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oiint

Represents a double surface integral over a closed surface in multivariable calculus and vector analysis.

Overview

Essential for expressing flux integrals and working with vector fields in three-dimensional space, particularly in electromagnetic theory and fluid dynamics.

  • Commonly used in Gauss's divergence theorem calculations
  • Appears frequently in physics when analyzing closed surfaces
  • Found in advanced mathematics for computing flux across boundary surfaces
  • Requires the 'amsmath' or 'esint' package for proper rendering

Examples

Surface integral over a closed surface in 3D space, commonly used in physics for Gauss's law.

 ⁣SEdS=Qϵ0\oint\!\oint_S \vec{E} \cdot d\vec{S} = \frac{Q}{\epsilon_0}
\oint\!\oint_S \vec{E} \cdot d\vec{S} = \frac{Q}{\epsilon_0}

Double surface integral for calculating the total flux through a closed boundary.

VFdS=V(F)dV\oiint_{\partial V} \vec{F} \cdot d\vec{S} = \int_V (\nabla \cdot \vec{F})\,dV
\oiint_{\partial V} \vec{F} \cdot d\vec{S} = \int_V (\nabla \cdot \vec{F})\,dV

Divergence theorem applied to an electromagnetic field.

SBdS=0\oiint_S \vec{B} \cdot d\vec{S} = 0
\oiint_S \vec{B} \cdot d\vec{S} = 0