 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/undated/prelims/mathematics-iii-resit/3/b/ii/solution

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).

 New to topics? Read the docs here!