\left( \dfrac{s}{r}\right) \dfrac{ \partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

Por lo tanto,

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial r^{2}} & = & \dfrac{2}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \tan \dfrac{r}{s}\dfrac{\partial f}{\partial x}+\dfrac{1 }{s^{2}}\sec ^{4}\left( \dfrac{r}{s}\right) \dfrac{\partial ^{2}f}{\partial x^{2}}+\dfrac{2}{r^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+\left( - \dfrac{2s}{r^{3}}\csc ^{2}\left( \dfrac{s}{r}\right) +\dfrac{2s^{2}}{r^{4}} \csc ^{2}\left( \dfrac{s}{r}\right) \cot \dfrac{s}{r}\right) \dfrac{\partial f}{\partial y}+ \\ & & +\dfrac{s^{2}}{r^{4}}\csc ^{4}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ \\ \dfrac{\partial ^{2}z}{\partial s\partial r} & = & \left( -\dfrac{1}{s^{2}}\sec ^{2}\left( \dfrac{r}{s}\right) -\dfrac{2r}{s^{3}}\sec ^{2}\left( \dfrac{r}{s} \right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{\partial x }-\dfrac{r}{s^{3}}\sec ^{4}\left( \dfrac{r}{s}\right) \dfrac{\partial ^{2}f}{ \partial x^{2}}-\dfrac{2}{rs}\sec ^{2}\left( \dfrac{r}{s}\right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y\partial x} + \\ & & + \left( \dfrac{1}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) -\dfrac{2s}{r^{3} }\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \right) \dfrac{\partial f}{\partial y}-\dfrac{s^{2}}{r^{3}}\csc ^{4}\left( \dfrac{s}{ r}\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ \\ \dfrac{\partial ^{2}z}{\partial s^{2}} & = & \left( \dfrac{2r}{s^{3}}\sec ^{2}\left( \dfrac{r}{s}\right) +\dfrac{2r^{2}}{s^{4}}\sec ^{2}\left( \dfrac{r }{s}\right) \tan \left( \dfrac{r}{s}\right) \right) \dfrac{\partial f}{ \partial x}+\dfrac{r^{2}}{s^{4}}\sec ^{4}\left( \dfrac{r}{s}\right) \dfrac{ \partial ^{2}f}{\partial x^{2}}+\dfrac{2}{s^{2}}\sec ^{2}\left( \dfrac{r}{s} \right) \csc ^{2}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+\dfrac{2}{r^{2}}\csc ^{2}\left( \dfrac{s}{r}\right) \cot \left( \dfrac{s}{r}\right) \dfrac{\partial f}{\partial y}+ \\ & & +\dfrac{1}{r^{2}}\csc ^{4}\left( \dfrac{s}{r}\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}