Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics IV / 3 / a / i

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- Solution i

Solution

The ellipsoid is the level set

F (x, y, z) = 2 x^{2} + y^{2} + z^{2} = 1

. Its gradient is

(4 x, 2 y, 2 z)

, which never vanishes on the ellipsoid. Half this gradient gives the outward normal vector

n (θ, ϕ) = (2 cos θ, sin θ cos ϕ, sin θ sin ϕ) .

This normal vector remains valid at

θ = 0, π

, where the given parametrization is singular.

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