 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/16/b/solution

The first field has vanishing curl and is the gradient of a monomial:

\nabla \times F_{1} = 0, ϕ_{1} = x^{3} y^{2} z + C, F_{1} = \nabla ϕ_{1} .

For the second field, direct component differentiation gives

\nabla \times F_{2} = (- 2 x^{3} y, 3 x^{2} z (2 y - z), 3 x^{2} z (2 y - z)) .

It is not identically zero. Thus $F_{1}$ is conservative and $F_{2}$ is not on

R^{3}

. The conditional path-integral requests therefore apply to

F_{1}

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