$$ z=f(x, y) $$ con derivadas parciales de segundo orden continuas: Si $$ x=r^{2}+s^{2}, \quad y=2 r s \quad $$ determine $$ \frac{\partial^{2} z}{\partial r^{2}}, \quad \frac{\partial^{2} z}{\partial s^{2}}, \quad \frac{\partial^{2} z}{\partial s \partial r} $$

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Davison Reid · Accepted Answer

Para encontrar las segundas derivadas parciales de $$ z $$ con respecto a $$ r $$ y $$ s $$, primero calculamos las primeras derivadas parciales de $$ z $$ con respecto a $$ x $$ y $$ y $$ usando la regla de la cadena. Luego, calculamos las segundas derivadas parciales. Las fórmulas resultantes son: $$ \frac{\partial^{2} z}{\partial r^{2}} = 2f_x + 2r\frac{\partial f_x}{\partial r} + 2s\frac{\partial f_y}{\partial r} $$ $$ \frac{\partial^{2} z}{\partial s^{2}} = 2f_x + 2s\frac{\partial f_x}{\partial s} + 2r\frac{\partial f_y}{\partial s} $$ $$ \frac{\partial^{2} z}{\partial s \partial r} = 2s\frac{\partial f_x}{\partial r} + 2f_y + 2r\frac{\partial f_y}{\partial r} $$

\( z=f(x, y) \) con derivadas parciales de segundo orden continuas: Si \( x=r^{2}+s^{2}, \quad y=2 r s \quad \) determine \( \frac{\partial^{2} z}{\partial r^{2}}, \quad \frac{\partial^{2} z}{\partial s^{2}}, \quad \frac{\partial^{2} z}{\partial s \partial r} \)

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\( z=f(x, y) \) con derivadas parciales de segundo orden continuas: Si \( x=r^{2}+s^{2}, \quad y=2 r s \quad \) determine \( \frac{\partial^{2} z}{\partial r^{2}}, \quad \frac{\partial^{2} z}{\partial s^{2}}, \quad \frac{\partial^{2} z}{\partial s \partial r} \)

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