 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/13/a/solution

Write

a = (a_{x}, a_{y}, a_{z})

v_{1} = (a_{y} z - a_{z} y, a_{z} x - a_{x} z, a_{x} y - a_{y} x)

\partial_{x} (2 x y) = 2 y

\partial_{y} (x^{2} + 3 y^{2} + z^{2}) = 6 y

, and

\partial_{z} (2 yz) = 2 y

. Thus

\nabla \cdot v_{1} = 0, \nabla \cdot v_{2} = 10 y .

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