I am reading a book on calculus of variation, and it talks about minimal surface, it gives the equation of minimal surface of $z(x,y)$, it should satisfy:
$$(1 + z_y^2)z_{xx} - 2 z_x z_y z_{xy} + (1+z_x^2)z_{yy} = 0$$
and then says the saddle surface:
$$z = x^3 - 2xy^2$$
can easily verify that this equation satisfy the above equation.
But I cann't verify it, here's my calculation process:
$$z_x = 3x^2 - 2y^2 $$
$$z_y = -4xy $$
$$z_{xx} = 6x$$
$$z_{yy} = -4x$$
$$z_{xy} = -4y $$
$$z_{yx} = -4y$$
Then plug into the above equation:
$$(1 + z_y^2)z_{xx} - 2 z_x z_y z_{xy} + (1+z_x^2)z_{yy} $$
$$ = \big( 1 + (-4xy)^2 \big) 6x - 2 (3x^2 - 2y^2)( -4xy)(-4y) + \big( 1 + (3x^2 - 2y^2)^2 \big) (-4x) $$
$$ = \big( 1 + 16x^2y^2 \big) 6x - 2 (3x^2 - 2y^2)(4xy)(4y) - \big( 1 + (3x^2 - 2y^2)^2 \big) 4x$$
$$= 6x + 96x^3y^2 - 32xy^2(3x^2 - 2y^2) - 4x \big( 1 + 9x^4 + 4y^4 - 12x^2y^2 \big) $$
$$ = 6x + 96x^3y^2 - 96x^3y^2 + 64xy^4 - 4x - 36x^5 - 16xy^4 + 48 x^3y^2 $$
$$ = 2x + 64xy^4 - 36x^5 - 16xy^4 + 48 x^3y^2 $$
$$ = -36x^5 + 48 x^3y^2 + 48xy^4 + 2x $$
I can't tell that $-36x^5 + 48 x^3y^2 + 48xy^4 + 2x = 0$, is there any wrong with the calculation?
=========== update, thanks for the comments.
it turns out the book have a typo on this equation, it should be
$$z = x^3 - 3xy^2$$
and it is called Monkey saddle. And this is still not a minimal surface.