 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/16/b/solution

Assume the components of

F

have continuous second partial derivatives. Expanding the divergence of its curl gives

\nabla \cdot (\nabla \times F) = \partial_{x} \partial_{y} F_{z} - \partial_{x} \partial_{z} F_{y} + \partial_{y} \partial_{z} F_{x} - \partial_{y} \partial_{x} F_{z} + \partial_{z} \partial_{x} F_{y} - \partial_{z} \partial_{y} F_{x} .

Each positive term cancels a negative term because the corresponding mixed derivatives commute. Consequently

\nabla \cdot G = \nabla \cdot (\nabla \times F) = 0 .

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