 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016-resit/6/solution

The field is the gradient of a globally defined polynomial:

ϕ (x, y, z) = 2 x y + y^{2} z, \nabla ϕ = (2 y, 2 x + 2 yz, y^{2}) = v .

R^{3}

\oint_{C} v \cdot d l = ϕ (C_{end}) - ϕ (C_{start}) = 0

for every closed piecewise smooth loop. Constructing the global vector-field potential avoids relying on a zero-curl test without checking the topology of the domain.

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