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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 tworepresents 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
 Keeran
 Calculus

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