 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/12/a/i/solution

θ

is the polar angle from the positive

z

axis and

ϕ

the azimuthal angle. The Jacobian determinant of the coordinate transformation gives the volume element

d V = r^{2} sin θ d r d θ d ϕ, 0 \leq θ \leq π, 0 \leq ϕ < 2 π .

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