I have seen several definitions of conservative fields, where we always assume that the domain of a vector field $ \pmb F $ is simply connected before giving the definition of a conservative field.
QUESTION:
If we find a function $ f $, where $ \pmb F = \pmb \nabla f $ , and also $ \pmb F $ and $ f $ have the same domain, then:
Can we ignore whether the domain is simply connected or not and label the vector field $ \pmb F $ as conservative ?
Thank you in advance