多元复合函数求导类.设函数z具有连续二阶偏导数,试求常数a,使得变换u=x-2y,v=x+ay可以把方程6Zxx+Zxy-Zyy=0化简为Zuv=0.
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![多元复合函数求导类.设函数z具有连续二阶偏导数,试求常数a,使得变换u=x-2y,v=x+ay可以把方程6Zxx+Zxy-Zyy=0化简为Zuv=0.](/uploads/image/z/5350859-35-9.jpg?t=%E5%A4%9A%E5%85%83%E5%A4%8D%E5%90%88%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC%E7%B1%BB.%E8%AE%BE%E5%87%BD%E6%95%B0z%E5%85%B7%E6%9C%89%E8%BF%9E%E7%BB%AD%E4%BA%8C%E9%98%B6%E5%81%8F%E5%AF%BC%E6%95%B0%2C%E8%AF%95%E6%B1%82%E5%B8%B8%E6%95%B0a%2C%E4%BD%BF%E5%BE%97%E5%8F%98%E6%8D%A2u%3Dx-2y%2Cv%3Dx%2Bay%E5%8F%AF%E4%BB%A5%E6%8A%8A%E6%96%B9%E7%A8%8B6Zxx%2BZxy-Zyy%3D0%E5%8C%96%E7%AE%80%E4%B8%BAZuv%3D0.)
多元复合函数求导类.设函数z具有连续二阶偏导数,试求常数a,使得变换u=x-2y,v=x+ay可以把方程6Zxx+Zxy-Zyy=0化简为Zuv=0.
多元复合函数求导类.
设函数z具有连续二阶偏导数,试求常数a,使得变换u=x-2y,v=x+ay可以把方程
6Zxx+Zxy-Zyy=0化简为Zuv=0.
多元复合函数求导类.设函数z具有连续二阶偏导数,试求常数a,使得变换u=x-2y,v=x+ay可以把方程6Zxx+Zxy-Zyy=0化简为Zuv=0.
你确定是化简为Zuv=0吗?我只能得到某个a,化简为Zuu=0
Zx=Zu*Ux+Zv*Vx
Zxx=(Zu*Ux+Zv*Vx)x=(Zu+Zv)x=(Zu)x+(Zv)x=Zuu*Ux+Zuv*Vx+Zvu*Ux+Zvv*Vx
=Zuu+Zuv+Zvu+Zvv
Zxy=(Zu*Ux+Zv*Vx)y=(Zu+Zv)y=(Zu)y+(Zv)y=Zuu*Uy+Zuv*Vy+Zvu*Uy+Zvv*Vy =-2Zuu+aZuv-2Zvu+aZvv
Zy=Zu*Uy+Zv*Vy
Zyy=(Zu*Uy+Zv*Vy)y=(-2Zu+aZv)y=-2(Zu)y+a(Zv)y=-2(Zuu*Uy+Zuv*Vy)+a(Zvu*Uy+Zvv*Vy)=-2(-2Zuu+aZuv)+a(-2Zvu+aZvv)=4Zuu-2aZuv-2aZvu+a^2Zvv
6Zxx+Zxy-Zyy
=6(Zuu+Zuv+Zvu+Zvv)+(2Zuu+aZuv-2Zvu+aZvv)-(4Zuu-2aZuv-2aZvu+a^2Zvv)
=4Zuu+(6+3a)Zuv+(4+2a)Zvu+(6+a-a^2)Zvv=0
6+3a=0 =>a=-2
4+2a=0 =>a=-2
6+a-a^2=0 => a=-2 或 a=3
故a=-2时,6Zxx+Zxy-Zyy=0化简为4Zuu=0,即Zuu=0