 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/7/ii/solution

\nabla v = (- u_{y}, u_{x})

. Therefore

\nabla u \cdot \nabla v = - u_{x} u_{y} + u_{y} u_{x} = 0.

The gradient is normal to a regular level set, so these curves intersect orthogonally wherever their gradients are nonzero. At a stationary point, where both gradients vanish, a regular curve tangent need not exist; the algebraic identity remains valid.

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