 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/20/e/solution

Take a piecewise linear path along three coordinate directions, passing through the origin and then

(0, 1, 0)

x (t) = ⎩ ⎨ ⎧ (1 - t, 0, 0), (0, t - 1, 0), (0, 1, t - 2), 0 \leq t \leq 1, 1 \leq t \leq 2, 2 \leq t \leq 3.

On the first segment

G_{x} = y^{2} + z^{2} = 0

, so its line integral is zero. On the second

G_{y} = x^{2} + z^{2} = 0

, so its contribution is also zero. On the last segment

G_{z} = x^{2} + y^{2} = 1

and

z

increases from zero to one. Therefore

\int G \cdot d x = 0 + 0 + \int_{0}^{1} 1 d z = 1 \neq = \frac{2}{3} .

This gives the requested concrete path dependence with the same ordered endpoints.

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