Question

高等数学偏导数问题

Answer 1

根据题意，du/dx=1 du/dy=a dv/dx=1 dv/dy=3
所以dz/dx=dz/du*du/dx+dz/dv*dv/dx=dz/du+dz/dv
dz/dy=dz/du*du/dy+dz/dv*dv/dy=a*dz/du+3*dz/dv
所以d^2z/dx^2=d(dz/du+dz/dv)/dx
=d(dz/du)/du*du/dx+d(dz/du)/dv*dv/dx+d(dz/dv)/du*du/dx+d(dz/dv)/dv*dv/dx
=d^2z/du^2+2*d^2z/dudv+d^2z/dv^2
d^2z/dy^2=d(a*dz/du+3*dz/dv)/dy
=a*d(dz/du)/du*du/dy+a*d(dz/du)/dv*dv/dy+3*d(dz/dv)/du*du/dy+3*d(dz/dv)/dv*dv/dy
=a^2*d^2z/du^2+6a*d^2z/dudv+9*d^2z/dv^2
d^z/dxdy=d(dz/du+dz/dv)/dy
=d(dz/du)/du*du/dy+d(dz/du)/dv*dv/dy+d(dz/dv)/du*du/dy+d(dz/dv)/dv*dv/dy
=a*d^2z/du^2+(a+3)*d^2z/dudv+3*d^2z/dv^2
所以6*d^2z/dx^2+d^2z/dxdy-d^2z/dy^2=0
6*d^2z/du^2+12*d^2z/dudv+6d^2z/dv^2+a*d^2z/du^2+(a+3)*d^2z/dudv+3*d^2z/dv^2-a^2*d^2z/du^2-6a*d^2z/dudv-9*d^2z/dv^2=0
(6+a-a^2)*d^2z/du^2+(15-5a)*d^2z/dudv=0
(a-3)(a+2)*d^2z/du^2+(a-3)*d^2z/dudv=0
(a+2)*d^2z/du^2+d^2z/dudv=0
所以当a=-2时，d^2z/dudv=0