 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2/7/a/i/solution

H (x, y, z) = x^{3} + x y + yz - 1

\nabla H = (3 x^{2} + y, x + z, y)

y \geq 1/2

on the given patch, division by its last component is valid and gives the unique normalized field

u = \frac{\nabla H}{y} = (\frac{3 x ^{2} + y}{y}, \frac{x + z}{y}, 1) .

k

is one, as required; it is not generally a unit vector.

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