E structures

... they were created. The short descriptions provided are general; therefore, we recommend further consideration of the structures to determine which ones are appropriate for your use. Sales ... here a list of the available information structures which are within the “SAP name space” structures numbered between S001 and S500. We list the structures according to the application area ... System E 4 Quality Management Information System E 4 Plant Maintenance Information System E 5 Retail Information System E 5 Where to Learn More E 5 E Appendix E: LIS Information Structures...

... LLCphysicalandtechnologicalinterestaswell;theclassofsurfacesforwhichana-lyticsolutionstothepotentialtheoryproblem(ofsolvingLaplace’sequation)canbefoundisthusconsiderablyenlarged,beyondthewell-establishedclassofsolutionsobtainedbyseparationofvariables(see,forexample[54]).Sinceitwillbecentraltolaterdevelopments,Sections1.1and1.2brieflydescribetheformofLaplace’sequationinsomeoftheseorthogonalcoordinatesys-tems,andthesolutionsgeneratedbytheclassicalmethodofseparationofvariables.TheformulationofpotentialtheoryforstructureswithedgesisexpoundedinSection1.3.Fortheclassofsurfacesdescribedabove,dual(ormultiple)seriesequationsarisenaturally,asdodual(ormultiple)integralequations.VariousmethodsforsolvingsuchdualseriesequationsaredescribedinSec-tion1.4,includingtheAbelintegraltransformmethodthatisthekeytoolemployedthroughoutthistext.ItexploitsfeaturesofAbel’sintegralequation(describedinSection1.5)andAbel-typeintegralrepresentationsofLegendrepolynomials,Jacobipolynomials,andrelatedhypergeometricfunctions(de-scribedinSection1.6).InthefinalSection(1.7),theequivalenceofthedualseriesapproachandthemoreusualintegralequationapproach(employingsingle-ordouble-layersurfacedensities)topotentialtheoryisdemonstrated.1.1Laplace’sequationincurvilinearcoordinatesThestudyofLaplace’sequationinvariouscoordinatesystemshasalonghistory,generating,amongstotheraspects,manyofthespecialfunctionsofappliedmathematicsandphysics(Besselfunctions,Legendrefunctions,etc.).Inthissectionwegathermaterialofareferencenature;foragreaterdepthofdetail,werefertheinterestedreadertooneofthenumeroustextswrittenonthesetopics,suchas[44],[32]or[74].HereweconsiderLaplace’sequationinthosecoordinatesystemsthatwillbeofconcreteinterestlaterinthisbook;inthesesystemsthemethodofseparationofvariablesisapplicable.Letu1,u2,andu3beasystemofcoor-dinatesinwhichthecoordinatesurfacesu1=constant,u2=constant,andu3=constantaremutuallyorthogonal(i .e. ,intersectorthogonally).Fixapoint(u1,u2,u3)andconsidertheelementaryparallelepipedformedalongthecoordinatesurfaces,asshowninFigure1.1.Thus ... LLCphysicalandtechnologicalinterestaswell;theclassofsurfacesforwhichana-lyticsolutionstothepotentialtheoryproblem(ofsolvingLaplace’sequation)canbefoundisthusconsiderablyenlarged,beyondthewell-establishedclassofsolutionsobtainedbyseparationofvariables(see,forexample[54]).Sinceitwillbecentraltolaterdevelopments,Sections1.1and1.2brieflydescribetheformofLaplace’sequationinsomeoftheseorthogonalcoordinatesys-tems,andthesolutionsgeneratedbytheclassicalmethodofseparationofvariables.TheformulationofpotentialtheoryforstructureswithedgesisexpoundedinSection1.3.Fortheclassofsurfacesdescribedabove,dual(ormultiple)seriesequationsarisenaturally,asdodual(ormultiple