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Post by mondo on Apr 7, 2022 0:40:10 GMT
Hello,
The second paragraph of a chapter "Winding Numbers and Vector Fields" on page 456 says "For example, in [6] we see that $\gamma[T_{s}]$ = -1"(I a not sure what Greek letter it is) My question is how was it calculated? The index definition says that it counts the net number of revolutions of $V(z)$ as z traverses given loop. However, how to decide what is the direction of these vectors on $\Gamma$? Some of them point straight up, some straight down, yet some seems to be canceled out by other vector... Also, there is a comment in the book that says these vectors on $\Gamma$ where only put there to ease the calculations - I am a bit lost, I can't count it with those vectors in place nor have an idea how to count them by looking only at phase portrait.
Thank you.
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Post by Admin on Apr 7, 2022 10:38:38 GMT
Mondo
First of all note that the sense of traversal of the loop $\Gamma_s$ is anticlockwise. The subscript $s$ means that the loop is around the singular point $s$.
In figure 6 on page 457, at every point of intersection of the loop with a streamline, the vector field has a direction which is tangential to the streamline at the point of intersection. If we imagine that as we traverse the loop we draw the vector $V$ at each point and notice how many revolutions $V$ makes as we go round the loop once, then this is the index of $s$.
So starting at any point on the loop in figure 6 we can see that as we traverse the loop in the given direction (anticlockwise) the base of the vector $V$ moves along the loop and the direction of $V$ changes and when it returns to its starting point the vector $V$ itself has rotated once in a clockwise sense.
So we can say that the index of the point is $-1$. One revolution of $V$, in a clockwise sense, for one traversal of the loop in an anticlockwise sense.
I find it useful to hold a pen or pencil in my hand to represent $V$ and rotate it as I go round the loop in my mind, using the example vectors in the figure as a guide to how it should move. Now try it for the other examples in figure 5. You need to draw a suitable loop $\Gamma$, of course, for each example. Notice that if the loop does not enclose the singular point $s$, then the index is zero and your pen does not make any complete rotations. Good luck!
Vasco
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Post by mondo on Apr 8, 2022 5:56:24 GMT
Vasco,
thank you, your answer and especially the pen/pencil suggestion helped me to see the idea. However, two things are still confusing:
1. On page 458 in point (i) They say "There is nothing to stop us applying this definition to a non-singular point but in this case the index must vanish". It is clear, the index will be 0 in this case as there will be no revolution (no point to revolve around). But, just 3 pages later author calculates index of a non singular loop $B_{2}$ of figure [9] and gets it at $1$. This loop does not encircle any singularity so index should be rather $0$?
2. Going back to page 458 and second paragraph author says what I could already sense that there is a connection between a winding number and index. His explanation is confusing to me: "If we think of V as a mapping, sending the points of $\tau_{s}$ to those of a new loop $V(\tau_{s})$ then a moment of thought reveals that the index of $\tau_{s}$ is just a new interpretation of the iwnding number of its image loop". So what I don't get here is how can $V(\tau_{s})$ be a mapping that sends points of $\tau_{s}$ to there images- what images? $V(z)$ is supposed to be a vector field and points on $\tau_{s}$ are just elements of this vector field - so how can we map them again?
Thank you.
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Post by Admin on Apr 9, 2022 14:53:40 GMT
Mondo
I understand why you are finding this chapter difficult to follow. I found it difficult when I first studied it a number of years ago now.
I will put together a reply to your specific question as soon as I can. In the meantime you may find it useful to read the other 2 threads on the board where your thread is posted. They are discussions I had with Gary about similar problems to yours and I think you may find them relevant.
Vasco
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