Rax Adaam

I'll definitely follow up with anything I find either way. Cheers -

Cool - thanks a lot. (*mathy)

Like really math?

Unfortunately I work tomorrow and so have to take off, but this has been bothering me for a while: I though the issue was the singularity, so I thought the domain restriction of the arctan example necessarily extended to the 1/r fields, which made no sense to me (what would it mean to have a locally conservative gravitational force?!).

If you have any suggestions for an intermediate text on this stuff, I'd be very interested. I have a solid undergrad math education, but with a science background (not a math major).

Oh yeah?

I'll have to flesh this out a bit more and look for more counter-examples, but I think that basically answers my question: effectively $\nabla \times \vec{F} = 0$ means that it is exceedingly likely that $\vec{F}$ actually is conservative; especially in applied contexts...

Well, that's another 4h of my life lost to the pedantry of mathematicians!

They were trying to make a point that the constraint is not strong enough - but actually, it is a pathological example that is not representative of "normal" functions?

I see, so the textbooks & notes that I read use $\nabla \big(-\arctan(x/y) \big)$ because it is a well-known exception to the otherwise reliable rule that $\nabla \times \vec{F} = 0 \quad \Rightarrow \quad \vec{F}$ is exact.

it's really all about defining the severity of the singularity.

so it doesn't even matter that the domain of the vector field isn't defined at the origin?

Wouldn't we expect that the gravitational field is either conservative or not conservative?

Ok, you said the first is pathological - so I can ignore it for the time being, but the others are so fundamental to our physical model, that it doesn't make sense to me that a) that the domain should be limited and b) what that would mean, physically?

$1/x^2$ is also fine?

Ok, then yes I think it may be a similar idea.

Simple singularities - is that a measure of severity (like in Frobenius Series solutions to ODEs)?

Ok - I'll figure that out later (assuming it compiles for you).

oh - no mathjax in the rooms?

I'm confused by the role of the domain in determining whether a field is exact or not. The working definition I have for being exact is that $\vec{F} = \nabla f$ for some scalar function $f$. So no surprise that $\nabla \times \vec{F} = 0$ is necessary but not sufficient; but I don't know how to make sense of fields like $\nabla (-\arctan\big(\frac{x}{y}\big)$ and $\nabla (\frac{k}{|\vec{r}|}$.

Thanks.