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How to calculate the winding number?
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I've been given the following loop $gamma$, which clearly divides the complex plain into 5 domains. For each of these domains, I have been asked to find the winding number of $a$ around $gamma$, where $a$ is a point in the domain.
Since I haven't been given any further information of the loop, I don't think I can use the formula $$n(gamma,a) = frac12pi iint_gammafracdzz-a$$ now intuitively I know that the winding numbers of the domain outside the loop is $0$, the domain in the centre is, I'm guessing, $2$ and for the remaining domains is $1$. But nowhere have I been able to find a good explanation for winding number apart from the aforementioned formula (and the proof that it will be an integer)[Resources consulted: Couple of Lecture Notes online, this website, and books by Lang, Ahlfors and Bak-Newman]. Can someone please give me an explanation on how I can find the winding number in situations like these? I haven't yet been taught the Cauchy Integral Formula.
complex-analysis winding-number
edited 52 mins ago
José Carlos Santos
125k17101188
asked 1 hour ago
junkquill
983219
add a comment |Â
up vote
2
down vote
favorite
I've been given the following loop $gamma$, which clearly divides the complex plain into 5 domains. For each of these domains, I have been asked to find the winding number of $a$ around $gamma$, where $a$ is a point in the domain.
Since I haven't been given any further information of the loop, I don't think I can use the formula $$n(gamma,a) = frac12pi iint_gammafracdzz-a$$ now intuitively I know that the winding numbers of the domain outside the loop is $0$, the domain in the centre is, I'm guessing, $2$ and for the remaining domains is $1$. But nowhere have I been able to find a good explanation for winding number apart from the aforementioned formula (and the proof that it will be an integer)[Resources consulted: Couple of Lecture Notes online, this website, and books by Lang, Ahlfors and Bak-Newman]. Can someone please give me an explanation on how I can find the winding number in situations like these? I haven't yet been taught the Cauchy Integral Formula.
complex-analysis winding-number
edited 52 mins ago
José Carlos Santos
125k17101188
asked 1 hour ago
junkquill
983219
1
Is math.stackexchange.com/questions/703695/… of some help?
– Michael Hoppe
1 hour ago
This is very similar to an application of Ampère's law (integral form) in which the path has the topology of a trefoil knot. When you have a current going through the central part of the path you get a different value from the lobes. You can then apply Ampere's law again to break the closed path in two disjoint open paths where you know the results.
– minimax
51 mins ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I've been given the following loop $gamma$, which clearly divides the complex plain into 5 domains. For each of these domains, I have been asked to find the winding number of $a$ around $gamma$, where $a$ is a point in the domain.
Since I haven't been given any further information of the loop, I don't think I can use the formula $$n(gamma,a) = frac12pi iint_gammafracdzz-a$$ now intuitively I know that the winding numbers of the domain outside the loop is $0$, the domain in the centre is, I'm guessing, $2$ and for the remaining domains is $1$. But nowhere have I been able to find a good explanation for winding number apart from the aforementioned formula (and the proof that it will be an integer)[Resources consulted: Couple of Lecture Notes online, this website, and books by Lang, Ahlfors and Bak-Newman]. Can someone please give me an explanation on how I can find the winding number in situations like these? I haven't yet been taught the Cauchy Integral Formula.
complex-analysis winding-number
edited 52 mins ago
José Carlos Santos
125k17101188
asked 1 hour ago
junkquill
983219
I've been given the following loop $gamma$, which clearly divides the complex plain into 5 domains. For each of these domains, I have been asked to find the winding number of $a$ around $gamma$, where $a$ is a point in the domain.
Since I haven't been given any further information of the loop, I don't think I can use the formula $$n(gamma,a) = frac12pi iint_gammafracdzz-a$$ now intuitively I know that the winding numbers of the domain outside the loop is $0$, the domain in the centre is, I'm guessing, $2$ and for the remaining domains is $1$. But nowhere have I been able to find a good explanation for winding number apart from the aforementioned formula (and the proof that it will be an integer)[Resources consulted: Couple of Lecture Notes online, this website, and books by Lang, Ahlfors and Bak-Newman]. Can someone please give me an explanation on how I can find the winding number in situations like these? I haven't yet been taught the Cauchy Integral Formula.
complex-analysis winding-number
complex-analysis winding-number
edited 52 mins ago
José Carlos Santos
125k17101188
asked 1 hour ago
junkquill
983219
edited 52 mins ago
José Carlos Santos
125k17101188
asked 1 hour ago
junkquill
983219
edited 52 mins ago
José Carlos Santos
125k17101188
edited 52 mins ago
José Carlos Santos
125k17101188
edited 52 mins ago
José Carlos Santos
125k17101188
125k17101188
asked 1 hour ago
junkquill
983219
asked 1 hour ago
junkquill
983219
asked 1 hour ago
junkquill
983219
983219
1
Is math.stackexchange.com/questions/703695/… of some help?
– Michael Hoppe
1 hour ago
This is very similar to an application of Ampère's law (integral form) in which the path has the topology of a trefoil knot. When you have a current going through the central part of the path you get a different value from the lobes. You can then apply Ampere's law again to break the closed path in two disjoint open paths where you know the results.
