 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-2/8/solution

In Cartesian coordinates, calculate the divergence by differentiating each component with respect to its matching coordinate:

\nabla \cdot F = 2 x + x + 6 z = 3 x + 6 z .

For the curl, take the alternating cross-derivatives:

\nabla \times F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x}) = (0, 2 x, 3 y) .

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