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 \(x=\cos \left( rs\right) ,\) \(y=\ \text{sen}\ \left( rs\right) .\)

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}\left( -s\ \text{sen}\ \left( rs\right) \right) +\dfrac{\partial f}{\partial y}s\cos \left( rs\right) \\ & = &-s\ \text{sen}\ \left( rs\right) \dfrac{\partial f}{\partial x}+s\cos \left( rs\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( -r\ \text{sen}\ \left( rs\right) \right) +\dfrac{\partial f}{\partial y}r\cos \left( rs\right) \\ & = &-r\ \text{sen}\ \left( rs\right) \dfrac{\partial f}{\partial x}+r\cos \left( rs\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( -s\ \text{sen}\ \left( rs\right) \dfrac{\partial f}{\partial x}+s\cos \left( rs\right) \dfrac{\partial f}{ \partial y}\right) \\ & = &\dfrac{\partial }{\partial r}\left( -s\ \text{sen}\ \left( rs\right) \right) \dfrac{\partial f}{\partial x}-s\ \text{sen}\ \left( rs\right) \dfrac{ \partial }{\partial r}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{ \partial }{\partial r}\left( s\cos \left( rs\right) \right) \dfrac{\partial f }{\partial y}+s\cos \left( rs\right) \dfrac{\partial }{\partial r}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &-s^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}-s\ \text{sen}\ \left( rs\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) -s^{2}\ \text{sen}\ \left( rs\right) \dfrac{\partial f}{ \partial y}+s\cos \left( rs\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) \\ & = &-s^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}-s\ \text{sen}\ \left( rs\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) -s^{2}\ \text{sen}\ \left( rs\right) \dfrac{ \partial f}{\partial y}+s\cos \left( rs\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) \\ & = &-s^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}-s\ \text{sen}\ \left( rs\right) \left( \dfrac{\partial ^{2}f}{\partial x^{2}}\left( -s \ \text{sen}\ \left( rs\right) \right) +\dfrac{\partial ^{2}f}{\partial y\partial x}s\cos \left( rs\right) \right) -s^{2}\ \text{sen}\ \left( rs\right) \dfrac{\partial f}{\partial y}+ \\ & & +s\cos \left( rs\right) \left( \dfrac{\partial ^{2}f}{\partial x\partial y}\left( -s\ \text{sen}\ \left( rs\right) \right) +\dfrac{\partial ^{2}f}{\partial y^{2}}s\cos \left( rs\right) \right) \\ & = &-s^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}+s^{2}\ \text{sen} ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}} -s^{2}\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-s^{2}\ \text{sen}\ \left( rs\right) \dfrac{ \partial f}{\partial y}-s^{2}\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial x\partial y}+s^{2}\cos ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &-s^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}+s^{2}\ \text{sen} ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}} -2s^{2}\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-s^{2}\ \text{sen}\ \left( rs\right) \dfrac{ \partial f}{\partial y}+s^{2}\cos ^{2}\left( rs\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( -s\ \text{sen}\ \left( rs\right) \dfrac{\partial f}{\partial x}+s\cos \left( rs\right) \dfrac{\partial f}{ \partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( -s\ \text{sen}\ \left( rs\right) \right) \dfrac{\partial f}{\partial x}-s\ \text{sen}\ \left( rs\right) \dfrac{ \partial }{\partial s}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{ \partial }{\partial s}\left( s\cos \left( rs\right) \right) \dfrac{\partial f }{\partial y}+s\cos \left( rs\right) \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &\left( -\ \text{sen}\ \left( rs\right) -sr\cos \left( rs\right) \right) \dfrac{\partial f}{\partial x}-s\ \text{sen}\ \left( rs\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) +\left( \cos \left( rs\right) -sr\ \text{sen}\ \left( rs\right) \right) \dfrac{\partial f}{ \partial y}+ \\ & &+s\cos \left( rs\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( -\ \text{sen}\ \left( rs\right) -sr\cos \left( rs\right) \right) \dfrac{\partial f}{\partial x}-s\ \text{sen}\ \left( rs\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( \cos \left( rs\right) -sr\ \text{sen}\ \left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & & +s\cos \left( rs\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( -\ \text{sen}\ \left( rs\right) -sr\cos \left( rs\right) \right) \dfrac{\partial f}{\partial x}-s\ \text{sen}\ \left( rs\right) \left( \dfrac{ \partial ^{2}f}{\partial x^{2}}\left( -r\ \text{sen}\ \left( rs\right) \right) +\dfrac{\partial ^{2}f}{\partial y\partial x}r\cos \left( rs\right) \right) +\left( \cos \left( rs\right) -sr\ \text{sen}\ \left( rs\right) \right) \dfrac{\partial f}{\partial y}+ \\ & & +s\cos \left( rs\right) \left( \dfrac{ \partial ^{2}f}{\partial x\partial y}\left( -r\ \text{sen}\ \left( rs\right) \right) +\dfrac{\partial ^{2}f}{\partial y^{2}}r\cos \left( rs\right) \right) \\ & = &\left( -\ \text{sen}\ \left( rs\right) -sr\cos \left( rs\right) \right) \dfrac{\partial f}{\partial x}+sr\ \text{sen} ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-sr\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+ \\ & & +\left( \cos \left( rs\right) - sr\ \text{sen}\ \left( rs\right) \right) \dfrac{\partial f}{ \partial y}-sr\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{ \partial ^{2}f}{\partial x\partial y}+sr\cos ^{2}\left( rs\right) \dfrac{ \partial ^{2}f}{\partial y^{2}} \\ & = &\left( -\ \text{sen}\ \left( rs\right) -sr\cos \left( rs\right) \right) \dfrac{\partial f}{\partial x}+sr\ \text{sen} ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-2sr\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}+\left( \cos \left( rs\right) -sr\ \text{sen}\ \left( rs\right) \right) \dfrac{\partial f}{ \partial y}+sr\cos ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}

