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An accurate mathematical definition of vortex vector and vortex are given，which is local，Galilean invariant,and unique.There is no other vortex vector orvortex mathematical definition.All others either wrong or inaccurateNot only the iso-surface of R can be used，vortex vector can also be used toshow the vortex structure including vortex vector,vortex Lines,vortex tube,vortex surface.Vortex lines are parallelpenetrate vortex structureto vortex structure unlike vorticity lines which canThere is not only vortex strength iso-surface which has same vortex strength，but also vortex strength change can be clearly shown along the vortex line orvortex structureIt is very convenient and fast to calculate vortex vector including the directionand strength。