MediumChange of variables

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §14.4 Cylindrical and Spherical Coordinates, #30 · Multivariable Calculus · Change of variables

Show as in Problem 29 that the average distance is …, from…

Problems, on the attraction of a sphere, use Figure 14.18 and the law of cosines $q^{2} = D^{2} - 2 ρ D cos ϕ + ρ^{2} = u$ .

Show as in Problem 29 that the average distance is $\overset{q}{ˉ} = D + \frac{1}{3} ρ^{2} / D$ , from the outside point $(0, 0, D)$ to points on the shell of radius $ρ$ . Then integrate $∭ q d V$ and divide by $4 π R^{3} /3$ to find $\overset{q}{ˉ}$ for the solid sphere.

This exercise builds on another exercise in its source book, which isn't shown here.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §14.4 Cylindrical and Spherical Coordinates, #30

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