 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2017-resit/5/solution

A vector line integral along an oriented parametrized curve

r (t)

\int a (r (t)) \cdot r^{'} (t) d t

; for a force field it is the work along that path. On the upper branch, set

y = t

x = t^{2}

0 \leq t \leq 1

. Then

a \cdot d r = (t^{2} - t) (2 t d t) + (t^{2} + t) d t = (2 t^{3} - t^{2} + t) d t,

\int_{C} a \cdot d r = [\frac{t ^{4}}{2} - \frac{t ^{3}}{3} + \frac{t ^{2}}{2}]_{0}^{1} = \frac{2}{3} .

The stated

x

endpoints do not distinguish the two branches of

y^{2} = x

. If the lower branch is intended, the same parametrization runs from

t = 0

t = - 1

, giving instead

4/3

. The two values correspond to different terminal points,

(1, 1)

and

(1, - 1)

; choosing a branch is necessary to make the question unambiguous.

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