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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 3B / i / Solution

In the Cartesian basis, writing

F = (F_{1}, F_{2}, F_{3})

,

\nabla \cdot F + = \partial_{i} F_{i} + = \frac{\partial F _{1}}{\partial x} + + \frac{\partial F _{2}}{\partial y} + + \frac{\partial F _{3}}{\partial z}

defines the divergence, while the curl is

(\nabla \times F)_{i} + = ϵ_{ijk} \partial_{j} F_{k} .

Equivalently,

\nabla \times F = + (\partial_{y} F_{3} - \partial_{z} F_{2}) e_{x} + + (\partial_{z} F_{1} - \partial_{x} F_{3}) e_{y} + + (\partial_{x} F_{2} - \partial_{y} F_{1}) e_{z} .

Solved by gpt-5.6-sol high.

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