Divergence Theorem Calculator

Use x, y, z variables and operators like +, -, *, /, ^. Example: x^2 + y*z
Enter bounds as comma-separated values. Example: -1,1

Surface Integral: -

Volume Integral: -

Understanding the Divergence Theorem

What is the Divergence Theorem?

The divergence theorem, also known as Gauss's theorem, relates the flux of a vector field through a closed surface to the divergence of the field over the volume enclosed by the surface:

∯S F·n dS = ∭V (∇·F) dV

Key Components

  • Surface Integral (Left side): Measures flux through boundary
  • Volume Integral (Right side): Measures total divergence
  • Vector Field F(x,y,z): Three-dimensional vector function
  • Divergence (∇·F): Scalar measure of field's "spread"

Applications

Fluid Dynamics

Analysis of fluid flow and incompressibility

Electromagnetics

Gauss's law in electrostatics

Heat Transfer

Heat flux through surfaces