Properties of Line Integrals question

[malawi_glenn]

What is C1 and C2 and how are those related to C?
Is f a general vector function?

 

[WMDhamnekar]

What is C1 and C2 and how are those related to C?
Is f a general vector function?

I think for both C₁ and C₂, x= cos(t), y=sin(t) where 0≤t≤2π and

and that's why
∮_C1f⋅dr = ∮_C2f⋅dr =0.
Am I correct?

 

[malawi_glenn]

I think for both C1 and C2, x= cos(t), y=sin(t) where 0≤t≤2π and

Eh no, C1 and C2 must be related to C in some way and C must be a general closed curve. It should be written in your book. Btw are you sure they should be closed? ∮ indicates so, but are you really sure the question is formulated in that way? Otherwise, if they are closed, they should be related like this (but I do not know for sure since you are not providing enough info)

 

[WMDhamnekar]

Eh no, C1 and C2 must be related to C in some way and C must be a general closed curve. It should be written in your book. Btw are you sure they should be closed? ∮ indicates so, but are you really sure the question is formulated in that way? Otherwise, if they are closed, they should be related like this (but I do not know for sure since you are not providing enough info)
View attachment 303692

Whatever information you require is given in this example and related figure.

For more general proof, you read below given study material.

 

[malawi_glenn]

So it is not a general theorem or such? Just for this example? Then you just calculate:
∮_C1f⋅dr₁ which is equal to 2π
and
∮_C2f⋅dr₂ which is equal to -2π

 

[Orodruin]

I mean, don’t lose track of the underlying purpose of that example: To demonstrate an application of Green’s theorem to a region with holes in it where the boundary is not a single closed curve but a union of several closed curves. In those cases Green’s theorem relates the sum of the line integrals over those closed curves to a surface integral over the contained region.

I do disagree somewhat with figure 4.3.4. There is no need to cut the region into several regions. It is enough to introduce enough cuts to make a single bounded region without holes. I used this myself last year when installing guiding cable for a lawnmower robot in my garden.

 

[WMDhamnekar]

I mean, don’t lose track of the underlying purpose of that example: To demonstrate an application of Green’s theorem to a region with holes in it where the boundary is not a single closed curve but a union of several closed curves. In those cases Green’s theorem relates the sum of the line integrals over those closed curves to a surface integral over the contained region.

I do disagree somewhat with figure 4.3.4. There is no need to cut the region into several regions. It is enough to introduce enough cuts to make a single bounded region without holes. I used this myself last year when installing guiding cable for a lawnmower robot in my garden.

Just one question:
Does author want to say Green's theorem holds for 'multiple connected regions' instead of ' multiply connected regions' ?
If no, would you distinguish between 'multiple connected regions' and 'multiply connected regions'

 

[Orodruin]

Just one question:
Does author want to say Green's theorem holds for 'multiple connected regions' instead of ' multiply connected regions' ?
If no, would you distinguish between 'multiple connected regions' and 'multiply connected regions'

No. Yes.

The concept of being multiply connected is a topological concept that effectively tells you that a region has holes in it. It is in contrast to simply connected region.

 

[mathwonk]

Imagine a connected region in the plane bounded by a finite number of closed curves. a transverse cut is a path in the region whose endpoints lie on the boundary. such a region is simply connected, or 1-conncted, if any transverse cut separates it into two pieces, and a region is n+1 connected if any transverse cut that does not separate it, changes it into an n connected region. E.g. a region is doubly connected if any transverse cut separates it into a simply connected region. I.e. an annulus, the region between two concentric circles, is doubly connected. Removing n disjoint discs from the interior of a disc gives an n+1 connected region.

these definitions are taken from Riemann's paper on the theory of abelian functions, where he introduced these ideas, (and apparently created the subject of topology).

see the simply connected (einfach zusammenhangende) and doubly connected (zweifach zusammenhangende) surfaces depicted on page 9 of his paper: (which are similar to figures in your posts 29, 34)

https://www.maths.tcd.ie/pub/HistMath/People/Riemann/AbelFn/AbelFn.pdf