 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/8/i/solution

On the straight path

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

0 \leq t \leq 1

, both fields restrict to

(3 t^{5}, 2 t^{5}, t^{5})

. Dotting with

r^{'} = (1, 1, 1)

gives

\int F \cdot d r = \int G \cdot d r = \int_{0}^{1} 6 t^{5} d t = 1.

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