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.
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).