 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/19/a/solution

\nabla \cdot F = \partial_{x} (y^{2}) + \partial_{y} (e^{x z}) + \partial_{z} (e^{x y}) = 0.

For any closed piecewise smooth surface bounding a volume, use the divergence theorem with the outward orientation:

\int_{S} F \cdot d S = \int_{V} \nabla \cdot F d V = 0.

Reversing the orientation still gives zero. The field is smooth everywhere, so there is no hidden singular source inside the surface.

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