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 Incoming links: Path dependence

Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 3 / b / ii / Solution 2026-10-25

Evaluate the line integral separately on the three oriented line segments. Their parametrizations and contributions are

r (t), 0 \leq t \leq 1 (t, 0, 0) (1, t, 0) (1, 1, t) F (r (t)) \cdot r^{'} (t) 00 t contribution 00 1/2

Thus

\int_{C} F \cdot d r = \frac{1}{2} .

The different value in i demonstrates path dependence. The vector field in this part is different from the conservative vector field in part (a).

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