– minimax
51 mins ago
add a comment |Â
1
Is math.stackexchange.com/questions/703695/… of some help?
– Michael Hoppe
1 hour ago
This is very similar to an application of Ampère's law (integral form) in which the path has the topology of a trefoil knot. When you have a current going through the central part of the path you get a different value from the lobes. You can then apply Ampere's law again to break the closed path in two disjoint open paths where you know the results.
– minimax
51 mins ago
1
1
Is math.stackexchange.com/questions/703695/… of some help?
– Michael Hoppe
1 hour ago
Is math.stackexchange.com/questions/703695/… of some help?
– Michael Hoppe
1 hour ago
This is very similar to an application of Ampère's law (integral form) in which the path has the topology of a trefoil knot. When you have a current going through the central part of the path you get a different value from the lobes. You can then apply Ampere's law again to break the closed path in two disjoint open paths where you know the results.
– minimax
51 mins ago
This is very similar to an application of Ampère's law (integral form) in which the path has the topology of a trefoil knot. When you have a current going through the central part of the path you get a different value from the lobes. You can then apply Ampere's law again to break the closed path in two disjoint open paths where you know the results.
– minimax
51 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
You left out a vital piece of information: that little arrow on the left, pointing down. It tells you in which direction the loop is travelled (only once, I assume). Therefore, the winding number of each point of the three lobes is $1$ and in the central region it is equal to $2$. I suggest that you read this intuitive description of the winding number.
answered 1 hour ago
José Carlos Santos
125k17101188
Thanks! This sort of an intuitive description is exactly what I was looking for, though I don't know how I missed the Wikipedia entry on the topic.
– junkquill
18 mins ago
@junkquill I'm glad I could help.
– José Carlos Santos
16 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
You left out a vital piece of information: that little arrow on the left, pointing down. It tells you in which direction the loop is travelled (only once, I assume). Therefore, the winding number of each point of the three lobes is $1$ and in the central region it is equal to $2$. I suggest that you read this intuitive description of the winding number.
answered 1 hour ago
José Carlos Santos
125k17101188
Thanks! This sort of an intuitive description is exactly what I was looking for, though I don't know how I missed the Wikipedia entry on the topic.
– junkquill
18 mins ago
@junkquill I'm glad I could help.
– José Carlos Santos
16 mins ago
add a comment |Â
up vote
4
down vote
accepted
You left out a vital piece of information: that little arrow on the left, pointing down. It tells you in which direction the loop is travelled (only once, I assume). Therefore, the winding number of each point of the three lobes is $1$ and in the central region it is equal to $2$. I suggest that you read this intuitive description of the winding number.
answered 1 hour ago
José Carlos Santos
125k17101188
Thanks! This sort of an intuitive description is exactly what I was looking for, though I don't know how I missed the Wikipedia entry on the topic.
– junkquill
18 mins ago
@junkquill I'm glad I could help.
– José Carlos Santos
16 mins ago
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
You left out a vital piece of information: that little arrow on the left, pointing down. It tells you in which direction the loop is travelled (only once, I assume). Therefore, the winding number of each point of the three lobes is $1$ and in the central region it is equal to $2$. I suggest that you read this intuitive description of the winding number.
answered 1 hour ago
José Carlos Santos
125k17101188
You left out a vital piece of information: that little arrow on the left, pointing down. It tells you in which direction the loop is travelled (only once, I assume). Therefore, the winding number of each point of the three lobes is $1$ and in the central region it is equal to $2$. I suggest that you read this intuitive description of the winding number.
answered 1 hour ago
José Carlos Santos
125k17101188
answered 1 hour ago
José Carlos Santos
125k17101188
answered 1 hour ago
José Carlos Santos
125k17101188
answered 1 hour ago
José Carlos Santos
125k17101188
125k17101188
Thanks! This sort of an intuitive description is exactly what I was looking for, though I don't know how I missed the Wikipedia entry on the topic.
– junkquill
18 mins ago
@junkquill I'm glad I could help.
– José Carlos Santos
16 mins ago
add a comment |Â
Thanks! This sort of an intuitive description is exactly what I was looking for, though I don't know how I missed the Wikipedia entry on the topic.
– junkquill
18 mins ago
@junkquill I'm glad I could help.
– José Carlos Santos
16 mins ago
Thanks! This sort of an intuitive description is exactly what I was looking for, though I don't know how I missed the Wikipedia entry on the topic.
– junkquill
18 mins ago
Thanks! This sort of an intuitive description is exactly what I was looking for, though I don't know how I missed the Wikipedia entry on the topic.
– junkquill
18 mins ago
@junkquill I'm glad I could help.
– José Carlos Santos
16 mins ago
@junkquill I'm glad I could help.
– José Carlos Santos
16 mins ago
add a comment |Â
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1
Is math.stackexchange.com/questions/703695/… of some help?
– Michael Hoppe
1 hour ago
This is very similar to an application of Ampère's law (integral form) in which the path has the topology of a trefoil knot. When you have a current going through the central part of the path you get a different value from the lobes. You can then apply Ampere's law again to break the closed path in two disjoint open paths where you know the results.
– minimax
51 mins ago