Copyright © Cengage Learning. All rights reserved. 16 Vector CalculusCopyright © Cengage Learning. All rights reserved. 16.8 Stokes’ Theorem3 3 Stokes' Theorem Learning objectives: § convert a line integral to a surface integral using Stokes’ Theorem, or vice versa (and then evaluate) § compare surface integrals over different surfaces with common boundary using Stokes’ Theorem4 4 Stokes' Theorem Stokes’ Theorem can be regarded as a higher-dimensional version of Green’s Theorem. Whereas Green’s Theorem relates a double integral over a plane region D to a line integral around its plane boundary curve, Stokes’ Theorem relates a surface integral over a surface S to a line integral around the boundary curve of S (which is a space curve). Figure 1 shows an oriented surface with unit normal vector n. Figure 15 5 Stokes' Theorem The orientation of S induces the positive orientation of the boundary curve C shown in the figure. This means that if you walk in the positive direction around C with your head pointing in the direction of n, then the surface will always be on your left.6 6 Stokes' Theorem Since ∫C F ! dr = ∫C F ! T ds and curl F ! dS = curl F ! n dS Stokes’ Theorem says that the line integral around the boundary curve of S of the tangential component of F is equal to the surface integral over S of the normal component of the curl of F. Stokes’ Theorem is often expressed as curl F ! dS = F ! dr7 7 Stokes' Theorem In the special case where the surface S is flat and lies in the xy-plane with upward orientation, then the unit normal vector is k, the surface integral becomes a double integral, and Stokes’ Theorem becomes ∫C F ! dr = curl F ! dS = (curl F) ! k dA This is precisely the vector form of Green’s Theorem. Thus we see that Green’s Theorem is really a special case of Stokes’ Theorem.8 8 Stokes' Theorem In general, if S1 and S2 are oriented surfaces with the same oriented boundary curve C and both satisfy the hypotheses of Stokes’ Theorem, then This fact is useful when it is difficult to integrate over one surface but easy to integrate over the other. We can use Stokes’ Theorem to throw additional light on the meaning of the curl vector.9 9 Stokes' Theorem Suppose that C is an oriented closed curve and v represents the velocity field in fluid flow. Consider the line integral ∫C v ! dr = ∫C v ! T ds and recall that v ! T is the component of v in the direction of the unit tangent vector T.10 10 Stokes' Theorem This means that the closer the direction of v is to the direction of T, the larger the value of v ! T. Thus ∫C v ! dr is a measure of the tendency of the fluid to move around C and is called the circulation of v around C. (See Figure 5.) Figure 5 (a) ∫C v ! dr > 0, positive circulation (b) ∫C v ! dr < 0, negative circulation11 11 Stokes' Theorem Now let P0(x0, y0, z0) be a point in the fluid and let Sa be a small disk with radius a and center P0. Then (curl F)(P) ≈ (curl F)(P0) for all points P on Sa because curl F is continuous. Thus, by Stokes’ Theorem, we get the following approximation to the circulation around the boundary circle Ca:12 12 Stokes' Theorem This approximation becomes better as a → 0 and we have Equation 4 gives the relationship between the curl and the circulation. It shows that curl v ! n is a measure of the rotating effect of the fluid about the axis n. The curling effect is greatest about the axis parallel to curl
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