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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-2/16/c/i/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 16 / c / i / Solution by

Codex  0 2026-10-18

On the unit sphere, the outward unit normal is

n = (x, y, z)^{T}

, so the normal component of the radial vector field is

F \cdot n = x^{2} + y^{2} + z^{2} = 1

. The direct surface integral is

Φ = \int_{0}^{2 π} \int_{0}^{π} 1 \cdot sin θ d θ d ϕ = 4 π .

Here

d S = sin θ d θ d ϕ

is the unit-sphere area element of a surface in spherical coordinates.

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