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Unformatted text preview: SECTION 4.4 GREEN’S THEOREM FOR CIRCULATION AND CURL 67 4.4. G REEN ’ S T HEOREM FOR C IRCULATION AND C URL While flux measures the amount of material leaving a region, circulation measures the amount of flow running in a counterclockwise direction along the boundary of the region. Our first goal for this section is to derive a line integral formula for circulation. This should remind the reader of our derivation of the formula for flux in Section 4.2, although the details in this case are a shade simpler. C IRCULATION Suppose we want to compute the circulation of the vector field F P i Q j along the boundary of the region R , which is parameterized, in counterclockwise orientation, by r t . r t T r t r t r t R We want to calculate the amount of material flowing along the boundary of R at each point r t on the boundary. At this point, the flow is given by the vector field F r t . For circulation, we are concerned in the amount of flow which is tangent to r t , which means that it is in the same direction as r t . Therefore the contribution to circulation at the point r t is the projection of F r t onto the unit tangent vector T r t r t , which is simply the dot product F T because T is a unit vector. The line integral (with respect to arc length) of F T gives us the total circulation along the boundary, Circulation C F T ds. As we did with flux, we now want to express this line integral in a way that is easier to deal with. Suppose that r t x t ,y t . Since ds x t ,y t dt, 68 CHAPTER 4 V ECTOR FIELDS IN TWO DIMENSIONS this will cancel with the denominator of T in the line integral above. Therefore we have that C F T ds C P,Q x t ,y t dt C Px t Qy t dt. By replacing x t dt with dx and y t dt with dy we arrive at the formula below. Circulation. Suppose that C is a piecewise smooth, simple, closed curve enclosing a region R in the plane and oriented in the counter- clockwise direction. Let F P i Q j be a vector field and suppose that P and Q have continuous first order partial derivatives. Then the circulation of F on R is C F T ds C P dx Qdy, where T is the unit tangent vector. C URL Using Green’s Theorem from the previous section, we see that Circulation C P dx Qdy C Qdy P dx R Q x P y dA, whenever the hypotheses about P , Q , R , and C are satisfied. Since the lefthand side of this equation has a physical interpretation (circulation), it is natural to ask what the quantity inside the double integral, Q x P y , measures. Adapting our approach from Section 4.2, it is easy to see that this quantity represents circulation per unit area (this is Exercise 1). We will call this quantity two-represents circulation per unit area (this is Exercise 1)....
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This note was uploaded on 12/10/2011 for the course MAC 2313 taught by Professor Keeran during the Fall '08 term at University of Florida.
- Fall '08