Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 115 / 2 / c

Table of contents
- Solution c

Solution

On the Euclidean unit ball

D^{n}

,

ω_{X} = d x^{1} \land \dots \land d x^{n} .

The outward unit normal along

S^{n - 1}

is the radial vector field

N = \sum_{i} x^{i} \partial_{i}

, so part b gives

ω_{\partial X} = \sum_{i = 1}^{n} (- 1)^{i - 1} x^{i} d x^{1} \land \dots \land d x^{i} \land \dots \land d x^{n} .

Let

R = \sum_{i} x^{i} \partial_{i}

on the ball and put

β = ι_{R} ω_{X}

. Direct use of the exterior derivative gives

d β = n ω_{X}

. Therefore the Generalized Stokes theorem yields

\int_{\partial X} ω_{\partial X} = \int_{\partial X} F^{*} β = \int_{X} d β = n \int_{X} ω_{X} .

This proves the volume of a Euclidean unit sphere formula.

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