Let zFx y be a smooth function of two variables and form the

Let z=F(x, y) be a smooth function of two variables and form the X(u,v)=(u,v f(u,v)) the mean curvature is represented by Suppose that F(x, y)=g(x)th(y) show that H= 0 if and only IF

Solution

This is an immediate consequence of the equation for H.

If F(x,y) = g(x)+h(y),that is

F(u,v) = g(u)+h(v)............................................................(1)

So the numerator of the equation for H

=(1+h\'(v)²) g\'\'(u) + (1+g\'(u)²) h\'\'(v)-2. 0 (as the mixed second partial derivative of F vanishes, in view of (1))

=(1+h\'(v)²) g\'\'(u) + (1+g\'(u)²) h\'\'(v)

So H=0 iff the condition

(1+h\'(y)²) g\'\'(x) + (1+g\'(x)²) h\'\'(y)=0

is satisfied.