I've always disliked the notation because used that unthinkingly results in doing two square roots which, just as you see here, cancel!
Instead think of as the "vector differential of surface area" and calculate it directly. We can write the surface in a vector equation with x and y as parameter: . The derivatives with respect to x and y give two tangent vectors, and . The cross product of those gives the "vector differential of surface area": where I have chosen the order of multiplication to get a negative value for , "oriented downward".
You are correct that so that [tex]= (0(2x)+ 0(2y)- 1(-1))dxdy= dxdy.
That gives , not .
Your normal vector, was oriented upward, not downward.