Let F be a conservative vector field on R3 with F = V. If a particle travels along a

Chapter 6, Problem 39

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Let F be a conservative vector field on R3 with F = V. If a particle travels along a path x, recall that its potential energy at time t is defined to be V(x(t)). Use line integrals to prove the law of conservation of energy: As a particle of mass m moves between any two points A and B in a conservative force field, the sum of the potential and kinetic energies of the particle remains constant. (Use Exercise 38 and Theorem 3.3.) The use of line integrals provides an alternative proof of Theorem 4.2 in 4.4.