Partiella derivator med kedjeregeln

Jag tycker det är enklast att uttrycka det i form av differentialoperatorer som t.ex

$\frac{\partial }{\partial s}=\frac{\partial x}{\partial s}\frac{\partial }{\partial x}+\frac{\partial y}{\partial s}\frac{\partial }{\partial y}$

Hur blir det för t? Den sökta andraderivatan är produkten av s- och t-operatorerna verkande på f.

Ah, okej, förstår nu.

Tack!

$$\begin{gathered} \frac{{\partial }^{2}f}{\partial s\partial t}=\frac{\partial }{\partial s}\left(\frac{\partial f}{\partial t}\right)=\frac{{\displaystyle \partial }}{{\displaystyle \partial s}}(\frac{{\displaystyle \partial f}}{{\displaystyle \partial x}}\frac{{\displaystyle \partial x}}{{\displaystyle \partial t}}+\frac{{\displaystyle \partial f}}{{\displaystyle \partial y}}\frac{{\displaystyle \partial y}}{{\displaystyle \partial t}})= \\ \frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial x^2}}\frac{{\displaystyle \partial x}}{{\displaystyle \partial s}}\frac{{\displaystyle \partial x}}{{\displaystyle \partial t}}+\frac{{\displaystyle {\partial }^{2}x}}{{\displaystyle \partial s\partial t}}\frac{{\displaystyle \partial f}}{{\displaystyle \partial x}}+\frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial y^2}}\frac{{\displaystyle \partial y}}{{\displaystyle \partial s}}\frac{{\displaystyle \partial y}}{{\displaystyle \partial t}}+\frac{{\displaystyle {\partial }^{2}y}}{{\displaystyle \partial s\partial t}}\frac{{\displaystyle \partial f}}{{\displaystyle \partial y}} \\ nu beräknar vi vad som behövs: \\ \frac{{\displaystyle \partial x}}{{\displaystyle \partial s}}=t \cos \left(s\right) , \frac{{\displaystyle \partial x}}{{\displaystyle \partial t}}=\sin \left(s\right) , \frac{{\displaystyle {\partial }^{2}x}}{{\displaystyle \partial s\partial t}}=\frac{{\displaystyle \partial }}{{\displaystyle \partial s}}\left(\frac{{\displaystyle \partial x}}{{\displaystyle \partial t}}\right)=\frac{{\displaystyle \partial }}{{\displaystyle \partial s}}(\sin s)=\cos \left(s\right) \\ \frac{{\displaystyle \partial y}}{{\displaystyle \partial s}}=-t \sin \left(s\right) , \frac{{\displaystyle \partial y}}{{\displaystyle \partial t}}=\cos \left(s\right) , \frac{{\displaystyle {\partial }^{2}y}}{{\displaystyle \partial s\partial t}}=\frac{{\displaystyle \partial }}{{\displaystyle \partial s}}\left(\frac{{\displaystyle \partial y}}{{\displaystyle \partial t}}\right)=\frac{{\displaystyle \partial }}{{\displaystyle \partial s}}(\cos s)=-\sin \left(s\right) \\ Svaret: \frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial s\partial t}}=\frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial x^2}}(t \cos s)(\sin s)+(\cos s)\frac{{\displaystyle \partial f}}{{\displaystyle \partial x}}+\frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial y^2}}(-t \sin t)(\cos s)+(-\sin s)\frac{{\displaystyle \partial f}}{{\displaystyle \partial y}}= \\ =\frac{t}{2}\sin \left(2s\right)\frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial x^2}}+(\cos s)\frac{{\displaystyle \partial f}}{{\displaystyle \partial x}}-\frac{t}{2}\sin \left(2s\right)\frac{{\displaystyle {\partial }^{2}f}}{{\displaystyle \partial y^2}}-\sin \left(s\right)\frac{{\displaystyle \partial f}}{{\displaystyle \partial y}} \end{gathered}$$