 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-v/2/b/iii/solution

Parametrize the straight segment by

r (t) = (t, t, t)

0 \leq t \leq 1

. Then the line integral is

\int_{C} F \cdot d r = \int_{0}^{1} (3 t^{2} + 6 t - 14 t^{2} + 20 t^{3}) d t = 3 - \frac{11}{3} + 5 = \frac{13}{3} .

The three different answers are possible because this vector field is not a conservative vector field; for example

\partial_{y} F_{x} = 6

whereas

\partial_{x} F_{y} = 0

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