![]() ![]() 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
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