I am reading Calculus of Variations by Gelfand and Fomin. On page 24 there is an example where they derive Euler's equation for a functional $$ J[z]=\int\int_{R} \sqrt{1 + z_{x}^{2}+z_{y}^{2}}\ dx dy. $$ They say that Euler's equation has the form $$ r(1+q^{2})-2spq+t(1+p^{2}), $$ where $$ p=z_{x}, q=z_{y}, r=z_{xx}, s=z_{xy}, t=z_{yy}. $$ My problem is that I cannot derive that!
Euler's equation in this case is form $$ F_{z}-\frac{\partial}{\partial x}F_{z_{x}}-\frac{\partial}{\partial y}F_{z_{y}}. $$
I get $F_{z}=0, F_{z_{x}}=z_{x}+z_{x}^{3}+z_{x}z_{y}^{2}$ and $F_{z_{y}}=z_{y}+z_{y}z_{x}^{2}+z_{y}^{3}$. After I take the derivatives from last two of those I get something that is not the form stated in the begin.
What do I miss? I do not have errata, so there might be a typo in the book, but I strongly think that I have made a mistake.
I am not really sure how you got those values for $ F_{z_x} $ and $ F_{z_y} $ so here is how I got the result of the book. $$ F(x,y,z,z_x,z_y)= \sqrt{1+z_x^2 + z_y ^2} $$
So as you correctly stated we can easily see $ F_z = 0 $ Let's calculate $ F_{z_x} $, we must consider only the derivative with respect to $z_x$ meaning we should treat $z_y$ as constant. $$F_{z_x} = \displaystyle \frac{z_x}{\sqrt{1+z_x^2+ z_y^2}} $$
So if we are now to proceed to calculate $ \frac{\partial}{\partial x} F_{z_x} $ we should use the quotient formula for derivation. So we get: $$ \frac{\partial}{\partial x} F_{z_x} = \displaystyle \frac{\sqrt{1+z_x^2+z_y^2}(z_{xx})-z_x \left[\frac{z_xz_{xx}+z_yz_{xy}}{\sqrt{1+z_x^2+z_y^2}}\right]}{1+z_x^2+z_y^2}$$
(Trick) We can also get the derivative $ \frac{\partial}{\partial y} F_{z_y} $ :
$$ \frac{\partial}{\partial y} F_{z_y} = \displaystyle \frac{\sqrt{1+z_x^2+z_y^2}(z_{yy})-z_y \left[\frac{z_yz_{yy}+z_xz_{xy}}{\sqrt{1+z_x^2+z_y^2}}\right]}{1+z_x^2+z_y^2}$$
Now we are interested in $ \frac{\partial}{\partial x} F_{z_x} + \frac{\partial}{\partial y} F_{z_y} = 0 $ so we can get rid of the denominators (as long as we first make the common) and we get the equation,
$$ (1+z_{x}^2+z_y^2) (z_{xx}) - z_x^2 z_{xx} - z_xz_yz_{xy} + (1+z_{x}^2+z_y^2)(z_{yy})-z_y^2z_{yy}-z_yz_xz_{xy} = 0 $$
This is actually the required formula (we just have to make the substitution the book makes and simplify).
So $p = z_x, q=z_y, r=z_{xx}, s=z_{xy}, t=z_{yy} $ using this in the formula above we get
$$ (1+p^2+q^2)(r)-p^2r-pqs + (1+p^2+q^2)(t)-q^2t-pqs = 0 $$ So finally we achieve the result:
$$ r(1+q^2) -2pqs + t(1+p^2) = 0 $$
Hope this helps.