Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 13 / d

The original PDF has angle

θ (t) = 2 π t^{n}

, with

n

t

. It is not

2 πn t

: each permitted value of

n

changes the speed but traverses the same circle once counterclockwise. Along the path,

r^{2} = 2

and

r^{'} (t) = θ^{'} (t) (- sin θ, cos θ, 0), θ^{'} = 2 πn t^{n - 1} .

The radial contribution to the line integral vanishes, since

(cos θ, sin θ, 1) \cdot (- sin θ, cos θ, 0) = 0

. The rotational contribution gives

F (r (t)) \cdot r^{'} (t) = a θ^{'} (t) .

Therefore

\int_{Γ} F \cdot d r = a [θ (1) - θ (0)] = 2 πa, n = 1, 2, \dots .

The answer is independent of the monotone parametrization, as an oriented line integral should be.

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