Question

设G（x+z*y^（-1），y+z*x^（-1））=0确定了z=f（x，y）证明：x*z对x的偏导数+y*z对y的偏导数=z-xy

Answer 1

G[x + z*y^(- 1)，y + z*x^(- 1)] = 0
证明x*∂z/∂x + y*∂z/∂y = z - xy？

Gz = (1/y)G1 + (1/x)G2 = L
Gx = G1 - (z/x²)G2
Gy = (- z/y²)G1 + G2
∂z/∂x = - Gx/Gz = [- G1 + (z/x²)G2]/L
∂z/∂y = - Gy/Gz = [(z/y²)G1 - G2]/L

x*∂z/∂x + y*∂z/∂y = [- xG1 + (z/x)G2 + (z/y)G1 - yG2]/L
= [(z - xy)*(1/y)G1 + (z - xy)*(1/x)G2]/L
= [(z - xy)L]/L
= z - xy