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Gauss's Divergence Theorem Calculator

Integrate divergence over a box to illustrate the divergence theorem.

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∯_S F·n dS = ∭_V div F dV
e.g. x
e.g. y
e.g. z

📖 Tutorial | Gauss Theorem

1. Theorem
Gauss's divergence theorem: the total outward flux through a closed surface S equals ∭ div F dV over V: ∯_S F·n dS = ∭_V div F dV.
2. Symbols
Fvector field
Sclosed surface
divdivergence
Vvolume
3. How it works
  • Pointwise divergence;
  • 3D Simpson integration;
  • Equals the total outward flux.
4. Steps
  1. Enter P, Q, R;
  2. Enter six box bounds;
  3. Click Calculate.
5. Example
Example: F=(x,y,z) over [0,1]³.
Solution: div=3, volume=1, so ∭div dV=3.
6. Pitfalls
This tool computes the volume integral side;
Use outward normal;
Constant div → div×volume.

❓ FAQ | Gauss Theorem

Use of the theorem?
Convert flux surface integrals to volume integrals.
Flux?
Net flow through the surface.
Outward normal?
Pointing out of the volume.
Why 3?
div F=3, volume=1.
Relation to Stokes?
Gauss is 3D; Stokes is the 2D circulation version.