Conversely, we can calculate the line integral of vector field F along the boundary of surface S by translating to a double integral of the curl of F over S. Stokes’ theorem says we can calculate the flux of curl F across surface S by knowing information only about the values of F along the boundary of S. We use Stokes’ theorem to derive Faraday’s law, an important result involving electric fields. Furthermore, the theorem has applications in fluid mechanics and electromagnetism. In addition to allowing us to translate between line integrals and surface integrals, Stokes’ theorem connects the concepts of curl and circulation. Therefore, just as the theorems before it, Stokes’ theorem can be used to reduce an integral over a geometric object S to an integral over the boundary of S. Stokes’ theorem relates a vector surface integral over surface S in space to a line integral around the boundary of S. This theorem, like the Fundamental Theorem for Line Integrals and Green’s theorem, is a generalization of the Fundamental Theorem of Calculus to higher dimensions.

In this section, we study Stokes’ theorem, a higher-dimensional generalization of Green’s theorem. 6.7.4 Use Stokes’ theorem to calculate a curl.6.7.3 Use Stokes’ theorem to calculate a surface integral.6.7.2 Use Stokes’ theorem to evaluate a line integral.6.7.1 Explain the meaning of Stokes’ theorem.