Essential Concepts
- 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]
Glossary
- 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
