Question

设函数z=f(x,y)由方程x²+y²+z²-6z=0确定,求∂²z/∂x∂y

Answer 1

x^2+y^2+z^2-6z=0
2x + 2z∂z/∂x -6∂z/∂x = 0
∂z/∂x = x/(3-z)
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x^2+y^2+z^2-6z=0
2y + 2z∂z/∂y -6∂z/∂y = 0
y + z∂z/∂y -3∂z/∂y = 0
(3-z)∂z/∂y =y
∂z/∂y =y/(3-z)
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y + z∂z/∂y -3∂z/∂y = 0
z.∂²z/∂x∂y+ ∂z/∂x .∂z/∂y - 3∂²z/∂x∂y = 0
(3-z)∂²z/∂x∂y = ∂z/∂x .∂z/∂y
= xy/(3-z)^2
∂²z/∂x∂y =xy/(3-z)