Assuming that $u = f(x,y)$, $x = e^s\sin(t)$, $y = e^s\sin(t)$
Show that $(\frac{\partial u}{\partial s})^2 \neq \frac{\partial^2 u}{\partial d s^2}$
I know what to do, but I don't know how to do it. The RHS gives me difficulties.
Assuming that $u = f(x,y)$, $x = e^s\sin(t)$, $y = e^s\sin(t)$
Show that $(\frac{\partial u}{\partial s})^2 \neq \frac{\partial^2 u}{\partial d s^2}$
I know what to do, but I don't know how to do it. The RHS gives me difficulties.
Since $x=y$, let $f(x,y)=g(x)\\ \frac{\mathrm{d}u}{\mathrm{d}s}=x\frac{\mathrm{d}g}{\mathrm{d}s}\\ \frac{\mathrm{d}^2u}{\mathrm{d}s^2}=\frac{\mathrm{d}g}{\mathrm{d}s}+x\frac{\mathrm{d}^2g}{\mathrm{d}s^2}$
Setting $h=\frac{\mathrm{d}g}{\mathrm{d}s} \ \, \ \frac{\mathrm{d}u}{\mathrm{d}s}=\frac{\mathrm{d}^2u}{\mathrm{d}s^2} \text{iff} \frac{\mathrm{d}h}{\mathrm{d}s}=xh^2-\frac{h}x$. Since a solution must exist, they are unequal only for almost all $u$.