Stokestheorema

Stokes' Theorem, also known as Kelvin-Stokes theorem, is a fundamental result in vector calculus that relates a surface integral of a vector field to a line integral of the same vector field. It is named after Sir George Gabriel Stokes, who published it in 1850, although it was known to William Thomson (Lord Kelvin) as early as 1846.

The theorem states that for a vector field F defined on a piecewise-smooth oriented surface Σ in three-dimensional Euclidean space, with boundary ∂Σ, the surface integral of the curl of F over Σ is equal to the line integral of F over the boundary of Σ. Mathematically, this is expressed as:

∮∂Σ F · dR = ∬Σ (∇ × F) · dS

where ∇ × F represents the curl of F, and dR and dS are the differential line and surface elements, respectively.

Stokes' Theorem is a generalization of both Green's Theorem in the plane and the Fundamental Theorem of Line Integrals in space. It has numerous applications in physics and engineering, particularly in electromagnetism, where it is used to derive Maxwell's equations from a surface integral form to a volume integral form.

The proof of Stokes' Theorem typically involves the Divergence Theorem (also known as Gauss's Theorem) and the fact that the curl of a gradient is zero. The theorem can be extended to higher dimensions and more general settings, such as manifolds and differential forms.