Chapter 15 - Differentiation in Several Variables - 15.6 The Chain Rule - Exercises - Page 810: 46

Using the Chain Rule, we evaluate the partial derivatives of $f$ with respect to $t$ and obtain Eq. (12): ${f_{tt}} = {f_{xx}}{\left( {\frac{{\partial x}}{{\partial t}}} \right)^2} + 2{f_{xy}}\left( {\frac{{\partial x}}{{\partial t}}} \right)\left( {\frac{{\partial y}}{{\partial t}}} \right) + {f_{yy}}{\left( {\frac{{\partial y}}{{\partial t}}} \right)^2} + {f_x}\frac{{{\partial ^2}x}}{{\partial {t^2}}} + {f_y}\frac{{{\partial ^2}y}}{{\partial {t^2}}}$

Let $f$ be a function of $x$ and $y$, where $x = g\left( {t,s} \right)$, $y = h\left( {t,s} \right)$. Thus, $f\left( {t,s} \right) = f\left( {g\left( {t,s} \right),h\left( {t,s} \right)} \right)$ Using the Chain Rule, we evaluate the partial derivative of $f$ with respect to $t$: ${f_t} = \frac{{\partial f}}{{\partial g}}\frac{{\partial g}}{{\partial t}} + \frac{{\partial f}}{{\partial h}}\frac{{\partial h}}{{\partial t}}$ Another derivative gives ${f_{tt}} = \frac{\partial }{{\partial t}}\left( {\frac{{\partial f}}{{\partial g}}\frac{{\partial g}}{{\partial t}}} \right) + \frac{\partial }{{\partial t}}\left( {\frac{{\partial f}}{{\partial h}}\frac{{\partial h}}{{\partial t}}} \right)$ ${f_{tt}} = \frac{\partial }{{\partial t}}\left( {\frac{{\partial f}}{{\partial g}}} \right)\frac{{\partial g}}{{\partial t}} + \frac{{\partial f}}{{\partial g}}\frac{\partial }{{\partial t}}\left( {\frac{{\partial g}}{{\partial t}}} \right) + \frac{\partial }{{\partial t}}\left( {\frac{{\partial f}}{{\partial h}}} \right)\frac{{\partial h}}{{\partial t}} + \frac{{\partial f}}{{\partial h}}\frac{\partial }{{\partial t}}\left( {\frac{{\partial h}}{{\partial t}}} \right)$ ${f_{tt}} = \left( {\frac{{{\partial ^2}f}}{{\partial {g^2}}}\frac{{\partial g}}{{\partial t}} + \frac{{{\partial ^2}f}}{{\partial h\partial g}}\frac{{\partial h}}{{\partial t}}} \right)\frac{{\partial g}}{{\partial t}} + \frac{{\partial f}}{{\partial g}}\frac{{{\partial ^2}g}}{{\partial {t^2}}}$ ${\ \ \ \ \ }$ $ + \left( {\frac{{{\partial ^2}f}}{{\partial g\partial h}}\frac{{\partial g}}{{\partial t}} + \frac{{{\partial ^2}f}}{{\partial {h^2}}}\frac{{\partial h}}{{\partial t}}} \right)\frac{{\partial h}}{{\partial t}} + \frac{{\partial f}}{{\partial h}}\frac{{{\partial ^2}h}}{{\partial {t^2}}}$ ${\ \ \ \ }$ $ = \frac{{{\partial ^2}f}}{{\partial {g^2}}}{\left( {\frac{{\partial g}}{{\partial t}}} \right)^2} + \frac{{{\partial ^2}f}}{{\partial h\partial g}}\frac{{\partial h}}{{\partial t}}\frac{{\partial g}}{{\partial t}} + \frac{{\partial f}}{{\partial g}}\frac{{{\partial ^2}g}}{{\partial {t^2}}}$ ${\ \ \ \ \ }$ $ + \frac{{{\partial ^2}f}}{{\partial g\partial h}}\frac{{\partial g}}{{\partial t}}\frac{{\partial h}}{{\partial t}} + \frac{{{\partial ^2}f}}{{\partial {h^2}}}{\left( {\frac{{\partial h}}{{\partial t}}} \right)^2} + \frac{{\partial f}}{{\partial h}}\frac{{{\partial ^2}h}}{{\partial {t^2}}}$ Since $x = g\left( {t,s} \right)$, $y = h\left( {t,s} \right)$, we can write ${f_{tt}} = \frac{{{\partial ^2}f}}{{\partial {x^2}}}{\left( {\frac{{\partial x}}{{\partial t}}} \right)^2} + \frac{{{\partial ^2}f}}{{\partial y\partial x}}\frac{{\partial y}}{{\partial t}}\frac{{\partial x}}{{\partial t}} + \frac{{\partial f}}{{\partial x}}\frac{{{\partial ^2}x}}{{\partial {t^2}}}$ ${\ \ \ \ \ }$ $ + \frac{{{\partial ^2}f}}{{\partial x\partial y}}\frac{{\partial x}}{{\partial t}}\frac{{\partial y}}{{\partial t}} + \frac{{{\partial ^2}f}}{{\partial {y^2}}}{\left( {\frac{{\partial y}}{{\partial t}}} \right)^2} + \frac{{\partial f}}{{\partial y}}\frac{{{\partial ^2}y}}{{\partial {t^2}}}$ ${\ \ \ \ }$ $ = {f_{xx}}{\left( {\frac{{\partial x}}{{\partial t}}} \right)^2} + {f_{xy}}\frac{{\partial y}}{{\partial t}}\frac{{\partial x}}{{\partial t}} + {f_x}\frac{{{\partial ^2}x}}{{\partial {t^2}}}$ ${\ \ \ \ \ }$ $ + {f_{yx}}\frac{{\partial x}}{{\partial t}}\frac{{\partial y}}{{\partial t}} + {f_{yy}}{\left( {\frac{{\partial y}}{{\partial t}}} \right)^2} + {f_y}\frac{{{\partial ^2}y}}{{\partial {t^2}}}$ We assume that ${f_{xy}} = {f_{yx}}$, so ${f_{tt}} = {f_{xx}}{\left( {\frac{{\partial x}}{{\partial t}}} \right)^2} + 2{f_{xy}}\left( {\frac{{\partial x}}{{\partial t}}} \right)\left( {\frac{{\partial y}}{{\partial t}}} \right) + {f_x}\frac{{{\partial ^2}x}}{{\partial {t^2}}} + {f_{yy}}{\left( {\frac{{\partial y}}{{\partial t}}} \right)^2} + {f_y}\frac{{{\partial ^2}y}}{{\partial {t^2}}}$ Hence, we obtain Eq. (12): ${f_{tt}} = {f_{xx}}{\left( {\frac{{\partial x}}{{\partial t}}} \right)^2} + 2{f_{xy}}\left( {\frac{{\partial x}}{{\partial t}}} \right)\left( {\frac{{\partial y}}{{\partial t}}} \right) + {f_{yy}}{\left( {\frac{{\partial y}}{{\partial t}}} \right)^2} + {f_x}\frac{{{\partial ^2}x}}{{\partial {t^2}}} + {f_y}\frac{{{\partial ^2}y}}{{\partial {t^2}}}$