{"knowledge_schema": {"broad_category": "Physics \u2192 Vector Calculus \u2192 Divergence Theorem", "refinement": "This problem involves evaluating the volume integral of the divergence of a vector field using the Divergence Theorem, which relates the volume integral of the divergence of a vector field to a surface integral over the boundary of the volume.", "specific_scope": "The vector field f(r) is given as f(r) = 1/r^2 in spherical coordinates, and we need to evaluate the integral of its divergence over a sphere of radius R centered at the origin.", "goal": "Calculate the volume integral of the divergence of the vector field f(r) within the specified spherical volume."}, "summary": "To solve this problem, we apply the Divergence Theorem, which states that the volume integral of the divergence of a vector field over a volume is equal to the surface integral of the field over the boundary of that volume. Given the vector field f(r) = 1/r^2, we first compute its divergence, which in this case results in a constant value. Evaluating the surface integral over the sphere of radius R leads to the conclusion that the volume integral of the divergence of the field inside the sphere is equal to 4\u03c0. Therefore, the most appropriate answer is 1. 4 \u03c0."}