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oiiint

Represents a triple surface integral in multivariable calculus, indicating integration over a closed surface in three dimensions.

Overview

Essential in advanced mathematics and physics for calculating flux through closed surfaces and working with conservative vector fields.

  • Commonly used in electromagnetic theory to express Gauss's law
  • Appears in fluid dynamics when analyzing flow through closed surfaces
  • Important in vector calculus for computing total flux across boundary surfaces
  • Often paired with vector fields to calculate outward flux through closed surfaces in 3D space

Examples

Triple surface integral over a closed surface in vector calculus.

∮∮∮S∇⋅F dV=0\oint\oint\oint_S \nabla \cdot F\,dV = 0 
\oint\oint\oint_S \nabla \cdot F\,dV = 0

Triple surface integral in spherical coordinates.

∰R3f(r,θ,ϕ) r2sin⁡ϕ dr dθ dϕ\oiiint_{\mathbb{R}^3} f(r,\theta,\phi)\,r^2\sin\phi\,dr\,d\theta\,d\phi 
\oiiint_{\mathbb{R}^3} f(r,\theta,\phi)\,r^2\sin\phi\,dr\,d\theta\,d\phi

Divergence theorem in three dimensions with triple integral.

∰V(∇×F) dV=∮SF⋅n dS\oiiint_V (\nabla \times F)\,dV = \oint_S F \cdot n\,dS 
\oiiint_V (\nabla \times F)\,dV = \oint_S F \cdot n\,dS