TeXipedia

oint

Represents a line integral around a closed path or contour in complex analysis and vector calculus.

Overview

Essential in advanced mathematics and physics for calculating circulation, flux, and complex path integrals.

  • Commonly used in electromagnetic theory to express Ampère's law and Faraday's law
  • Appears frequently in complex analysis for contour integration around poles
  • Critical in vector calculus for expressing Green's theorem and Stokes' theorem
  • Distinguished from regular integrals by indicating integration along a closed loop or surface

Examples

Line integral around a closed curve C in the complex plane.

∮Cf(z) dz=2πi\oint_C f(z)\,dz = 2\pi i 
\oint_C f(z)\,dz = 2\pi i

Magnetic field circulation using Ampère's law.

∮LB⃗⋅dl⃗=μ0I\oint_L \vec{B} \cdot d\vec{l} = \mu_0 I 
\oint_L \vec{B} \cdot d\vec{l} = \mu_0 I

Electric flux through a closed surface using Gauss's law.

∮SE⃗⋅dA⃗=Qϵ0\oint_S \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0} 
\oint_S \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0}