设函数z=f(sinx,xy),其中 具有二阶连续偏导数,求ε^2z/εxεy

题目

设函数z=f(sinx,xy),其中 具有二阶连续偏导数,求ε^2z/εxεy

答案

设u=sinx,v=xy
dz/dx=dz/du*du/dx+dz/dv*dv/dx=cosxf1'+yf2'
d^2z/dxdy=d(dz/dx)/dy=(-sinx)f1'+cosx*df1'/dx+y*df2'/dx
=-sinxf1'+cosx(df1'/du*du/dx+df1'/dv*dv/dx)+y(df2'/du*du/dx+df2'/dv*dv/dx)
=-sinxf1'+cosx(cosxf11''+yf12'')+y(cosxf21''+yf22'')
=-sinxf1'+(cosx)^2f11''+(y+ycosx)f12''+y^2f22''
f1',f11'',f12'',f22''分别指df/du,d^2f/du^2,d^2f/dudv,d^2f/dv^2