 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/9/solution

The gradient components are

u_{x} = x^{2} (3 - 2 x^{2}) e^{- x^{2} - y^{2}}, u_{y} = - 2 y x^{3} e^{- x^{2} - y^{2}} .

Away from the vertical axis,

x \neq = 0

, so

u_{y} = 0

y = 0

. The other stationary point equation then gives

x^{2} = 3/2

. Therefore the two requested points are

(\frac{3}{2}, 0), (- \frac{3}{2}, 0) .

Every point with

x = 0

also has zero gradient, but those points are explicitly excluded.

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