field through a surface is proportional to the solid angle it subtends. For a closed surface, the total flux is
Arnold uses the solid angle to prove qualitatively: Point Inside : If is inside a closed surface , the surface surrounds entirely. The total solid angle subtended by is the full surface area of the unit sphere, which is Result : Point Outside : If is outside , any ray from
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
This write-up covers section ("Solid Angle") from V.I. Arnold’s Mathematical Methods of Classical Mechanics . In this section, Arnold provides a geometric interpretation of Newton's potential using the concept of solid angle, leading to a simplified understanding of Gauss's Theorem . Problem Context
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
dΩ=dS⋅cos(θ)r2=r⃗⋅n⃗dSr3d cap omega equals the fraction with numerator d cap S center dot cosine open paren theta close paren and denominator r squared end-fraction equals the fraction with numerator modified r with right arrow above center dot modified n with right arrow above space d cap S and denominator r cubed end-fraction is the angle between the normal n⃗modified n with right arrow above and the radius vector r⃗modified r with right arrow above Arnold demonstrates that the gravitational acceleration g⃗modified g with right arrow above produced by a mass (or charge) at point
: 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