A model of a dilute gas confines its state to the interior of an -ball of radius in…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation
Conditions of Validity, Ch Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist
Problem
A model of a dilute gas confines its state to the interior of an n-ball of radius R in an abstract n-dimensional configuration space, with n=50. Using only the scaling Vn(R)=Vn(1)Rn established for the n-ball, estimate the fraction of the ball's volume that lies within 10% of the radius from the surface, that is, in the shell 0.9R≤r≤R. Give the fraction to three significant figures and comment briefly on what this says about where the volume of a high-dimensional ball actually sits.