Equations · equation
Divergence Theorem
Calculus · 1813
- Equation
- ∮_S F · dA = ∫_V (∇ · F) dV
- Field
- Calculus
- Named After
- Carl Friedrich Gauss
- Year
- 1813
Symbols
F a vector field · S a closed surface enclosing the volume V · dA an outward-pointing patch of surface · ∇ · F the divergence: net outflow per unit volume at a point
Significance
What flows out through a closed surface equals what is created inside it. That single sentence is Gauss's law for electricity (flux out equals charge within), Gauss's law for magnetism (no flux, no monopoles) and the continuity equation of fluid flow. It is the fundamental theorem of calculus in three dimensions: an integral over a boundary traded for one over the interior.
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