Minton and Smith, in "Calculus" define the chain rule for full derivatives $\frac {dz} {dt}$ as it follows:
Vretblad, however, in "Fourier Analysis and its Applications", mentions an "easy exercise in applying the chain rule" in an expansion of a partial derivative
:
The question is: can the chain rule, originally defined only on $\frac {dz} {dt}$, be extended to $\frac {\partial z} {\partial t}$, or is Vretblad applying the chain rule on a full derivative somehow?
Best Answer
Let me put it in this way: $u$ is a function of $x,t$, but can be thought of as a function of the new variables $\xi$ and $\eta$ after the change. Vretblad is using the standard physical formalism and keeps the same name for the function $u(x,t)$ and $u(\xi,\eta)$, so we get the (terrible from the mathematical point of view) identity $$ u(x,t)=u(\xi(x,t),\eta(x,t)). $$ Derivating both sides wrt $x$ (using the chain rule in the RHS) we get $$ u_x=\frac{\partial u}{\partial \xi}\underbrace{\frac{\partial\xi}{\partial x}}_{=1}+\frac{\partial u}{\partial \eta}\underbrace{\frac{\partial\eta}{\partial x}}_{=1}= \frac{\partial u}{\partial \xi}+\frac{\partial u}{\partial \eta}. $$ Doing it once again and applying the chain rule to both terms in the RHS gives you $$ u_{xx}=\color{red}{\frac{\partial}{\partial x}\Bigl(\frac{\partial u}{\partial \xi}\Bigr)}+\color{blue}{\frac{\partial}{\partial x}\Bigl(\frac{\partial u}{\partial \eta}\Bigr)}=\color{red}{\frac{\partial^2 u}{\partial \xi^2}\frac{\partial\xi}{\partial x}+\frac{\partial^2 u}{\partial\eta\partial\xi}\frac{\partial\eta}{\partial x}}+\color{blue}{\frac{\partial^2 u}{\partial \xi\partial\eta}\frac{\partial\xi}{\partial x}+\frac{\partial^2 u}{\partial\eta^2}\frac{\partial\eta}{\partial x}}=... $$ I hope you can continue after that.