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So then he wrote KE=lineintegral from 0-S, d/dt(gamma*mv)ds

and then from there you end up with gamma*mc^2-mc^2 after a whole bunch of math, which you take to mean the total energy - the "rest" energy? So...shouldn't up there you write change in KE=lineintegral etc...? If it IS just KE, I dunno why, since the line integral definition is for work anyways...well that may be a little confusing in just words, sorry