 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/17/b/ii/solution

Choose

f (z) = z,

or add any constant. Then

f^{'} (z) = 1

, so the vector field is exactly

r

and necessarily has the same flux as in part (a). The formula from part (b)(i) confirms this:

2 π (z_{0} + 1) + π z_{0} (1 - z_{0}^{2}) = π (2 + 3 z_{0} - z_{0}^{3}) .

 New to topics? Read the docs here!