▁Alternatively ▁and ▁equival ently , ▁in ▁this ▁case ▁every ▁equilibrium ▁load ▁on ▁the ▁framework ▁may ▁be ▁resolved ▁by ▁a ▁stress ▁that ▁generates ▁an ▁equal ▁and ▁opposite ▁set ▁of ▁forces , ▁and ▁the ▁framework ▁is ▁said ▁to ▁be ▁stat ically ▁rig id . ▁ ▁Kin emat ics ▁If ▁the ▁vertices ▁of ▁a ▁framework ▁are ▁in ▁a ▁motion , ▁then ▁that ▁motion ▁may ▁be ▁described ▁over ▁small ▁scales ▁of ▁distance ▁by ▁its ▁gradient , ▁a ▁vector ▁for ▁each ▁vertex ▁specifying ▁its ▁speed ▁and ▁direction . ▁The ▁gradient ▁describes ▁a ▁linear ized ▁approximation ▁to ▁the ▁actual ▁motion ▁of ▁the ▁points , ▁in ▁which ▁each ▁point ▁moves ▁at ▁constant ▁velocity ▁in ▁a ▁straight ▁line . ▁The ▁gradient ▁may ▁be ▁described ▁as ▁a ▁row ▁vector ▁that ▁has ▁one ▁real ▁number ▁coordinate ▁for ▁each ▁pair ▁ ▁where ▁ ▁is ▁a ▁vertex ▁of ▁the ▁framework ▁and ▁ ▁is ▁the ▁index ▁of ▁one ▁of ▁the ▁Cart esian ▁coordinates ▁of ▁- dimensional ▁space ; ▁that ▁is , ▁the ▁dimension ▁of ▁the ▁gradient ▁is ▁the ▁same ▁as ▁the ▁width ▁of ▁the ▁rig id ity ▁matrix . ▁ ▁If ▁the ▁edges ▁of ▁the ▁framework ▁are ▁assumed ▁to ▁be ▁rig id ▁bars ▁that ▁can ▁neither ▁expand ▁nor ▁contract ▁( but ▁can ▁freely ▁rotate ) ▁then ▁any ▁motion ▁respect ing ▁this ▁rig id ity ▁must ▁preserve ▁the ▁lengths ▁of ▁the ▁edges : ▁the ▁derivative ▁of ▁length , ▁as ▁a ▁function ▁of ▁the ▁time ▁over ▁which ▁the ▁motion ▁occurs , ▁must ▁remain ▁zero . ▁This ▁condition ▁may ▁be ▁expressed ▁in ▁linear ▁algebra ▁as ▁a ▁constraint ▁that ▁the