ɪɾ ɪz stɹeɪtfˈɔːɹwɚd tə ʃˈoʊ ðˌæɾə vˈɛktɚ fˈiːld ɪz pˈæθˌɪndᵻpˈɛndənt ɪf ænd ˈoʊnli ɪf ðɪ ˈɪntᵻɡɹəl ʌvðə vˈɛktɚ fˈiːld ˌoʊvɚɹ ˈɛvɹi klˈoʊzd lˈuːp ɪn ɪts dəmˈeɪn ɪz zˈiəɹoʊ.
It is straightforward to show that a vector field is path-independent if and only if the integral of the vector field over every closed loop in its domain is zero.