The Stokes theorem relates the line integral of a closed curve C in a vector field F to the surface integral of the curl of F over the boundary of the surfa...
The Stokes theorem relates the line integral of a closed curve C in a vector field F to the surface integral of the curl of F over the boundary of the surface.
Formally, if F is a continuous vector field on a closed surface S, then the Stokes theorem establishes the following equality:
∫_C F · ds = ∫_S (∇ × F) · dS,
where:
ds is the differential of the curve C in the plane.
dS is the surface element of the boundary S.
∇ is the gradient operator, represented by a nabla symbol.
F is the vector field.
∇ × F is the curl of the vector field, represented by a cross product symbol.
The Stokes theorem tells us that the line integral of a closed curve C in a vector field F is equal to the surface integral of the curl of F over the boundary of the surface. This theorem allows us to evaluate surface integrals of vector fields by evaluating line integrals of simpler curves