{"knowledge_schema": {"broad_category": "Mathematics \u2192 Vector Calculus \u2192 Divergence Theorem", "refinement": "This problem involves calculating the volume integral of the divergence of a vector field within a specified geometric region (a sphere).", "specific_scope": "The focus is on a radial vector field defined in spherical coordinates, where the divergence needs to be computed and integrated over a sphere of radius R.", "goal": "To find the volume integral of the divergence of the vector field g(r) within the sphere and determine the correct value from the given options."}, "summary": "This problem requires applying the divergence theorem to calculate the volume integral of the divergence of a radial vector field g(r) = 1/r^2 within a sphere of radius R. The divergence theorem relates the volume integral of the divergence of a vector field to a surface integral over the boundary of the volume. In this case, the result of the integral is found to be 4\u03c0, which corresponds to the total flux of the vector field through the surface of the sphere."}