Line Integrals
Table of Contents
- Scalar line integral
Let \( \gamma:[a,b]\mapsto X \) be a path of class \( C^{1} \) where \( X\subset \mathbb{R}^{n} \). Let \( f:X\mapsto \mathbb{R} \) be a continuous function. The scalar line integral of \( f \) along \( \gamma \) is defined by
\[ \int_{\gamma}f\,ds = \int_{a}^{b} f(\gamma(t))\|\gamma'(t)\| \,dt \]
- Vector line integral
Let \( \gamma:[a,b]\mapsto X \) be a path of class \( C^{1} \) where \( X\subset \mathbb{R}^{n} \). Let \( \mathbf{F}:X\mapsto \mathbb{R}^{n} \) be a continuous function. The scalar line integral of \( \mathbf{F} \) along \( \gamma \) is defined by
\[ \int_{\gamma}\mathbf{F}\cdot d\mathbf{s} = \int_{a}^{b} \mathbf{F}(\gamma(t))\cdot \gamma'(t) \,dt \]
The lefthand side is expanded to \( \int_{\gamma}\mathbf{F}\cdot d\mathbf{s} = \int_{\gamma}(F_{1}dx_{1} + F_{2}dx_{2} + \dots + F_{n}dx_{n}) \)
1. (Re-)Parameterization
Proposition. Let \( \gamma_{1}:[a,b]\mapsto\mathbb{R}^{n} \) and \( \gamma_{2}:[c,d]\mapsto\mathbb{R}^{n} \) be paths of class \( C^{1} \) where \( \gamma_{2} \) is a reparameterization of \( \gamma_{1} \). Let \( f:X\mapsto\mathbb{R} \) be a continuous function where \( X\subset \mathbb{R}^{n} \) contains the image of \( \gamma_{1} \). Then
\begin{equation} \int_{\gamma_{1}} f\,ds = \int_{\gamma_{2}} f\,ds \end{equation}Proposition. Let \( \gamma_{1}:[a,b]\mapsto\mathbb{R}^{n} \) and \( \gamma_{2}:[c,d]\mapsto\mathbb{R}^{n} \) be paths of class \( C^{1} \) where \( \gamma_{2} \) is a reparameterization of \( \gamma_{1} \). Let \( \mathbf{F}:X\mapsto\mathbb{R}^{n} \) be a continuous function where \( X\subset \mathbb{R}^{n} \) contains the image of \( \gamma_{1} \). Then
- If the reparameterization is orientation-preserving, then \( \int_{\gamma_{1}}\mathbf{F}\cdot d\mathbf{s} = \int_{\gamma_{2}}\mathbf{F}\cdot d\mathbf{s} \)
- If the reparameterization is orientation-reversing, then \( \int_{\gamma_{1}}\mathbf{F}\cdot d\mathbf{s} = -\int_{\gamma_{2}}\mathbf{F}\cdot d\mathbf{s} \)
Based on these propositions, we can define a scalar line integral along a curve \( C \) by choosing a parameterization \( \gamma \) of \( C \) and then define:
\[ \int_{C} f\,ds = \int_{\gamma} f\, ds \]
If \( C \) is a closed curve or can be decomposed as finite number of closed curves, we sometimes use the notations:
\[ \oint_{C} f\,ds \quad\text{and}\quad \oint_{C} \mathbf{F}\cdot d\mathbf{s} \]
to mean \( \int_{C} f\,ds \) and \( \int_{C}\mathbf{F}\cdot d\mathbf{s} \) respectively.