Practice question · Match the pairs
Match each integral theorem to what it equates.
- Fundamental theorem for line integrals
- Green’s theorem
- Stokes’ theorem
- Divergence theorem
- Loop circulation in the plane ↔ curl over the enclosed area
- Loop circulation in 3D ↔ curl flux through a spanning surface
- Flux through a closed surface ↔ divergence over the volume
- Endpoint values ↔ integral of gradient along a curve
Hints
- All four share one skeleton: behaviour on the boundary equals a derivative summed over the interior.
- Order them by dimension: endpoints (0D) → plane loops (1D boundary in 2D) → space loops → closed surfaces.
Show the answer
- Fundamental theorem for line integrals → Endpoint values ↔ integral of gradient along a curve
- Green’s theorem → Loop circulation in the plane ↔ curl over the enclosed area
- Stokes’ theorem → Loop circulation in 3D ↔ curl flux through a spanning surface
- Divergence theorem → Flux through a closed surface ↔ divergence over the volume
Why
These are one theorem climbing a ladder of dimensions, each equates a boundary quantity with an interior derivative. Recognizing which rung a problem sits on tells you which conversion will crack it.
Practise Stokes' Theorem
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