▁transformation ▁law ▁for ▁two - point ▁tensor ▁Suppose ▁we ▁have ▁two ▁coordinate ▁systems ▁one ▁prim ed ▁and ▁another ▁un prim ed ▁and ▁a ▁vectors ' ▁components ▁transform ▁between ▁them ▁as ▁ ▁. ▁For ▁tens ors ▁suppose ▁we ▁then ▁have ▁. ▁A ▁tensor ▁ ▁in ▁the ▁system ▁. ▁In ▁another ▁system , ▁let ▁the ▁same ▁tensor ▁be ▁given ▁by ▁ ▁. ▁We ▁can ▁say ▁. ▁Then ▁▁ ▁is ▁the ▁routine ▁tensor ▁transformation . ▁But ▁a ▁two - point ▁tensor ▁between ▁these ▁systems ▁is ▁just ▁▁ ▁which ▁transform s ▁as ▁ ▁. ▁ ▁The ▁most ▁m und ane ▁example ▁of ▁a ▁two - point ▁tensor ▁The ▁most ▁m und ane ▁example ▁of ▁a ▁two - point ▁tensor ▁is ▁the ▁transformation ▁tensor , ▁the ▁Q ▁in ▁the ▁above ▁discussion . ▁Note ▁that ▁ ▁. ▁Now , ▁writing ▁out ▁in ▁full , ▁ ▁and ▁also ▁. ▁This ▁then ▁requires ▁Q ▁to ▁be ▁of ▁the ▁form ▁ ▁. ▁By ▁definition ▁of ▁tensor ▁product , ▁ ▁So ▁we ▁can ▁write ▁▁ ▁Thus ▁▁ ▁In cor por ating ▁( ), ▁we ▁have ▁. ▁ ▁In ▁the ▁equation ▁following ▁( 1 ) ▁there ▁are ▁four ▁q ' s ▁! ? ▁ ▁See ▁also ▁ ▁M ixed ▁tensor ▁ ▁Cov ar iance ▁and ▁contra var iance ▁of ▁vectors ▁ ▁References ▁ ▁External ▁links ▁ ▁Mathemat ical ▁found ations ▁of ▁el astic ity ▁By ▁Jer ro ld ▁E . ▁Mars den , ▁Thomas ▁J . ▁R . ▁Hugh es ▁ ▁Two - point ▁T ens ors ▁at ▁i Me chan ica ▁ ▁Category