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# 2313-4-3 - 58 C HAPTER 4 4.3 V ECTOR F IELDS IN T WO D...

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58 CHAPTER 4V ECTOR FIELDS IN TWO DIMENSIONS 4.3. G REEN S T HEOREM FOR F LUX AND D IVERGENCE In terms of our interpretation of vector ±elds as ²ows of material, outward ²ux measures the amount of material leaving a region, while divergence measures the pressure at points within the region. Because matter can be neither created nor destroyed, the only way to have an overall decrease in pressure (increase in divergence) within a region is for mate- rial to leave the region (positive ²ux). Therefore, we should expect that the ²ux of a ±eld over a region is equal to the double integral of divergence over the region. The following theorem, named for George Green 1 shows that this intuition is correct, not just for vector ±elds with physical interpretations, but in general for all well-behaved vector ±elds over well-behaved regions. The de±nition of “well-behaved region” is that it should have a piecewise smooth boundary, a term we’ve already encountered, and also be simply con- nected ,whichmeanstwothings : between every two points in the region, there is a path (in other words, the region isn’t split into two or more disconnected pieces), and every closed curve in the region encloses only other points inthereg ion( ino ther words, the region can’t contain a hole). For example, the region on the left, below, is simply connected, but the two other regions are not simply connected. With the hypotheses explained, we can now state Green’s Theorem. Green’s Theorem for Flux and Divergence. Let C be a piecewise smooth curve enclosing a simply-connected region R in the xy - plane. Let F P i Q j be a vector ±eld, where P and Q have continuous ±rst partial derivatives in an open region containing R . Then C F n ds C Pdy Qdx outward ²ux R div F dA R F dA integral of divergence . 1 George Green (1793–1841) published Green’s Theorem in his 1828 article An essay on the application of mathematical analysis to the theories of electricity and magnetism ,o fwh ichheso ld51cop ies . Uptoth ispo in t , Green had attended school for precisely one year, when he was 8, and worked maintaining his father’s mill. When his father died in 1829, Green sold the mill and used the proceeds to attend Cambridge, in 1833, at the age of 40.

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SECTION 4.3 GREEN’S THEOREM FOR FLUX AND DIVERGENCE 59 In short, outward fux (which is a line integral) is the double integral o± divergence (±or 2 -dimensional vector ²elds), when the hypotheses of Green’s Theorem hold .Be±oredemonstrat- ing the uses o± Green’s Theorem, we check that it holds ±or our running example ±rom the previous section. Example 1. Use Green’s Theorem to veri±y that the outward fux o± F xy 2 x i x 2 j on the disc o± radius a centered at the origin is 2 πa 2 (as computed in Example 1 o± Section 4.2).
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2313-4-3 - 58 C HAPTER 4 4.3 V ECTOR F IELDS IN T WO D...

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