 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2015-resit/6/b/solution

Read the given vector field as the three-dimensional field

F = (x, y, 0)

. Its divergence is the sum of the three matching-coordinate partial derivatives:

\nabla \cdot F = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial 0}{\partial z} = 2 .

The absence of a

z

component does not make the three-dimensional flux vanish: the field still flows outward through surface normals with horizontal components.

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