 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-v/3/a/i/solution

The curl is

\nabla \times F = (x, y, - 2 z)

. For the upward orientation of the paraboloid, the vector surface area element is

(2 x, 2 y, 1) d x d y

. Therefore Stokes theorem gives

\oint_{C} F \cdot d r = \iint_{x, y \geq 0, x^{2} + y^{2} \leq 4} {2 x^{2} + 2 y^{2} - 2 (4 - x^{2} - y^{2})} d x d y = \int_{0}^{π /2} \int_{0}^{2} (4 r^{2} - 8) r d r d θ = 0 .

The upward normal gives the prescribed anticlockwise boundary orientation viewed from above.

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