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3.6: Лінійні інтеграли
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$\dfrac{df(\gamma (t))}{dt} = \nabla f(\gamma (t)) \cdot \gamma '(t) = F(\gamma (t)) \cdot y'(t).$ \[\begin{array} {rcl} {\int_{\gamma} F \cdot dr} & = & {\int_a^b F (\gamma (t)) \cdot y' (t)\ dt} \... $\dfrac{df(\gamma (t))}{dt} = \nabla f(\gamma (t)) \cdot \gamma '(t) = F(\gamma (t)) \cdot y'(t).$ $\begin{array} {rcl} {\int_{\gamma} F \cdot dr} & = & {\int_a^b F (\gamma (t)) \cdot y' (t)\ dt} \\ {} & = & {\int_a^b \dfrac{df(\gamma (t))}{dt} dt} \\ {} & = & {f(\gamma (b)) - f(\gamma (a))} \\ {} & = & {f(P) - f(Q)} \end{array}$ Якщо $F$ векторне поле є градієнтним полем, з $F = \nabla f$ , то ми викликаємо $f$ потенційну функцію для $F$ .