Proper velocity by Hermann Minkowski's flat-space metric equation . Here a reference map frame of yardsticks and synchronized clocks define map position x and map time "t" respectively, and the "d" preceding a coordinate means infinitesimal change. A bit of manipulation allows one to show that proper velocity where as usual coordinate velocity . Thus finite "w" ensures that "v" is less than lightspeed "c". By grouping "γ" with v in the expression for relativistic momentum p, proper velocity also extends the Newtonian form of momentum as mass times velocity to high speeds without a need for relativistic mass. The proper velocity
Cross product formula_123 is its velocity and formula_124 is the body's angular velocity. Since position formula_110 and velocity formula_123 are "true" vectors, the angular velocity formula_124 is a "pseudovector" or "axial vector". The cross product is used to describe the Lorentz force experienced by a moving electric charge formula_128: Since velocity formula_123, force formula_118 and electric field formula_132 are all "true" vectors, the magnetic field formula_133 is a "pseudovector". In vector calculus, the cross product is used to define the formula for the vector operator curl. The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in