Question about conditions for conservative field

[kelsiu]

In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is simply connected and open.

Usually in textbooks there is not much explanations on why these assumptions are necessary, no proof is given on why conservative field is not possible if the region is not simply connected or not open.

I wonder whether these two assumptions are just for computational convenience or it is really logically not possible to have a conservative field in region that is not simply connected or is not open?

 

[Ishuda]

In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is simply connected and open.

Usually in textbooks there is not much explanations on why these assumptions are necessary, no proof is given on why conservative field is not possible if the region is not simply connected or not open.

I wonder whether these two assumptions are just for computational convenience or it is really logically not possible to have a conservative field in region that is not simply connected or is not open?

It is not necessary that the region be simply connected, see
http://en.wikipedia.org/wiki/Conservative_vector_field
for example where an example is given,

 

[kelsiu]

It is not necessary that the region be simply connected, see
http://en.wikipedia.org/wiki/Conservative_vector_field
for example where an example is given,

I have go through the example but it is an example that if the region is not simply connected the vector field cannot be conservative.

Could you give a vector function in a non-simply connected region that is conservative?

 

[Ishuda]

I have go through the example but it is an example that if the region is not simply connected the vector field cannot be conservative.

Could you give a vector function in a non-simply connected region that is conservative?

Actually I was thinking of an irrotational vector field not needing to be simply connected but look at example 3.4 at
http://www-home.math.uwo.ca/~mfrankla/Math241/Math241_163_Conservative.pdf

 

[kelsiu]

Actually I was thinking of an irrotational vector field not needing to be simply connected but look at example 3.4 at
http://www-home.math.uwo.ca/~mfrankla/Math241/Math241_163_Conservative.pdf

Very useful, thanks.