The surface is oriented by the shown normal vector moveable cyan arrow on surface , and the curve is oriented by the red arrow. More information about applet. Stokes' theorem allows us to do even more. Note that moving the green point on the top slider does not change the value of either integral in the above formulas.
The important point is that, even in this case, the left line integral and the right surface integral are always equal. There is one more subtlety that you have to get correct, or else you'll may be off by a sign. You need to orient the surface and boundary properly.
The cyan normal vector to the surface and the orientation of the curve shown by red arrow in the above applet are chosen with the proper relative orientations so that Stokes' theorem applies. You can read some examples here. By Figure ,. As a line integral, you can parameterize C by. Both integrals give.
Therefore, the theorem allows us to compute surface integrals or line integrals that would ordinarily be quite difficult by translating the line integral into a surface integral or vice versa. We now study some examples of each kind of translation. Calculate surface integral where S is the surface, oriented outward, in Figure and. Use of this equation requires a parameterization of S.
Surface S is complicated enough that it would be extremely difficult to find a parameterization. Therefore, the methods we have learned in previous sections are not useful for this problem. In Figure , we calculated a surface integral simply by using information about the boundary of the surface. In general, let and be smooth surfaces with the same boundary C and the same orientation.
In Figure , we could have calculated by calculating where is the disk enclosed by boundary curve C a much more simple surface with which to work. Figure shows that flux integrals of curl vector fields are surface independent in the same way that line integrals of gradient fields are path independent.
Recall that if F is a two-dimensional conservative vector field defined on a simply connected domain, is a potential function for F , and C is a curve in the domain of F , then depends only on the endpoints of C.
Therefore, the flux integral of G does not depend on the surface, only on the boundary of the surface. Flux integrals of vector fields that can be written as the curl of a vector field are surface independent in the same way that line integrals of vector fields that can be written as the gradient of a scalar function are path independent.
The boundary curve, C , is oriented clockwise. Parameterize the boundary of S and translate to a line integral. Calculate the line integral where and C is the boundary of the parallelogram with vertices and. To calculate the line integral directly, we need to parameterize each side of the parallelogram separately, calculate four separate line integrals, and add the result. This is not overly complicated, but it is time-consuming. Let S denote the surface of the parallelogram.
Note that S is the portion of the graph of for varying over the rectangular region with vertices and in the xy -plane. This triangle lies in plane. Recall that if C is a closed curve and F is a vector field defined on C , then the circulation of F around C is line integral If F represents the velocity field of a fluid in space, then the circulation measures the tendency of the fluid to move in the direction of C.
Let F be a continuous vector field and let be a small disk of radius r with center Figure. If is small enough, then for all points P in because the curl is continuous. The quantity is constant, and therefore. This equation relates the curl of a vector field to the circulation. Since the area of the disk is this equation says we can view the curl in the limit as the circulation per unit area. Recall that if F is the velocity field of a fluid, then circulation is a measure of the tendency of the fluid to move around The reason for this is that is a component of F in the direction of T , and the closer the direction of F is to T , the larger the value of remember that if a and b are vectors and b is fixed, then the dot product is maximal when a points in the same direction as b.
Therefore, if F is the velocity field of a fluid, then is a measure of how the fluid rotates about axis N. The effect of the curl is largest about the axis that points in the direction of N , because in this case is as large as possible.
To see this effect in a more concrete fashion, imagine placing a tiny paddlewheel at point Figure. The paddlewheel achieves its maximum speed when the axis of the wheel points in the direction of curl F. Let C be a closed curve that models a thin wire. In the context of electric fields, the wire may be moving over time, so we write to represent the wire.
At a given time t , curve may be different from original curve C because of the movement of the wire, but we assume that is a closed curve for all times t. Let be a surface with as its boundary, and orient so that has positive orientation. Suppose that is in a magnetic field that can also change over time.
The equation of this plane is,. Okay, we now need to find a couple of quantities. Recall that this comes from the function of the surface. Finishing this out gives,. Notes Quick Nav Download.
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