Line Integral over a Vector Field

Subjects: Vector Analysis
Links: Rectifiable Curves in Rn, Riemann Integral in R, Vector Valued Functions of Rn

Def: Let $U \subseteq R^{n}$ be an open an connected set, a curve $Γ \subseteq U$ and $γ : [a, b] \to R^{n}$ be a piecewise smooth function such that $Γ = γ [[a, b ∥ a, b]]$ , and the function $F = (F_{k})_{k = 1}^{n} : U \to R^{n}$ a function such that for any $F_{k} \circ γ$ be integrable over $[a, b]$ for any $k \in {1, \dots, n}$ . We define the integral of $F$ over the curve $Γ$ with the parametrization $γ$ as

\int_{Γ} F \cdot d r = \int_{Γ} F \cdot d γ = \int_{a}^{b} F (γ (t)) \cdot γ^{'} (t) d t

Where $d r$ is representing the differential of position, in a weireder notation we get that

\int_{Γ} F \cdot d r = \int_{Γ} \sum_{k = 1}^{n} F_{i} d x_{i}

If the curve is closed it can be denoted as follows, but it doesn’t change the nature of the calculation:

\oint_{Γ} f \cdot d γ = \oint_{Γ} f \cdot d r

Prop:* Let $U \subseteq R^{n}$ be a open and connected set, and $Γ \subseteq U$ be piecewise smooth and $γ : [a, b] \to R^{n}$ be a piecewise smooth parametrization of $Γ$ . Let $F, G : U \to R^{n}$ be such that $F_{k} \circ γ$ and $G_{k} \circ γ$ be integrable over $[a, b]$ for any $k \in {1, \dots, n}$ , and $α, β \in R$ , then

\int_{Γ} (α F + β G) \cdot d γ = α \int_{Γ} F \cdot d γ + β \int_{Γ} G \cdot d γ

Prop:************ Let $U \subseteq R^{n}$ be a open and connected set, and $Γ, Δ \subseteq U$ and piecewise smooth curves parametrized by $γ : [a, b] \to R^{n}$ and $δ : [c, d] \to R^{n}$ respectively, such that $Γ \cup Δ$ be piecewise smooth parametrized by $γ * δ$ . If $F : U \to R$ is such that $F_{k} \circ γ$ and $F_{k} \circ δ$ be integrable over their respective intervals for any $k \in {1, \dots, n}$ then

\int_{Γ \cup Δ} f \cdot d (γ * δ) = \int_{Γ} f \cdot d γ + \int_{Δ} f \cdot d δ

Cor:************ Let $Γ \subseteq R^{n}$ be a piecewise smooth nonsimple curve with a finite number of intersections parametrized by $γ : [a, b] \to R^{n}$ , then we can split $γ$ into $γ_{1}, \dots, γ_{k}$ simple curves such that $γ = γ_{1} * \dots * γ_{k}$

\int_{Γ} f \cdot d γ = \sum_{i = 1}^{k} \int_{Γ_{i}} f \cdot d γ_{i}

Prop:************ Let $U \subseteq R^{n}$ be a open and connected set, and $Γ \subseteq U$ be piecewise smooth and $γ : [a, b] \to R^{n}$ and $δ : [c, d] \to R^{n}$ be piecewise smooth parametrizations of $Γ$ , and $F : U \to R^{n}$ be such that $F_{k} \circ γ$ and $F_{k} \circ δ$ be integrable for in their respective intervals for any $k \in {1, \dots, n}$ . If there’s and $α : [c, d] \to [a, b]$ be a $C^{1}$ bijection with the property that $δ = γ \circ α$ , then

If $α$ preserves orientation, then
$\int_{Γ} F \cdot d γ = \int_{Γ} F \cdot d δ$
If $α$ changes orientation, then
$\int_{Γ} F \cdot d γ = - \int_{Γ} F \cdot d δ$
In particular
$\int_{Γ} F \cdot d γ = - \int_{Γ} F \cdot d γ^{-}$