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 \(x=r^{3}\ \text{sen}\ 2s,\) \(y=r^{2}\cos 3s.\)

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

y

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

Por último

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

En resumen

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