Let there be a given mass , acted by a net force . If the mass travels a small distance , prove that the infinitesimal change in energy equals the work done by .
Note that are scalars while are vectors.
We begin by defining and , where is the velocity-vector.
In relativistic mechanics,
This long expression can be reduced to , which can be further reduced to .
Assembling the above results yield .
Check out my other notes at Proof, Disproof, and Derivation