Gravitoelectromagnetic Field Equations II: A Vector Calculus Approach

We develop the first two equations of a gravitoelectromagnetic framework constructed by analogy with Maxwell’sequations, using vector calculus as the primary language. The first equation is a static Gauss’s Law for Gravity,expressed via the divergence of a gravitational field vector defined over a gradient-based position vector, derivedand verified both inside and outside a uniform-density spherical body. The second equation is a curvature-modifiedMaxwell–Faraday analogue in which the Ricci scalar R acts as a curvature weight onthe gravitoelectric field. In the flat spacetime limit, the equation reduces to a magnetostatic constraint, consistent withthe absence of gravitational sources. All derivations are verified symbolically using SymPy.