TeXipedia

iiint

Represents a triple integral in multivariable calculus, used for integrating over three-dimensional regions or volumes.

Overview

Essential for advanced mathematical calculations involving three-dimensional spaces and volume computations in physics and engineering applications.

  • Commonly used in vector calculus for volume integrals
  • Appears frequently in electromagnetic theory for volume charge distributions
  • Important in fluid dynamics for analyzing three-dimensional flow fields
  • Often combined with differential volume elements like dV or dx dy dz
  • Typically requires the amsmath package for proper rendering

Examples

Triple integral over a volume V in Cartesian coordinates.

∭Vf(x,y,z) dz dy dx\iiint_V f(x,y,z)\,dz\,dy\,dx 
\iiint_V f(x,y,z)\,dz\,dy\,dx

Computing electric flux through a closed surface using Gauss's law.

∭V∇⋅E dV=Qε0\iiint_V \nabla \cdot \mathbf{E}\,dV = \frac{Q}{\varepsilon_0} 
\iiint_V \nabla \cdot \mathbf{E}\,dV = \frac{Q}{\varepsilon_0}

Triple integral for mass calculation of a solid object with varying density.

M=∭Vρ(x,y,z) dVM = \iiint_V \rho(x,y,z)\,dV 
M = \iiint_V \rho(x,y,z)\,dV