 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2015/12/c/ii/solution

Interpret the axis-directed route as successive coordinate-axis-parallel segments

(0, 0, 0) ⟶ (1, 0, 0) ⟶ (1, 2, 0) ⟶ (1, 2, 1) .

On the first segment,

y = z = 0

, so the field vanishes. On the second,

z = 0

, so it again vanishes. On the last,

x = 1

y = 2

and only

d z

is nonzero, giving

F \cdot d r = 8 z d z

. Thus

\int_{C} F \cdot d r = 0 + 0 + \int_{0}^{1} 8 z d z = 4 .

The second and third segments are parallel to the corresponding global axes; they do not return to the global

y

- or

z

-axis before reaching the endpoint.

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