 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/ii/solution

For

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

, the velocity is

(1, 2 t, 2 t)

. Because

y = z

throughout this path, both fields again agree pointwise, now as

(3 t^{8}, 2 t^{7}, t^{7})

. Thus

\int F \cdot d r = \int G \cdot d r = \int_{0}^{1} (3 t^{8} + 4 t^{8} + 2 t^{8}) d t = 1.

These equal values alone do not establish path independence for either field on all of space.

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