)integralequations.VariousmethodsforsolvingsuchdualseriesequationsaredescribedinSec-tion1.4,includingtheAbelintegraltransformmethodthatisthekeytoolemployedthroughoutthistext.ItexploitsfeaturesofAbel’sintegralequation(describedinSection1.5)andAbel-typeintegralrepresentationsofLegendrepolynomials,Jacobipolynomials,andrelatedhypergeometricfunctions(de-scribedinSection1.6).InthefinalSection(1.7),theequivalenceofthedualseriesapproachandthemoreusualintegralequationapproach(employingsingle-ordouble-layersurfacedensities)topotentialtheoryisdemonstrated.1.1Laplace’sequationincurvilinearcoordinatesThestudyofLaplace’sequationinvariouscoordinatesystemshasalonghistory,generating,amongstotheraspects,manyofthespecialfunctionsofappliedmathematicsandphysics(Besselfunctions,Legendrefunctions,etc.).Inthissectionwegathermaterialofareferencenature;foragreaterdepthofdetail,werefertheinterestedreadertooneofthenumeroustextswrittenonthesetopics,suchas[44],[32]or[74].HereweconsiderLaplace’sequationinthosecoordinatesystemsthatwillbeofconcreteinterestlaterinthisbook;inthesesystemsthemethodofseparationofvariablesisapplicable.Letu1,u2,andu3beasystemofcoor-dinatesinwhichthecoordinatesurfacesu1=constant,u2=constant,andu3=constantaremutuallyorthogonal(i .e. ,intersectorthogonally).Fixapoint(u1,u2,u3)andconsidertheelementaryparallelepipedformedalongthecoordinatesurfaces,asshowninFigure1.1.Thus ... LLCAsalreadyindicated,theboundaryconditionsmustbesupplementedbyade-cayconditionatinfinityaswellasfiniteenergyconstraintsnearedges,sothatauniqueandphysicallyrelevantsolutioncanbefound.Sinceedgesintroducedistinctivefeaturesintothetheory,letusdistinguishbetweenclosedsurfaces,thosepossessingnoboundaryoredge,andopenshells,whichhaveoneormoreboundaries.Asphericalsurfaceisclosed,whilstthehemisphericalshellisopenwithacircularboundary.Amoresophisti-cateddistinctioncanbeformulatedintopologicalterms,butthisisunneces-saryforourpurposes.Thesmoothnessofthesurface,includingthepresenceofsingularitiessuchascornersorconicaltips,isimportantinconsideringtheexistenceanduniquenessofsolutions.ThistopichasbeenextensivelyinvestigatedbyKellogg[32].However,thesurfacesunderinvestigationinthisbookareportionsofcoordinatesurfacesasdescribedintheIntroduction,andboththesurfacesandboundingcurvesareanalyticorpiecewiseanalytic.Thesmoothnessconditions,whichmustbeimposedontheclosedoropensurfacesinamoregeneralformulationofpotentialtheory,areautomaticallysatisfiedandwillbeomittedfromfurtherdiscussionexceptfortwocases,theconicalshellsconsideredinChapter6,andthetwo-dimensionalaxially-slottedcylin-dersofarbitrarycross-sectionalprofileconsideredinSection7.5;appropriatesmoothnessconditionsareconsideredintherespectivesections.Thissectionoutlinesgenericaspectsofpotentialtheoryapplicabletobothopen...

... Cu(NO3)2 + nước.b. MnO2 + HCl → MnCl2 + Cl2 + H2O c. Fe2O3 + HCl → FeCl3 + H2Od. FeSO4 + H2SO4 + KMnO4 → Fe2(SO4)3 + MnSO4 + K2SO4 + H2OCâu 15 : Cho phản ... Fe3+ trong 16g Fe2O3 thành Fe.a. 0,3 b. 0,6 c. 0,45 d. 0,9.Câu 11 : Trong các ion sau , Ion nào chỉ đong vai trò oxi hóa trong cácPƯ ôxihóa-khử.a. Cl- b. S2 C. Fe2+ d. Cu2+Câu ... lái xe tác dụng vừa hết với 35 ml dung dịch K2Cr2O7 0,06M. Tính % khối lượng cồn trong huyết thanh.a. 0,017% b. 0,17% c. 0,034% d. 0,34%Câu 10 : Cân bao nhiêu mol e đẻ khử toàn bộ Fe3+...

...