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Stokes' theorem states ∮_C F · dr = ∬_S (∇ × F) · n dS, where S is a surface bounded by closed curve C. It generalizes Green's theorem to 3D: circulation around C equals flux of curl through S. Boundary orientation uses right-hand rule.
You know Green's theorem: for a flat region D in ℝ² bounded by a closed curve C, the circulation of a 2D vector field around C equals the double integral of the 2D curl over D. Stokes' theorem generalizes this to three dimensions: the circulation of a 3D vector field around a closed curve C equals the flux of the curl ∇ × F through any surface S bounded by C. The formula ∮_C F · dr = ∬_S (∇ × F) · n dS encodes the same fundamental idea — boundary information equals interior information — but now the boundary is a curve in 3D and the interior is a surface.
The leap from Green's theorem to Stokes is replacing the flat region D with a curved surface S in ℝ³. The boundary of S is the closed curve C, and the surface can be any shape you like — a flat disk, a hemisphere, a saddle — as long as it spans C. The integrand on the right is the flux of the curl: you compute ∇ × F at each point on the surface (a vector measuring local rotation of the field), take its component perpendicular to the surface (dot with the unit normal n), and integrate against the surface area element dS. The remarkable non-obvious fact is that the result is the same regardless of which spanning surface you choose — this follows because the curl is divergence-free (∇ · (∇ × F) = 0), so switching surfaces changes the double integral by zero.
Orientation is the essential bookkeeping issue. The direction of traversal around C and the direction of the surface normal n must be consistent via the right-hand rule: curl the fingers of your right hand in the direction you traverse C, and your thumb points in the direction of n. Getting orientation wrong introduces a sign error, flipping the sign of the entire answer. In practice: fix the orientation of C, then choose n consistently; or fix n first, then traverse C in the direction given by the right-hand rule.
The strategic use of Stokes' theorem mirrors Green's theorem — trade a hard integral for an easier one. A complicated line integral in 3D can become a flux integral of the curl if the curl is simple (or zero). A complicated flux integral of a curl can become a line integral if the boundary curve is manageable. A powerful special case: when ∇ × F = 0 everywhere (the field is irrotational and hence conservative on simply connected domains), Stokes' theorem gives ∮_C F · dr = 0 for any closed curve — recovering path-independence. Green's theorem, Stokes' theorem, and the divergence theorem are all instances of one master result from differential geometry: the integral of a differential form over a boundary equals the integral of its exterior derivative over the interior.