Line Integral
A line integral (also called a path integral or curve integral) computes the integral of a function along a curve or path in two or three dimensions. There are two types: scalar line integrals (integrating a scalar field along a curve) and vector line integrals (computing work done by a vector field along a path). Line integrals are fundamental in physics for calculating work, circulation, and flux in electromagnetic and fluid mechanics contexts.
Key Formula
∫_C F · dr = ∫_a^b F(r(t)) · r'(t) dt
LaTeX: \int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)\,dt
| Symbol | Meaning | Unit |
|---|---|---|
| F | Vector field | N or appropriate unit |
| C | Curve (path) of integration | dimensionless (path) |
| r(t) | Parametric representation of the curve | m |
| a, b | Parameter limits | dimensionless |
Worked Example
Problem
Compute the work done by the force field F(x,y) = (y, x) along the straight line from (0,0) to (1,1).
Solution
Step 1: Parametrize the path: r(t) = (t, t) for t ∈ [0,1], so r'(t) = (1, 1). Step 2: Evaluate F on the path: F(r(t)) = F(t,t) = (t, t). Step 3: Compute the dot product: F(r(t)) · r'(t) = (t)(1) + (t)(1) = 2t. Step 4: Integrate: ∫₀¹ 2t dt = [t²]₀¹ = 1.
Answer
Work W = 1 joule (assuming SI units)
Types of Line Integrals and Their Applications
| Type | Notation | Physical Meaning | Application |
|---|---|---|---|
| Scalar field | ∫_C f ds | Integral of scalar along arc length | Mass of a wire with linear density |
| Vector field (work) | ∫_C F·dr | Work done by force field | Work by gravity, electrostatic force |
| Circulation | ∮_C F·dr | Line integral around a closed loop | Fluid circulation, magnetic flux |
| Flux | ∫_C F·n ds | Flow across a curve | Fluid flux across a boundary |
| Complex line integral | ∫_C f(z) dz | Integral of complex function | Contour integration in complex analysis |
Interactive Tools
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Related Terms
Surface Integral
A surface integral extends integration to a two-dimensional surface embedded in three-dimensional space, computing the total value of a scalar or vector field over that surface. Scalar surface integrals find quantities like surface area or total mass of a thin shell, while vector surface integrals (flux integrals) measure how much of a vector field passes through a surface. Surface integrals are central to Maxwell's equations, fluid dynamics, and the theorems of Gauss and Stokes.
Triple Integral
A triple integral extends the concept of integration to three dimensions, computing the accumulation of a function over a three-dimensional region. It is used to find volumes, masses, and other physical quantities distributed throughout a solid region in space. Applications include calculating mass of a solid with variable density, electric charge distributions, and gravitational potential fields.
From Latin "linea" (line) and "integrare" (to make whole). The term was established in the 19th century as mathematicians like Cauchy and Riemann formalized complex analysis and multivariable calculus.