Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 3 / a / i

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- Solution i

Solution

For the given wrapped hyperbolic cosine curve,

x^{2} + y^{2} = cos^{2} t + sin^{2} t = 1, z = cosh t \geq 1 > 0,

so

C \subset S

. The curve makes one full turn around the circular cylinder, descending from

(- 1, 0, cosh π)

to

(1, 0, 1)

and ascending to its initial point. Its two parameter endpoints coincide; their tangent vectors

(0, - 1, - sinh π)

and

(0, - 1, sinh π)

do not agree, so the resulting closed curve has a corner there.

Figure 1.
The hyperbolic cosine curve wrapped around a unit circular cylinder, and the same surface unrolled
. The shaded finite patch extends from $z = 0$ to $z = cosh t$ . On the unrolled cylinder, its horizontal coordinate is $t \in [- π, π]$ ; the two vertical edges are identified. Details.

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