 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/13/c/solution

Differentiate the components directly:

(\nabla \times F)_{x} (\nabla \times F)_{y} (\nabla \times F)_{z} = \partial_{y} (- 3 x z^{2} - 1) - \partial_{z} (x^{2} - 2 y) = 0, = \partial_{z} (2 x y - z^{3}) - \partial_{x} (- 3 x z^{2} - 1) = - 3 z^{2} + 3 z^{2} = 0, = \partial_{x} (x^{2} - 2 y) - \partial_{y} (2 x y - z^{3}) = 2 x - 2 x = 0.

Thus

\nabla \times F = 0

. The field is smooth on all of

R^{3}

, a simply connected domain, so the zero-curl condition is sufficient for a conservative vector field. The domain condition matters: zero curl alone need not imply a global potential on a region with holes.

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