▁each ▁of ▁their ▁indices ▁and ▁are ▁used ▁in ▁continu um ▁mechan ics ▁to ▁transform ▁between ▁reference ▁(" material ") ▁and ▁present ▁(" configuration ") ▁coordinates . ▁Ex amples ▁include ▁the ▁de formation ▁gradient ▁and ▁the ▁first ▁Pi ola – K ir ch hoff ▁stress ▁tensor . ▁ ▁As ▁with ▁many ▁applications ▁of ▁tens ors , ▁Ein stein ▁summ ation ▁notation ▁is ▁frequently ▁used . ▁To ▁clarify ▁this ▁notation , ▁capital ▁indices ▁are ▁often ▁used ▁to ▁indicate ▁reference ▁coordinates ▁and ▁lower case ▁for ▁present ▁coordinates . ▁Thus , ▁a ▁two - point ▁tensor ▁will ▁have ▁one ▁capital ▁and ▁one ▁lower - case ▁index ; ▁for ▁example , ▁A j M . ▁ ▁Cont inu um ▁mechan ics ▁A ▁conventional ▁tensor ▁can ▁be ▁viewed ▁as ▁a ▁transformation ▁of ▁vectors ▁in ▁one ▁coordinate ▁system ▁to ▁other ▁vectors ▁in ▁the ▁same ▁coordinate ▁system . ▁In ▁contrast , ▁a ▁two - point ▁tensor ▁transform s ▁vectors ▁from ▁one ▁coordinate ▁system ▁to ▁another . ▁That ▁is , ▁a ▁conventional ▁tensor , ▁ ▁, ▁act ively ▁transform s ▁a ▁vector ▁u ▁to ▁a ▁vector ▁v ▁such ▁that ▁▁ ▁where ▁v ▁and ▁u ▁are ▁measured ▁in ▁the ▁same ▁space ▁and ▁their ▁coordinates ▁representation ▁is ▁with ▁respect ▁to ▁the ▁same ▁basis ▁( den oted ▁by ▁the ▁" e "). ▁ ▁In ▁contrast , ▁a ▁two - point ▁tensor , ▁G ▁will ▁be ▁written ▁as ▁▁ ▁and ▁will ▁transform ▁a ▁vector , ▁U , ▁in ▁E ▁system ▁to ▁a ▁vector , ▁v , ▁in ▁the ▁e ▁system ▁ ▁as ▁▁▁▁ ▁. ▁ ▁The