lea, s m mathematics for physicists 2e solutions manual [1]外语英文电子书外语英文电子书

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1、Chapter1:Describingtheuniverse1.Circularmotion.AparticleismovingaroundacirclewithangularvelocityWriteitsvelocityvectorasavectorproductofandthepositionvectorwithrespecttothecenterofthecircle.Justifyyourexpression.Differentiateyourrelation,andhencederivetheangularformofNew

2、ton'ssecondlaw(fromthestandardform(equation1.8).Thedirectionofthevelocityisperpendiculartoandalsototheradiusvectorandisgivenbyputtingyourrightthumbalongthevector:yourfingersthencurlinthedirectionofthevelocity.ThespeedisThusthevectorrelationwewantis:Differentiating,weget:

3、sinceisperpendiculartoThesecondtermistheusualcentripetalterm.Thenandsinceisperpendiculartoandforaparticle2.Findtwovectors,eachperpendiculartothevectorandperpendiculartoeachother.Hint:Usedotandcrossproducts.Determinethetransformationmatrixthatallowsyoutotransformtoanewcoo

4、rdinatesystemwithaxisalongandandaxesalongyourothertwovectors.WecanfindavectorperpendiculartobyrequiringthatAvectorsatsifyingthisis:NowtofindthethirdvectorwechooseTofindthetransformationmatrix,firstwefindthemagnitudeofeachvectorandthecorrespondingunitvectors:andTheelement

5、softhetransformationmatrixaregivenbythedotproductsoftheunitvectorsalongtheoldandnewaxes(equation1.21)Tocheck,weevaluate:asrequired.Similarlyandfinally:3.Showthatthevectors(15,12,16),(-20,9,12)and(0,-4,3)aremutuallyorthogonalandrighthanded.Determinethetransformationmatrix

6、thattransformsfromtheoriginalcordinatesystem,toasystemwithaxisalongaxisalongandaxisalongApplythetransformationtofindcomponentsofthevectorsandintheprimesystem.DiscusstheresultforvectorTwovectorsareorthogonaliftheirdotproductiszero.andFinallySothevectorsaremutuallyorthogon

7、al.InadditionSothevectorsformaright-handedset.Tofindthetransformationmatrix,firstwefindthemagnitudeofeachvectorandthecorrespondingunitvectors.SoSimilarlyandTheelementsofthetransformationmatrixaregivenbythedotproductsoftheunitvectorsalongtheoldandnewaxes(equation1.21)Thus

8、thematrixis:Check:asrequired.Then:andSincethecomponentsofthevectorremainunchanged,thisvectormustliealon