 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/13/c/solution

A continuously differentiable conservative vector field has zero curl. Here

\nabla \times F = (\frac{\partial F _{z}}{\partial y} - \frac{\partial F _{y}}{\partial z}, \frac{\partial F _{x}}{\partial z} - \frac{\partial F _{z}}{\partial x}, \frac{\partial F _{y}}{\partial x} - \frac{\partial F _{x}}{\partial y}) = (β (2 y - 1), 0, 0) .

This vanishes everywhere if and only if

β = 0

. Since the field is smooth on the simply connected domain

R^{3}

, zero curl is also sufficient; the explicit scalar potential in the next part confirms it.

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