Physics I / The Divergence Theorem
Practice question · Match the pairs

The divergence theorem, ∯SF⃗⋅dS⃗=∭V∇⋅F⃗ dV\oiint_S \vec{F}\cdot d\vec{S} = \iiint_V \nabla\cdot\vec{F}\,dV, match each symbol to its physical meaning.

  • ∯SF⃗⋅dS⃗\oiint_S \vec{F}\cdot d\vec{S}
  • ∇⋅F⃗\nabla\cdot\vec{F}
  • ∭V  dV\iiint_V \; dV
  • The theorem itself
  • Sum over every point of the enclosed region
  • Local source strength at a point
  • Net outflow through the closed boundary
  • Total outflow = total of all interior sources
Hints
  1. Read the equation as accounting: what leaves through the walls must have been produced inside.
  2. Divergence is per-unit-volume production; the triple integral adds it all up.
Show the answer
  • ∯SF⃗⋅dS⃗\oiint_S \vec{F}\cdot d\vec{S} → Net outflow through the closed boundary
  • ∇⋅F⃗\nabla\cdot\vec{F} → Local source strength at a point
  • ∭V  dV\iiint_V \; dV → Sum over every point of the enclosed region
  • The theorem itself → Total outflow = total of all interior sources
Why

The theorem is conservation bookkeeping: summing every point’s production (divergence) gives exactly what crosses the boundary. Gauss’s law in electromagnetism is this statement with charge as the source.

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