 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/i/solution

Parametrize the straight path by

r (t) = (t, t, 2 t)

for

0 \leq t \leq 1

. The conservative field along it is

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

and the tangent is

(1, 1, 2)

. Thus

\int_{A}^{B} F_{1} \cdot d r = \int_{0}^{1} 12 t^{5} d t = 2 .

The associated potential has

ϕ_{1} (B) - ϕ_{1} (A) = 2

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