 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2015/12/a/solution

Compute all three components of the Cartesian curl explicitly:

(\nabla \times F)_{x} (\nabla \times F)_{y} (\nabla \times F)_{z} = \partial_{y} (4 x yz) - \partial_{z} (2 x z^{2}) = 4 x z - 4 x z = 0, = \partial_{z} (2 y z^{2}) - \partial_{x} (4 x yz) = 4 yz - 4 yz = 0, = \partial_{x} (2 x z^{2}) - \partial_{y} (2 y z^{2}) = 2 z^{2} - 2 z^{2} = 0.

Thus

\nabla \times F = 0

throughout

R^{3}

. There are no excluded singular points in this polynomial field.

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