 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2010/5/i/solution

The complex line integral along the smooth parametrized curve is

\int_{γ} f (z) d z := \int_{a}^{b} f (γ (t)) γ^{'} (t) d t .

For the reversed curve, the chain rule gives

γ^{'} (t) = - γ^{'} (a + b - t)

. Substitution

s = a + b - t

yields

\int_{γ} f (z) d z = - \int_{a}^{b} f (γ (a + b - t)) γ^{'} (a + b - t) d t = - \int_{a}^{b} f (γ (s)) γ^{'} (s) d s .

Consequently reversing orientation negates the complex line integral.

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