 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/16/b/ii/solution

For

r (t) = (t, t, 2 t^{3})

, the conservative field is

(6 t^{7}, 4 t^{7}, t^{5})

and the tangent is

(1, 1, 6 t^{2})

. Therefore

\int_{A}^{B} F_{1} \cdot d r = \int_{0}^{1} 16 t^{7} d t = 2 .

The two different paths give the same value because their line integrals depend only on the potential difference

ϕ_{1} (B) - ϕ_{1} (A) = 2

. There is no scalar potential for

F_{2}

to which the conditional request could be applied.

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