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I am studying the book "Complex variables and applications" by James Ward Brown, Ruel Vance Churchill and I don't understand what they mean by the domain to lie left to a path.

How can we tell that the domain is left to the path from the drawing ?

enter image description here

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I think you should have met them in Green's Theorem. It's just the orientation of surface with respect to path. It simply means that when you move along the curve, in direction pointed by arrow, the domain will be towards the lefthand side. –  007resu Dec 27 '12 at 0:37
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Imagine walking along the path in the direction in which it’s oriented. Put your arms out; the left arm points into the region. –  Brian M. Scott Dec 27 '12 at 0:38
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By the way, it doesn't say 'lie left to the path' as you have it in your question, but 'lie to the left of the path'. –  Tara B Dec 27 '12 at 0:40
    
@BrianM.Scott - thanks a lot! Since this was the explanasion I understood I suggest you make it an answer so I can accept it –  Belgi Dec 27 '12 at 0:41
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Of course, all of this assumes the (natural) convention that when we walk around the curve, we are standing so that our head is pointed upward from the printed page (or outward from the monitor). :) –  Ted Dec 27 '12 at 0:51

4 Answers 4

up vote 4 down vote accepted

Imagine walking along the path in the direction in which it’s oriented. Put your arms out; the left arm points into the region.

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Oh good, I was going to suggest you write your comment as an answer, since I thought you explained it better than either of us. –  Tara B Dec 27 '12 at 1:49

When you walk in one direction around a simple closed curve, the "inside" of the curve is either always on your left or always on your right. This simply means that the integration along the curves follow the path around the curve so that the interiors are on the left of the direction in which you traverse the curve.

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It means that if you are walking around the path in the direction shown by the arrow, then the domain will be on your left.

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The accepted answer is good enough from the perspective of a mathematician visualizing the complex plane containing a path, spread out before her. But that is a human situation. A mathematical definition of "left" in the complex plane is slightly tricky. In going from the negative to the positive part of the real line, the half plane to the left of our path is the one containing $\mathrm i$. Unfortunately, there is no way of telling which of $\mathrm i$ and$-\mathrm i$ is which; it is an arbitrary choice. As long as we place $\mathrm i$ in what from our own perspective is the "upper" half of the plane, the orientive conventions of the plane will accord with our native ones.

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