Stokes’ Theorem

Stokes’ theorem is a central result in vector calculus that connects the line integral of a vector field around a closed curve C to the surface integral of the curl of the vector field over a surface S bounded by C. It is named after the Irish mathematician Sir George Gabriel Stokes.

Stokes’ theorem states that for a continuously differentiable vector field F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k defined over a surface S with an outward-pointing normal vector n and its boundary C, the following relationship holds:

∮C F · dr = ∬S curl(F) · n dS

where ∮_C F · dr is the line integral of the vector field F along the curve C, ∬_S curl(F) · n dS is the surface integral of the curl of the vector field F over the surface S, and n is the outward-pointing normal vector to the surface S.

Stokes’ theorem is a generalization of Green’s theorem to three dimensions and plays a crucial role in various areas of mathematics, physics, and engineering, such as fluid dynamics, electromagnetism, and the study of conservative vector fields.