g⃗=−GMr2r⃗rmodified g with right arrow above equals negative the fraction with numerator cap G cap M and denominator r squared end-fraction the fraction with numerator modified r with right arrow above and denominator r end-fraction The flux of this field through a surface is directly proportional to the solid angle subtended by . Specifically, for a point mass at the origin, the flux through
that enters the volume must also leave it. The "entry" and "exit" patches of the surface subtend the same solid angle but have opposite flux signs (due to the orientation of the normal vector). : The net solid angle (and thus net flux) is 4. Physical Implications Ángulo sólido – Arnold 2.2.3
: The "Solid Angle" method serves as a bridge between the physical "force at a distance" and the geometric properties of space (specifically, exterior calculus and differential forms later in the book). ✅ Summary The flux of a central : The net solid angle (and thus net flux) is 4
times the enclosed mass if the source is inside, and zero if the source is outside. force) where the potential is related to the
force) where the potential is related to the surface area of a unit sphere "covered" by an object when viewed from a point. The solid angle Ωcap omega subtended by a surface at a point is defined as the area of the projection of onto the unit sphere centered at Mathematically, for a small surface element at a distance , the differential solid angle