 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/13/e/solution

For

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

r^{'} (t) = (1, 1, 2 t)

. Substitute into the field:

F (r (t)) = (2 t^{2} - t^{6}, t^{2} - 2 t, - 3 t^{5} - 1) .

3 t^{2} - 4 t - 7 t^{6}

. Hence the explicitly parametrized line integral is

\int_{C} F \cdot d r = \int_{1}^{2} (3 t^{2} - 4 t - 7 t^{6}) d t = [t^{3} - 2 t^{2} - t^{7}]_{1}^{2} = - 126 .

As a check, the potential from part (d) has values

- 128 + C

(2, 2, 4)

and

- 2 + C

(1, 1, 1)

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