Final answer:
To evaluate the surface integral of G(r) over the surface S, we set up a double integral in Cartesian coordinates or convert to polar coordinates, considering the surface and bounds given, and then integrate G(r) over the area element dA.
Step-by-step explanation:
To evaluate the surface integral of function G(r) over the surface S where G = arctan(y/x) and S is defined by z = x2 + y2 for 0 < z < 9, and 0 < x,y, we need to express dA in terms of dx and dy, and then integrate the function over the relevant surface. Considering the bounds given, we need to set up the double integral in Cartesian coordinates and evaluate it. This calculation involves converting the given G(r) and the surface equation to the appropriate form for the surface integral.
For practical computation of surface integrals, we generally convert to polar or cylindrical coordinates due to the symmetry of the surface involved, especially when integrating over surfaces such as the given paraboloid. For the given surface and function, we would likely switch to polar coordinates, find the limits of integration over the radius and angle, and then integrate G(r) over the surface area element dA. The bounds for the integral would be derived from the conditions 0 < z < 9 and 0 < x,y.