 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-2/16/solution

F_{1} = (z - y, x, - x)

, so direct curl calculation gives

\nabla \times F_{1} = (0, 2, 2) = 2 a .

It is not conservative because its curl is nonzero. The second is a radial gradient field:

F_{2} = \nabla [- \frac{1}{1 + r ^{2}}], \nabla \times F_{2} = 0 .

The scalar expression is a smooth function and single-valued on all of space, so

F_{2}

Φ = - (1 + r^{2})^{- 1/2}

. Parts (a) and (b) check path dependence directly.

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