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**ThePerfectHacker** What is the formal definition of "orientation" of a curve? Or is this a topology issue?

Let $\displaystyle \gamma: [a,b] \to \mathbb{C}$ be a smooth function. And $\displaystyle f(z)$ is continous on $\displaystyle \gamma^*$ where $\displaystyle \gamma^* = \{\gamma(t)|t\in [a,b]\}$, i.e. the image. And $\displaystyle \eta:[c,d]\to \mathbb{C}$ is another such function. With the property that $\displaystyle \eta^* = \gamma^*$ and have the same orientation. Then show that $\displaystyle \int_{\gamma}f(z) dz = \int_{\eta}f(z) dz$.

Basically, what I am trying to show that if we want to integrate a complex function along some path it makes no difference how we parametrize it (as long the orientation is the same).