Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 3 / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

A parametrization of the straight line segment is

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

,

0 \leq t \leq 1

. Then

F (r (t)) = (t, t^{2}, t)

and

r^{'} (t) = (1, 1, 1)

. By the line integral formula,

\int_{C} F \cdot d r = \int_{0}^{1} (2 t + t^{2}) d t = \frac{4}{3} .

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).

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