Differential element velocity vectors

To sum up, the basic idea of the Doppler-selected TOF technique is to cast the differential cross-section S ajdv3 in a Cartesian coordinate, and to combine three dispersion techniques with each independently applied along one of the three Cartesian axes. As both the Doppler-shift (vz) and ion velocity (vy) measurements are essentially in the center-of-mass frame, and the (i j-componcnl, associated with the center-of-mass velocity vector can be made small and be largely compensated for by a slight shift in the location of the slit, the measured quantity in the Doppler-selected TOF approach represents directly the center-of-mass differential cross-section in terms of per velocity volume element in a Cartesian coordinate, d3a/dvxdvydvz. As such, the transformation of the raw data to the desired doubly differential cross-section becomes exceedingly simple and direct, Eq. (11). 

The moving volume can be described in terms of surface, S, to which each point on it is given a velocity, v. That velocity is also defined over the spatial coordinates and time v(x, v, z, t). To complete the description of V in time, we need to not just describe its surface velocity but also the orientation of the surface at each point. This is done by defining a unit vector n which is always oriented outward from the volume and normal to the surface. This is illustrated in Figure 3.1, where AS represents a small surface element. As we shrink to the point (x,y,z), we represent AS as a differential element, d,S. ... 

Figure 6.13 Melting differential elements for a rotating screw a) melt film in Zone C (barrel velocity is zero), and b) melt film in Zone D with the screw moving with a velocity of and l/ . The red colored vectors are moving out of the plane of the page towards the upstream direction while the black vectors in the z direction are moving into the page and in the downstream direction...

We can arrive at this result from the consideration that the velocity defined by the components dxjdt, dyjdt, dzjdt (or the momentum obtained from this velocity on multiplication by the mass) cannot, in view of Lorentz s transformation, be regarded as a vector, since the differential dt in the denominator is also transformed. We obtain a serviceable covariant definition if we replace dt by dt, where dt is the element of the proper time of the particle, i.e. the time measured in the system of reference in which the particle is at rest. The relation between dt and dt is found by taking the derivative of t y... 

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