Rozwiązanie:
Funkcja:
[tex]g(r,s,t)=\sin (rst) \ln(r^{2}+t^{2})[/tex]
Mamy:
[tex]$\frac{\partial }{\partial r} g(r,s,t)=st \cos(rst) \ln (r^{2}+t^{2})+ \frac{2r\sin (rst) }{r^{2}+t^{2}} [/tex]
[tex]$\frac{\partial }{\partial s} g(r,s,t)=\ln(r^{2}+t^{2}) \cdot \frac{\partial g}{\partial s} \sin (rst)=rt \cos(rst) \ln(r^{2}+t^{2})[/tex]
[tex]$\frac{\partial ^{2}}{\partial s \partial r} g(r,s,t)=\frac{\partial g}{\partial s} \Big(\frac{\partial g}{\partial r} \Big)=\frac{\partial g}{\partial s} \Big(st \cos(rst) \ln (r^{2}+t^{2})+ \frac{2r\sin (rst) }{r^{2}+t^{2}} \Big)=[/tex]
[tex]$=\frac{\partial g}{\partial s} \Big(st \cos(rst) \ln (r^{2}+t^{2})\Big)+\frac{\partial g}{\partial s} \Big(\frac{2r\sin (rst) }{r^{2}+t^{2}} \Big)=[/tex]
[tex]$=t \ln(r^{2}+t^{2}) \cdot \frac{\partial g}{\partial s}\Big(s \cos(rst)\Big) +\frac{2r^{2}t \cos(rst)}{r^{2}+t^{2}} =[/tex]
[tex]$=t \ln(r^{2}+t^{2})\Big(\cos(rst)-rst\sin (rst)\Big)+\frac{2r^{2}t \cos(rst)}{r^{2}+t^{2}} [/tex]
