16 8 Stokes Theorem Copyright Cengage Learning All rights reserved Stokes Theorem Stokes Theorem can be regarded as a higher dimensional version of Green s Theorem 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 2 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 3 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 The positively oriented boundary curve of the oriented surface S is often written as S so Stokes Theorem can be expressed as curl F dS F dr 4 Stokes Theorem There is an analogy among Stokes Theorem Green s Theorem and the Fundamental Theorem of Calculus As before there is an integral involving derivatives on the left side of Equation 1 recall that curl F is a sort of derivative of F and the right side involves the values of F only on the boundary of S 5 Stokes Theorem In fact in the special case where the surface S is flat and lies in the xy plane with upward orientation the unit normal 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 6 Example 1 Evaluate C F dr where F x y z y2 i x j z2 k and C is the curve of intersection of the plane y z 2 and the cylinder x2 y2 1 Orient C to be counterclockwise when viewed from above Solution The curve C an ellipse is shown in Figure 3 Figure 3 7 Example 1 Solution cont d Although C F dr could be evaluated directly it s easier to use Stokes Theorem We first compute Although there are many surfaces with boundary C the most convenient choice is the elliptical region S in the plane y z 2 that is bounded by C 8 Example 1 Solution cont d If we orient S upward then C has the induced positive orientation The projection D of S onto the xy plane is the disk x2 y2 1 and so using equation with z g x y 2 y we have 9 Example 1 Solution cont d 10 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 now use Stokes Theorem to throw some light on the meaning of the curl vector 11 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 12 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 a C v dr 0 positive circulation b C v dr 0 negative circulation Figure 5 13 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 14 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 15 Stokes Theorem We know that F is conservative if C F dr 0 for every closed path C Given C suppose we can find an orientable surface S whose boundary is C Then Stokes Theorem gives A curve that is not simple can be broken into a number of simple curves and the integrals around these simple curves are all 0 Adding these integrals we obtain C F dr 0 for any closed curve C 16 MIDTERM 3 REVIEW 15 4 Double Integrals in Polar Coordinates Area element dA in polar coordinates Converting double integrals from rectangular coordinates to polar coordinates Evaluating double integrals in polar coordinates 17 MIDTERM 3 REVIEW 15 5 Applications of double integrals Volume under a surface the graph of a function of two variables Area of region R Average value of function over a region R Applications to physics computing mass and electrical charge given mass or charge density function center of mass centroid etc see the list of computational tasks below for details 18 MIDTERM 3 REVIEW 15 7 15 8 and 15 9 Triple integrals Triple integrals Application to computing volume mass electric charge moments center of mass centroid moments of inertia etc 19 MIDTERM 3 REVIEW 15 7 15 8 and 15 9 Triple integrals Cylindrical coordinates Spherical coordinates Simple surfaces in cylindrical and spherical coordinates Converting equations from rectangular to cylindrical or spherical coordinates and vice versa Computing triple integrals in cylindrical and spherical coordinates 20 MIDTERM 3 REVIEW 16 4 Green s theorem Evaluate line integrals using Green s theorem and double integrals or vice versa Find area using Green s theorem and line integrals Green s Theorem for regions with holes Reduce line integrals over complicated curves to regions using Green s theorem 21 MIDTERM 3 REVIEW 16 5 Divergence Gradient and Curl Divergence div gradient grad and curl of a vector field Identities for curl grad f and div curl F Curl test for a conservative vector field 3D version of mixed partials test Vector forms of Green s Theorem tangential version involving vertical component of curl and normal version involving divergence 22 MIDTERM 3 REVIEW 16 6 Parametric Surfaces Surfaces defined by parametrization r u v Surfaces that are the graph of a function z f x y …
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