Thepurposeistoreestablishthecoupledconservationlaws,thelocalconservationequationsandthejumpconditionsofmassandinertiaforpolarcontinuumtheories.Inthisconnectionthenewmaterialderivativesofthedeformationgradient,thelineelement,thesurfaceelementandthevolumeelementwerederivedandthegeneralizedReynoldstransporttheoremwaspresented.Combiningtheseconservationlawsofmassandinertiawiththebalancelawsofmomentum,angularmomentumandenergyderivedinourpreviouspapersofthisseries,arathercompletesystemofcoupledbasiclawsandprinciplesforpolarcontinuumtheoriesisconstitutedonthewhole.Fromthissystemthecouplednonlocalbalanceequationsofmass,inertia,momentum,angularmomentumandenergymaybeobtainedbytheusuallocalization.
