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For coordinates $x=x(q)$, the Jacobian is $J_{ij}=\partial x_i/\partial q_j$. Expanding after separating $x_1=r\cos {\theta }_1$ from the remaining coordinates givesConsequentlyand on $r=R$ the surface element is obtained by omitting $dr$ and replacing $r$ by $R$.

Since $Y={\int }_0^{\infty }{1}_{\{Y\ge y\}}dy$, Tonelli's theorem givesFor the integer-valued minimum,Markov's inequality now gives