PX153 - I9 - line integrals

split curve $C$ in $n$ sections, each with displacement vector:

Δ {\vec{l}}_{r} = Δ l_{r, x} \hat{i} + Δ l_{r, y} \hat{j} - Δ l_{r, z} \hat{k}

lim_{n \to \infty} \sum_{r = 1}^{n} F_{x} ({\bar{x}}_{r}, {\bar{y}}_{r}, {\bar{z}}_{r}) Δ l_{r, x} + F_{y} ({\bar{x}}_{r}, {\bar{y}}_{r}, {\bar{z}}_{r}) Δ l_{r, y} + F_{z} ({\bar{x}}_{r}, {\bar{y}}_{r}, {\bar{z}}_{r}) Δ l_{r, z}

= \int_{C} \vec{F} (\vec{r}) \cdot d \vec{l}

guide
eg: the integral could be the work down, $W$ , when moving a particle along a path, $C$ , when subjected to a force, $\vec{F}$ , that may be position dependent along path $C :$

W = \int_{C} \vec{F} \cdot d \vec{l}

eg: $I = \int_{C} \vec{P} \cdot d \vec{l}$ , where, $\vec{P} = 5 y^{2} \hat{i} + 2 x y \hat{j}$ , along $C_{1} :$ a straight line joining the origin and $(1, 1)$
- parameterizing the path:
$\vec{l} = x \vec{i} + y \hat{j} $ $ w i t h $ C_{1} : y = x $$ $d \vec{l} = \frac{d}{d x} (x, y) = (1, 1) d x = (\hat{i} + \hat{j}) d x$

\begin{aligned} \int_{C} \vec{P} \cdot d \vec{l} & = \int_{x = 0}^{x = 1} (5 y^{2} \hat{i} + 2 x y \hat{j}) \cdot (d x \hat{i} + d x \hat{j}) \\ = \int_{0}^{1} 7 x^{2} d x \\ = \frac{7}{3} \end{aligned}

eg: $I = \int_{C_{2}} \vec{P} \cdot d \vec{l}$ , where $C_{2} :$ two straight lines, one with $x$ from $0 \to 1$ , other with $y$ from $0 \to 1$

\begin{aligned} I & = \int_{x = 0, y = 0}^{x = 1} (5 y^{2} \hat{i} + 2 x y \hat{j}) \cdot (d x \hat{i} + d y \hat{j}) + \int_{y = 0, x = 1}^{y = 1} (5 y^{2} \hat{i} + 2 x y \hat{j}) \cdot (d x \hat{i} + d y \hat{j}) \\ = \int_{0}^{1} 0 d x + \int_{0}^{1} 2 y d y \\ = 1 \end{aligned}

this shows that the result depends on the path taken