Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Feb 5, 2018 16:02:44 GMT
Vasco,
On p. 460, it states The index of a simple loop is the sum of the indices of the singular points it contains. Then Figure [9] illustrates a region with two holes in it with boundary curves $B_1$ and $B_2$. It does not appear that $B_2$ contains a singular point, yet on p. 461, paragraph 1, we find $\mathcal{I}[B_2] = 1$. Can you explain this?
Gary
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Post by Admin on Feb 5, 2018 16:59:10 GMT
Vasco, On p. 460, it states The index of a simple loop is the sum of the indices of the singular points it contains. Then Figure [9] illustrates a region with two holes in it with boundary curves $B_1$ and $B_2$. It does not appear that $B_2$ contains a singular point, yet on p. 461, paragraph 1, we find $\mathcal{I}[B_2] = 1$. Can you explain this? Gary Gary To me it does appear that $B_2$ contains a singular point. Looking at figure 9 on page 461 we can see that the vector field on $B_2$, at the points where the streamlines illustrated in figure 9 emerge from $B_2$, is tangential to the streamlines and rotates once round $B_2$, and so the index is 1. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Feb 5, 2018 21:18:48 GMT
Vasco, On p. 460, it states The index of a simple loop is the sum of the indices of the singular points it contains. Then Figure [9] illustrates a region with two holes in it with boundary curves $B_1$ and $B_2$. It does not appear that $B_2$ contains a singular point, yet on p. 461, paragraph 1, we find $\mathcal{I}[B_2] = 1$. Can you explain this? Gary Gary To me it does appear that $B_2$ contains a singular point. Looking at figure 9 on page 461 we can see that the vector field on $B_2$, at the points where the streamlines illustrated in figure 9 emerge from $B_2$, is tangential to the streamlines and rotates once round $B_2$, and so the index is 1. Vasco Vasco, Are you suggesting that because there are arrows drawn on the streamlines, and the arrows rotate by $2\pi$ about the "hole", that there must be a hidden critical point within the hole?! Would such a point be a source? Gary |
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Post by Admin on Feb 5, 2018 22:33:38 GMT
Gary To me it does appear that $B_2$ contains a singular point. Looking at figure 9 on page 461 we can see that the vector field on $B_2$, at the points where the streamlines illustrated in figure 9 emerge from $B_2$, is tangential to the streamlines and rotates once round $B_2$, and so the index is 1. Vasco Vasco, Are you suggesting that because there are arrows drawn on the streamlines, and the arrows rotate by $2\pi$ about the "hole", that there must be a hidden critical point within the hole?! Would such a point be a source? Gary Gary $C$, $B_1$ and $B_2$ are just drawn on a vector field and so $B_1$ and $B_2$ are not holes in the sense that the field inside is not there. The inside of $C$ with holes $B_1$ and $B_2$ is a multiply connected region defined on the vector field. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Feb 5, 2018 23:52:43 GMT
Vasco, Are you suggesting that because there are arrows drawn on the streamlines, and the arrows rotate by $2\pi$ about the "hole", that there must be a hidden critical point within the hole?! Would such a point be a source? Gary Gary $C$, $B_1$ and $B_2$ are just drawn on a vector field and so $B_1$ and $B_2$ are not holes in the sense that the field inside is not there. The inside of $C$ with holes $B_1$ and $B_2$ is a multiply connected region defined on the vector field. Vasco Vasco, On p. 460, he describes the region as outside the boundary curves and the vector field as being "on such a region". The only singular points are within the region outside the boundary curves. And his definition of an index in sentence 1 requires a singular point. Is this not a contradiction? Or can we say that the boundary itself is source for the vector field? Gary |
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Post by Admin on Feb 6, 2018 17:15:16 GMT
Gary $C$, $B_1$ and $B_2$ are just drawn on a vector field and so $B_1$ and $B_2$ are not holes in the sense that the field inside is not there. The inside of $C$ with holes $B_1$ and $B_2$ is a multiply connected region defined on the vector field. Vasco Vasco, On p. 460, he describes the region as outside the boundary curves and the vector field as being "on such a region". The only singular points are within the region outside the boundary curves. And his definition of an index in sentence 1 requires a singular point. Is this not a contradiction? Or can we say that the boundary itself is source for the vector field? Gary Gary For me sentence one is not the definition of an index. The definition of the index of a loop is on page 456 in terms of the rotation of the vector field as $z$ traverses the loop. Sentence one just tells us that the sum of the indices of a singular point turns out to be equal to the index of a simple loop round the singular points it contains. It is a reinterpretation of the Topological Argument Principle. This is similar to (17) on page 364 which tells us that the difference between the number of $p$-points and poles inside $\Gamma$ is equal to the winding number of $f(\Gamma)$ round $p$. This is not a definition of winding number, it just tells us that one quantity happens to be equal to the other. The drawing of loops like $B_1$ and $B_2$ in figure 9 is no different than the drawing of the loop in figure 6, figure 7, and figure 8 and of course also the loop $C$ in figure 9. They just happen to be inside $C$. Another way to look at this: in figure 9 if we drew $B_1$ on it's own you probably wouldn't turn a hair. Then we could add $B_2$ and again I ask: would you turn a hair? Finally draw $C$ enclosing the others. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Feb 6, 2018 22:33:41 GMT
Vasco,
I think I'm getting the picture. A rereading of pp. 456-9 helps. It begins with a discussion of the index of singular points and ends comparing that to the winding number of L around a general point a.
Gary
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