16 Vector Calculus Copyright Cengage Learning All rights reserved 16 8 Stokes Theorem Copyright Cengage Learning All rights reserved 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 Theorem 3 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 1 4 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 5 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 dr 6 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 7 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 8 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 9 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 b C v dr 0 negative circulation a C v dr 0 positive circulation Figure 5 10 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 11 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 v 12
View Full Document
Unlocking...