What is the formal definition of "orientation" of a curve? Or is this a topology issue?
Let be a smooth function. And is continous on where , i.e. the image. And is another such function. With the property that and have the same orientation. Then show that .
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).
I would agree that it should not be so easy to show, however I do not know how to show it. Furthermore, my book assumes that it is okay without even stating that it is valid.
As I mentioned this is not completely true. Take, for example, the semi-unit upper circle. We can express it as and .
And consider the function .
Then, (note )
What is wrong? The answer is the orientation. One is clockwise and one is counterclockwise.
The problem is I do not know how to formally define this notion.
\eta here does not parametrize the same portion of the circle as \gamma, that's why the integrals are different
Because actually, the only change of orientation on a curve that can occur, is following it from the endpoint to the startpoint. And that just gives you a change of sign in the integral.