多元复合函数求导问题δ代表偏导号,2是上角标的意思,求详解设函数z=(u,v)具有二阶连续偏导数,变换u=x-2y,v=x+ay可把6*δ2z/δx2+δ2z/δxδy-δ2z/δy2=0简化为δ2z/δuδv=0求a的值

Question

多元复合函数求导问题δ代表偏导号,2是上角标的意思,求详解
设函数z=(u,v)具有二阶连续偏导数,变换u=x-2y,v=x+ay可把6*δ2z/δx2+δ2z/δxδy-δ2z/δy2=0简化为δ2z/δuδv=0求a的值

Answer 1

你那偏导号δ太难打了……我就用d代替了哦.
由变换,du/dx=1,du/dy=-2,dv/dx=1,dv/dy=a.
因此得：
dz/dx=(dz/du)*(du/dx)+(dz/dv)*(dv/dx)=(dz/du)+(dz/dv)
dz/dy=(dz/du)*(du/dy)+(dz/dv)*(dv/dy)=-2*(dz/du)+a*(dz/dv)
再求二阶导.注意函数z具有二阶连续偏导,因此求二阶偏导顺序可交换.
d2z/dx2=(d2z/du2)*(du/dx)+(d2z/dudv)*(dv/dx)+(d2z/dudv)*(du/dx)+(d2z/dv2)*(dv/dx)
=(d2z/du2)+(d2z/dudv)+(d2z/dudv)+(d2z/dv2)
=(d2z/du2)+2*(d2z/dudv)+(d2z/dv2)
d2z/dy2=-2*(d2z/du2)*(du/dy)-2*(d2z/dudv)*(dv/dy)+a*(d2z/dudv)*(du/dy)+a*(d2z/dv2)*(dv/dy)
=4*(d2z/du2)-2a*(d2z/dudv)-2a*(d2z/dudv)+a^2*(d2z/dv2)
=4*(d2z/du2)-4a*(d2z/dudv)+a^2*(d2z/dv2)
求d2z/dxdy时,用dz/dx对y求偏导或者用dz/dy对x求偏导都一样.下面用第一种.
d2z/dxdy=(d2z/du2)*(du/dy)+(d2z/dudv)*(dv/dy)+(d2z/dudv)*(du/dy)+(d2z/dv2)*(dv/dy)
=-2*(d2z/du2)+a*(d2z/dudv)-2(d2z/dudv)+a*(d2z/dv2)
=-2*(d2z/du2)+(a-2)*(d2z/dudv)+a*(d2z/dv2)
代入要化简的式子并整理,得：
6*d2z/dx2+d2z/dxdy-d2z/dy2
=(6-2-4)*(d2z/du2)+(12+a-2+4a)*(d2z/dudv)+(6+a-a^2)*(d2z/dv2)
=(10+5a)*(d2z/dudv)+(6+a-a^2)*(d2z/dv2)
=0
和d2z/dudv=0对比发现,要求6+a-a^2=0
即a=-2或3