If x is a flow line of a gradient vector field F = ∇ f , show that the function G(t) = f (x(t)) is an increasing function of t. (Hint: Show that G’(t) is always nonnegative.) Thus, we see that a particle traveling along a flow line of the gradient field F = ∇ f will move from lower to higher values of the potential function f . That’s why physicists define a potential function of a gradient vector field F to be a function g such that F = −∇g (i.e., so that particles traveling along flow lines move from higher to lower values of g).
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