The gradient theorem states that if the vector field  is the gradient of some scalar-valued function (i.e., if  is conservative), then  is a path-independent vector field (i.e., the integral of  over some piecewise-differentiable curve is dependent only on end points).
ðə ɡɹˈeɪdiənt θˈiəɹəm stˈeɪts ðæt ɪf ðə vˈɛktɚ fˈiːld ɪz ðə ɡɹˈeɪdiənt ʌv sˌʌm skˈeɪlɚvˈæljuːd fˈʌŋkʃən ˈaɪ.ˈiː., ɪf ɪz kənsˈɜːvətˌɪv, ðˈɛn ɪz ɐ pˈæθˌɪndᵻpˈɛndənt vˈɛktɚ fˈiːld ˈaɪ.ˈiː., ðɪ ˈɪntᵻɡɹəl ʌv ˌoʊvɚ sˌʌm pˈiːswaɪzdˈɪfɚɹˌɛnʃɪəbəl kˈɜːv ɪz dᵻpˈɛndənt ˈoʊnli ˌɔn ˈɛnd pˈɔɪnts.