[Verse 1] From the boundary to the heart, there's a story to be told Integration holds the key to mysteries of old When we take derivatives inside a region's space Equals what we integrate around the boundary's face [Chorus] Stokes theorem unifies them all Integral M d-omega equals integral boundary omega call From one dimension up to any space you find The boundary tells the story of what lies inside D squared zero, boundary twice is empty Local to global, the pattern's so plenty [Verse 2] In one dimension it's calculus fundamental and true Two dimensions brings us Green's theorem breaking through Three dimensions splits in two - divergence and the curl But the general form shows how all these truths unfurl [Chorus] Stokes theorem unifies them all Integral M d-omega equals integral boundary omega call From one dimension up to any space you find The boundary tells the story of what lies inside D squared zero, boundary twice is empty Local to global, the pattern's so plenty [Bridge] Start with rectangles and cubes where the proof is crystal clear Partition of unity helps patch the pieces here What's locally trivial becomes globally profound Conservation laws in physics keep this truth around [Verse 3] Maxwell's equations dance to this eternal song D of F equals zero shows the field is strong Exact forms on closed manifolds must integrate to nil Birth of cohomology from this simple drill [Chorus] Stokes theorem unifies them all Integral M d-omega equals integral boundary omega call From one dimension up to any space you find The boundary tells the story of what lies inside D squared zero, boundary twice is empty Local to global, the pattern's so plenty [Outro] From differential geometry to topology's embrace Homological algebra finds its rightful place The boundary of boundary vanishes away Stokes theorem lights the mathematical way
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