y ahora 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( -r\ \text{sen}\ \left( rs\right) \dfrac{\partial f}{\partial x}+r\cos \left( rs\right) \dfrac{\partial f}{ \partial y}\right) \\ & = &\dfrac{\partial }{\partial s}\left( -r\ \text{sen}\ \left( rs\right) \right) \dfrac{\partial f}{\partial x}-r\ \text{sen}\ \left( rs\right) \dfrac{ \partial }{\partial s}\left( \dfrac{\partial f}{\partial x}\right) +\dfrac{ \partial }{\partial s}\left( r\cos \left( rs\right) \right) \dfrac{\partial f }{\partial y}+r\cos \left( rs\right) \dfrac{\partial }{\partial s}\left( \dfrac{\partial f}{\partial y}\right) \\ & = &-r^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}-r\ \text{sen}\ \left( rs\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) -r^{2}\ \text{sen}\ \left( rs\right) \dfrac{\partial f}{ \partial y}+r\cos \left( rs\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) \\ & = &-r^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}-r\ \text{sen}\ \left( rs\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) -r^{2}\ \text{sen}\ \left( rs\right) \dfrac{ \partial f}{\partial y}+r\cos \left( rs\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) \\ & = &-r^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}-r\ \text{sen}\ \left( rs\right) \left( \dfrac{\partial ^{2}f}{\partial x^{2}}\left( -r \ \text{sen}\ \left( rs\right) \right) +\dfrac{\partial ^{2}f}{\partial y\partial x}\left( r\cos \left( rs\right) \right) \right) -r^{2}\ \text{sen}\ \left( rs\right) \dfrac{\partial f}{\partial y}+ \\ & &+r\cos \left( rs\right) \left( \dfrac{\partial ^{2}f}{\partial x\partial y}\left( -r\ \text{sen}\ \left( rs\right) \right) +\dfrac{\partial ^{2}f}{\partial y^{2}}\left( r\cos \left( rs\right) \right) \right) \\ & = &-r^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}+r^{2}\ \text{sen} ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}} -r^{2}\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-r^{2}\ \text{sen}\ \left( rs\right) \dfrac{ \partial f}{\partial y}-r^{2}\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial x\partial y}+r^{2}\cos ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ & = &-r^{2}\cos \left( rs\right) \dfrac{\partial f}{\partial x}+r^{2}\ \text{sen} ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}} -2r^{2}\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-r^{2}\ \text{sen}\ \left( rs\right) \dfrac{ \partial f}{\partial y}+r^{2}\cos ^{2}\left( rs\right) \dfrac{\partial ^{2}f }{\partial y^{2}} \end{eqnarray*}

En resumen

\begin{eqnarray*} \dfrac{\partial ^{2}z}{\partial r^{2}} & = &-s^{2}\cos \left( rs\right) \dfrac{ \partial f}{\partial x}+s^{2}\ \text{sen} ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-2s^{2}\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-s^{2} \ \text{sen}\ \left( rs\right) \dfrac{\partial f}{\partial y}+s^{2}\cos ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s\partial r} & = &\left( -\ \text{sen}\ \left( rs\right) -sr\cos \left( rs\right) \right) \dfrac{\partial f}{\partial x}+sr \ \text{sen}\ ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-2sr\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{ \partial ^{2}f}{\partial y\partial x}+\left( \cos \left( rs\right) -sr \ \text{sen}\ \left( rs\right) \right) \dfrac{\partial f}{\partial y}+sr\cos ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \\ \dfrac{\partial ^{2}z}{\partial s^{2}} & = &-r^{2}\cos \left( rs\right) \dfrac{ \partial f}{\partial x}+r^{2}\ \text{sen} ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial x^{2}}-2r^{2}\ \text{sen}\ \left( rs\right) \cos \left( rs\right) \dfrac{\partial ^{2}f}{\partial y\partial x}-r^{2} \ \text{sen}\ \left( rs\right) \dfrac{\partial f}{\partial y}+r^{2}\cos ^{2}\left( rs\right) \dfrac{\partial ^{2}f}{\partial y^{2}} \end{eqnarray*}