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I'm wondering if there is some intuitive way of understanding why the specified domain has the "power" to make a vector-field conservative or not.

e.g. $\quad \displaystyle \boldsymbol{F}(x,y) = \frac{-y \, \boldsymbol{ i} + x \, \boldsymbol{ j}}{x^2 + y^2}$ is conservative on the plane minus a ray coming out of the origin (let's say the $x$-axis for specificity).

The fact that including / excluding such a ray would affect something that appear to be a property of the field itself is really counter-intuitive to me; especially when applied to physical fields like the gravitational or electromagnetic fields - aren't these conservative regardless of what points we include in space?

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  • $\begingroup$ As a further point to illustrate my confusion further: if we consider the above vector-valued function to be defined on $\mathbb{R}^3$, then wouldn't it be conservative? And yet wouldn't some of the possible paths overlap with paths that could be defined only in $\mathbb{R}^2$? I don't see how the two situations would differ, mathematically, and yet the definition seems to imply that the line integral of such a field would (could?) differ in these two situations... $\endgroup$ – Rax Adaam Feb 19 '16 at 1:16
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Simply, your definition of "conservative" uses more than just the vector field, since you also need to use paths or something that is related to them. However, some properties of paths depend a lot on the domain.

Namely, not all closed paths are homotopic to one point without crossing points where the vector field is not defined. For example, on a plane minus a ray all paths (that don't cross the ray!) are homotopic to one point, without the need to cross the origin when constructing the homotopy. However, a circle around the origin is not homotopic to one point using some homopoty that does not cross the origin.

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  • $\begingroup$ I don't really see how this is more than an even more technical way of saying "because it depends on the domain." It's my understanding that vector calculus was originally created to handle computations related to certain physical situations, but I don't see how the constraints on the domain reflect physical reality: don't we have vector force fields that are conservative? shouldn't that be independent of our domain? Is this just a limitation of our chosen coordinate system? $\endgroup$ – Rax Adaam Feb 19 '16 at 1:04
  • $\begingroup$ Ah, you want to discuss physics. Then check physics.stackexchange.com. $\endgroup$ – John B Feb 19 '16 at 1:21
  • $\begingroup$ :D Not really (at least I don't think so - hard to tell without having an understanding yet!) - the physics is just what prompted the question. Nonetheless, I did find an interesting discussion that begins to address some of the points: physics.stackexchange.com/questions/134975/… Besides a reference to 'nature' this doesn't seem particularly physical, in argument ... but I'm still uncomfortable with the points raised in the comment above. Anyhow, thanks for your reply. $\endgroup$ – Rax Adaam Feb 19 '16 at 15:22

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