Summary of Stokes’ Theorem

Stokes’ theorem relates a flux integral over a surface to a line integral around the boundary of the surface. Stokes’ theorem is a higher dimensional version of Green’s theorem, and therefore is another version of the Fundamental Theorem of Calculus in higher dimensions.
Stokes’ theorem can be used to transform a difficult surface integral into an easier line integral, or a difficult line integral into an easier surface integral.
Through Stokes’ theorem, line integrals can be evaluated using the simplest surface with boundary [latex]C[/latex].
Faraday’s law relates the curl of an electric field to the rate of change of the corresponding magnetic field. Stokes’ theorem can be used to derive Faraday’s law.

Key Equations

Stokes’ theorem
[latex]\displaystyle\int_{C} {\bf{F}}\cdot{d{\bf{r}}}=\displaystyle\iint_{S}\text{curl } {\bf{F}}\cdot{d{\bf{S}}}[/latex]

Stokes’ theorem: relates the flux integral over a surface [latex]S[/latex] to a line integral around the boundary [latex]C[/latex] of the surface [latex]S[/latex]

surface independent: flux integrals of curl vector fields are surface independent if their evaluation does not depend on the surface but only on the boundary of the surface