Calculus×Computing
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Divergence theorem (Gauss)

Level AdvancedDifficulty ★★★★★Theorem⌖ Open in the map

What is it?

The total divergence inside a region equals the flux out through its boundary: what is produced inside must leave through the surface. It is the foundation of conservation laws and finite-volume methods.

Statement

∭V∇⋅F dV=∯∂VF⋅n dS.\iiint_V \nabla\cdot F\,\dd V = \oiint_{\partial V}F\cdot n\,\dd S.

Idea of the proof

Chop VV into tiny boxes. For each box, flux out ≈ divergence × volume. Adding all boxes, fluxes through shared internal faces cancel, leaving only the outer boundary.

Where it shows up in computing

  • Fluid dynamics and CFD★★★★★fundamentalPhysics and simulation

    Finite-volume CFD is the divergence theorem applied cell by cell — the proof idea above, as an algorithm.

  • Electromagnetism (Maxwell's equations)★★★★★fundamentalPhysics and simulation

    Gauss's law in integral form: the flux of EE through a closed surface is the enclosed charge over ε0\varepsilon_0.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚛ Physics and simulation

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