When a vector field has no “curl” [...], the vector field can be thought of as the gradient of a scalar field.
In case you weren’t aware, this is no longer true if the state space has “holes” (formally: if its first cohomology group is non-zero). For example, if the state space is the Euclidean plane without the origin, you can have a vector field on that space which has no curl but isn’t conservative (and thus is not the gradient of any utility function).
Why this might be relevant:
1. Maybe state spaces with holes actually occur, in which case removing the curl of the PVF wouldn’t always be sufficient to get a utility function
2. The fact that zero curl only captures the concept of transitivity for certain state spaces could be a hint that conservative vector fields are a better concept to think about here than irrotational ones (even if it turns out that we only care about simply connected state spaces in practice)
EDIT: an example of an irrotational 2D vector field which is not conservative is v(x,y)=(−yx2+y2,xx2+y2) defined for (x,y)∈R2∖{0}
In case you weren’t aware, this is no longer true if the state space has “holes” (formally: if its first cohomology group is non-zero). For example, if the state space is the Euclidean plane without the origin, you can have a vector field on that space which has no curl but isn’t conservative (and thus is not the gradient of any utility function).
Why this might be relevant:
1. Maybe state spaces with holes actually occur, in which case removing the curl of the PVF wouldn’t always be sufficient to get a utility function
2. The fact that zero curl only captures the concept of transitivity for certain state spaces could be a hint that conservative vector fields are a better concept to think about here than irrotational ones (even if it turns out that we only care about simply connected state spaces in practice)
EDIT: an example of an irrotational 2D vector field which is not conservative is v(x,y)=(−yx2+y2,xx2+y2) defined for (x,y)∈R2∖{0}