{"knowledge_schema": {"broad_category": "Mathematics \u2192 Vector Calculus \u2192 Surface Integrals", "refinement": "This problem involves evaluating a surface integral of a vector field over a closed surface (a sphere) in three-dimensional space.", "specific_scope": "The vector field g(r) has a specific form, with a magnitude that varies with distance from the origin and points radially outward, which affects the evaluation of the integral over the sphere.", "goal": "Calculate the surface integral of the vector field g over a sphere of radius R to determine the net flux through the surface."}, "summary": "In this problem, we are tasked with evaluating the surface integral of a vector field defined by g(r) = 1/r^3, which points radially outward, over a sphere of radius R centered at the origin. The integral represents the net flux of the vector field through the surface of the sphere. Due to the nature of the vector field and the symmetry of the sphere, the contributions to the integral from opposite points on the sphere cancel each other out, leading to a total flux of 0. Therefore, the most appropriate answer for the integral is 0."}