ˈiːvən ɪf ðə ɡɹˈeɪdiənt θˈiəɹəm ˈɔːlsoʊ kˈɔːld fˌʌndəmˈɛntəl θˈiəɹəm ʌv kˈælkjʊləs fɔːɹ lˈaɪn ˈɪntᵻɡɹəlz hˈæzbiːn pɹˈuːvd fɚɹə dˈɪfɚɹˌɛnʃɪəbəl sˌoʊ lˈʊkt æz smˈuːð kˈɜːv sˈoʊ fˌɑːɹ, ðə θˈiəɹəm ɪz ˈɔːlsoʊ pɹˈuːvd fɚɹə pˈiːswaɪzsmˈuːð kˈɜːv sˈɪns ðɪs kˈɜːv ɪz mˌeɪd baɪ dʒˈɔɪnɪŋ mˌʌltɪpəl dˈɪfɚɹˌɛnʃɪəbəl kˈɜːvz sˌoʊ ðə pɹˈuːf fɔːɹ ðɪs kˈɜːv ɪz mˌeɪd baɪ ðə pɹˈuːf pɜː dˈɪfɚɹˌɛnʃɪəbəl kˈɜːv kəmpˈoʊnənt.
Even if the gradient theorem (also called fundamental theorem of calculus for line integrals) has been proved for a differentiable (so looked as smooth) curve so far, the theorem is also proved for a piecewise-smooth curve since this curve is made by joining multiple differentiable curves so the proof for this curve is made by the proof per differentiable curve component.