If i had a vector field $F$ defined in $R^3-\{\vec 0\}$ can it still be conservative?
If so, with what proof can I prove it?
2026-04-06 15:56:05.1775490965
Can a vector field undefined in a point be conservative?
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Conservative means its curl vanishes. Independent of path means the line integral depends only on the endpoints of the path. The two notions are equivalent in the case that we consider a vector field defined on a simply connected region. For example a region in the plane with no missing point, or a region in $\mathbb{R}^3$ that doesn't have a line removed (one dimensional singularity).
Conversely, in the plane with a point removed, or $\mathbb{R}^3$ with a line removed, a vector field can be conservative but not independent of path (paths that wind on one side of the singularity can have different line integral than paths on the other).
Similarly, a vector field in $\mathbb{R}^3$ can be divergenceless, or it can be "independent of surface". The two notions are equivalent for vector fields defined in a region of $\mathbb{R}^3$ with no zero-dimensional singularity removed.
Conversely, in a region with a point removed (and this is the question you asked), surface integral can depend on which surface you choose.
Note that you ask about conservativity, but I think what you really want to ask is about "independence of surface".