Гѓngulo Sгіlido Вђ“ Arnold 2.2.3 -

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

Arnold explores the property of the gravitational field (or any

: This geometric approach explains why a hollow spherical shell exerts no gravitational force on a particle inside it: the solid angles subtended by opposite parts of the shell cancel out exactly because the force falls off as while the surface area grows as r2r squared Ángulo sólido – Arnold 2.2.3

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 Arnold uses the solid angle to prove qualitatively:

times the enclosed mass if the source is inside, and zero if the source is outside.

field through a surface is proportional to the solid angle it subtends. For a closed surface, the total flux is The "entry" and "exit" patches of the surface

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