Mathematica(@mathemetica):∇ × F records the infinitesimal circulation of a vector field at every point. Integrated over a surface the same quantity becomes net rotation through that surface, and Stokes’ theorem converts it back into a line integral around the rim: ∮_∂S F · dr = ∬_S (∇ × F) · n dS. Green had already written the planar version in 1828; the three-dimensional statement first appeared as an examination question set by Stokes in 1854. The identical local-to-global translation governs the divergence theorem and the four compact field equations Maxwell assembled a decade later.

2026.09.22 17:18

∇ × F records the infinitesimal circulation of a vector field at every point. Integrated over a surface the same quantity becomes net rotation through that surface, and Stokes’ theorem converts it back into a line integral around the rim: ∮_∂S F · dr = ∬_S (∇ × F) · n dS. Green had already written the planar version in 1828; the three-dimensional statement first appeared as an examination question set by Stokes in 1854. The identical local-to-global translation governs the divergence theorem and the four compact field equations Maxwell assembled a decade later.