Изменения

Пусть <tex> V: O \to \mathbb{R}^m </tex> потенциально, <tex> f </tex> — потенциал <tex> V </tex>, <tex> \gamma[a;b] \to 0 </tex>— кусочно гладкий.  Тогда <tex> \int\limits_{\gamma} (V_1 dx_1 + ... V_m dx_m) = f(\gamma(b)) - f(\gamma(a)) </tex>.|proof=1) <tex> \int\limits_{\gamma} \sum V_k d x_k = \int\limits_{a}^{b} (V_1(\gamma(t))\cdot\gamma'_1 + \ldots + V_m(\gamma(t))\cdot\gamma'_m) = f(\gamma(t))|_a^b </tex> — доказано для гладкого пути \\ <tex> V_1(\gamma(t))\cdot\gamma'_1 + \ldots + V_m(\gamma(t))\cdot\gamma'_m = f(\gamma(t))' </tex> <tex> = f(\gamma_1(t)\ldots\gamma_m(t))' = \frac{\partial f}{\partial x_1}\cdot\gamma'_1 + \ldots + \frac{\partial f}{\partial x_m}\cdot\gamma'_m </tex> \\ <tex> \frac{\partial f}{\partial x_1} = V_1; \ldots; \frac{\partial f}{\partial x_m} = V_m </tex> 2) <tex> a = t_0 < t_1 < \ldots < t_n = b </tex> <tex> \gamma|_{[t_{k-1}, t_{k}]} </tex> — гладкий <tex> \int\limits_{\gamma}\sum_k V_k d x_k = \sum_k (\int\limits_{t_k-1}^{t_k} \sum_i V_i d \gamma_i) = </tex><tex> \sum(f(\gamma(t_k)) - f(\gamma(t_{k-1}))) = f(\gamma(b)) - f(\gamma(a)) </tex>