(b) The line integral along any of the sides of E cancels out with the line integral along a side of an adjacent approximating square. In the limit, as the areas of the approximating squares go to zero, this approximation gets arbitrarily close to the flux.įigure 6.81 (a) The line integral along E l E l cancels out the line integral along F r F r because E l = − F r.

Therefore, the sum of all the fluxes (which, by Green’s theorem, is the sum of all the line integrals around the boundaries of approximating squares) can be approximated by a line integral over the boundary of S. After all this cancelation occurs over all the approximating squares, the only line integrals that survive are the line integrals over sides approximating the boundary of S. These three line integrals cancel out with the line integral of the lower side of the square above E, the line integral over the left side of the square to the right of E, and the line integral over the upper side of the square below E ( Figure 6.81). The same goes for the line integrals over the other three sides of E. On the square, we can use the flux form of Green’s theorem:Īs we add up all the fluxes over all the squares approximating surface S, line integrals ∫ E l F This square has four sides denote them E l, E l, E r, E r, E u, E u, and E d E d for the left, right, up, and down sides, respectively. Let D inherit its orientation from S, and give E the same orientation. We choose D to be small enough so that it can be approximated by an oriented square E. Let S be a surface and let D be a small piece of the surface so that D does not share any points with the boundary of S. This proof is not rigorous, but it is meant to give a general feeling for why the theorem is true. Proofįirst, we look at an informal proof of the theorem. We look at an intuitive explanation for the truth of the theorem and then see proof of the theorem in the special case that surface S is a portion of a graph of a function, and S, the boundary of S, and F are all fairly tame. The complete proof of Stokes’ theorem is beyond the scope of this text. Green’s theorem can only handle surfaces in a plane, but Stokes’ theorem can handle surfaces in a plane or in space. However, this is the circulation form of Green’s theorem, which shows us that Green’s theorem is a special case of Stokes’ theorem. In this special case, Stokes’ theorem gives ∫ C F d S is actually the double integral ∬ S curl F.

Then the unit normal vector is k and surface integral ∬ S curl F Suppose surface S is a flat region in the xy-plane with upward orientation. Note that the orientation of the curve is positive. With this definition in place, we can state Stokes’ theorem.įigure 6.79 Stokes’ theorem relates the flux integral over the surface to a line integral around the boundary of the surface. The orientation of S induces the positive orientation of C if, as you walk in the positive direction around C with your head pointing in the direction of N, the surface is always on your left. Furthermore, suppose the boundary of S is a simple closed curve C. Let S be an oriented smooth surface with unit normal vector N. 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.

Flux integral formula