 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013/3/a/solution

For

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

, the endpoints correspond to

t = 0, 1

and

\dot{r} = (1, 2 t, 3 t^{2})

. The vector field on the path is

A = (9 t^{2}, - 14 t^{5}, 20 t^{7})

\int_{C} A \cdot d r = \int_{0}^{1} (9 t^{2} - 28 t^{6} + 60 t^{9}) d t = 3 - 4 + 6 = 5.

Including the derivative of each path coordinate is essential; merely substituting the path into the field does not supply the line element.

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