1.B Appendix to Chapter 1

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1 Secon.B.B Append o Chper.B. The Ordnr Clcl Here re led ome mporn concep rom he ordnr clcl. The Dervve Conder ncon o one ndependen vrble. The dervve o dened b d d lm lm.b. where he ncremen n de o n ncremen The Derenl The derenl o dened b n. d.b. B conderng he pecl ce o one h d d o he derenl o he ndependen vrble eqvlen o he ncremen. d. Th n generl he derenl cn be wren d d. The derenl o nd ncremen n re onl ppromel eql d nd pproch one noher. Th llred n Fg..B.. lope derenl d ncremen d d Fgre.B.: he derenl I el ncon o noher vrble derenon gve hen he chn rle o Sold Mechnc Pr III Kell

2 Secon.B d d d.b.3 d Arc Lengh The lengh o n rc mered rom ed pon on he rc o rom he denon o he negrl d ec d d / d d.b.4 where he ngle he ngen o he rc mke wh he Fg.B. wh d / d n nd d d d d he lengh o he doed lne n Fg..B.b. Alo cn be een h lm pq pq chord rc lm lm.b.5 d d d d o h he ncremen mll. d q p d d b Fgre.B.: rc lengh The Clcl o Two or More Vrble Conder now wo ndependen vrble. We cn dene prl dervve o h or emple Sold Mechnc Pr III Kell

3 Secon.B lm lm.b.6 conn The ol derenl d de o ncremen n boh nd cn n h ce be hown o be whch wren d d / d d.b.7 / b eng d d. Agn he derenl d onl n ppromon o he cl ncremen he ncremen nd derenl re hown n Fg..B.3 or he ce d. I cn be hown h h epreon or he derenl d hold wheher nd re ndependen or wheher he re ncon hemelve o n ndependen vrble n whch ce one h he ol dervve o wh repec o d d d d.b.8 d d lope derenl d ncremen d Fgre.B.3: he prl dervve The Chn rle or Two or More Vrble Conder he ce where ncon o he wo vrble b lo. h nd re ncon o he wo ndependen vrble nd Then Sold Mechnc Pr III Kell

4 Secon.B Sold Mechnc Pr III Kell 3 d d d d d d d d d.b.9 B lo d d d.b. Comprng he wo nd nce d d re ndependen nd rbrr one obn he chn rle.b. In he pecl ce when nd re ncon o onl one vrble o h he bove redce o he ol dervve gven erler. One cn rher pecle: In he ce when ncon o nd wh one h d d d d.b. When ncon o one vrble onl o h he bove redce o he chn rle or ordnr derenon. Tlor Theorem Sppoe he vle o ncon known. I vle neghborng pon hen gven b.b.3

5 Secon.B The Men Vle Theorem I conno over n nervl b hen b.b.4 b Geomercll h eqvlen o ng h here e le one pon n he nervl or whch he ngen lne prllel o he lne onng nd b. Th rel known he men vle heorem. b b Fgre.B.4: he men vle heorem The lw o he men cn lo be wren n erm o n negrl: here le one pon n he nervl b ch h b d l.b.5 where l he lengh o he nervl l b. The rgh hnd de here cn be nerpreed he verge vle o over he nervl. The heorem hereore e h he verge vle o he ncon le omewhere n he nervl. The eqvlen epreon or doble negrl h here le one pon n regon R ch h d d.b.6 A R where A he re o he regon o negron R nd mlrl or rple/volme negrl. Sold Mechnc Pr III 4 Kell

6 Secon.B.B. Trnormon o Coordne Sem Le he coordne o pon n pce be 3. Inrodce econd e o coordne reled o he r e hrogh he rnormon eqon wh he nvere eqon 3 3.B.7 3 g.b.8 A rnormon ermed n dmble rnormon he nvere rnormon e nd n one-o-one correpondence n cern regon o he vrble 3 h ech e o nmber 3 dene nqe e 3 n he regon nd vce ver. Now ppoe h one h pon wh coordne whch.b.7. Eqn..B.7 wll be n generl non-lner b derenng led o d d.b.9 whch em o hree lner eqon. From bc lner lgebr h em cn be olved or he d nd onl he deermnn o he coecen doe no vnh.e J de.b. wh he prl dervve evled he one dmenonl on hown n Fg..B.5. I J one cn olve or he d : d A d.b.. Th lner ppromon o he nvere eqon.b.8 nd o he nvere e n mll regon ner 3. Th rgmen cn be eended o oher neghborng pon nd he regon n whch J wll be he regon or whch he rnormon wll be dmble. I he Jcobn pove everwhere hen rgh hnded e wll be rnormed no noher rgh hnded e nd he rnormon d o be proper. Sold Mechnc Pr III 5 Kell

7 Secon.B lner ppromon d Fgre.B.5: lner ppromon Sold Mechnc Pr III 6 Kell

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