General Proof of the Work-Energy Theorem Consider a particle that moves along a curved path in space from (x1, y1, z1) to (X2, Y2,Z2) At the initial point, the particle bas velocity it = v,i + v,, + vi. The path that the particle follow~ may be divided into infinitesimal segments d1 = dxi + djj + dzk. As the particle moves, it is acted on by a net force F = Fxi; + Fyj + Fzk. The force components F .. Fy , and F, are in general functions of position. By the same sequence of steps used in Eqs (6.11) through (6.13), prove the work-energy theorem for this general case. That is, prove that

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where K-Ihnalp.a-に(F.ds + r,dy + tot