MATH 2242: Extra Credit # 1 – KEY
1. (Marsden 4.3.18) Let c(t) be a flow line of a gradient field F=−∇V. Prove that V(c(t)) is a decreasing
function of t.
Proof. On the one hand, c(t) being a flow line of F=−∇Vmeans by definition that
c0(t) = F(c(t)) = −∇V(c(t)).
On the other hand, to show that V(c(t)) is a decreasing function of time, we need only show that its total
time derivative is nonpositive (i.e. ≤0). Thus using the chain rule and the equation above, we get
d
dt[V(c(t))] = ∇V(c(t)) ·c0(t)
=−c0(t)·c0(t)
=−||c0(t)||2
≤0.
Just for fun, what’s the physical meaning of this result? Well, if a free particle is moving in a potential
field (i.e. F=−∇V) then it will flow “down” the potential hill. In fact, from what we know about the
gradient vector, we can say more: the particle moves in the direction to achieve the fastest decrease in
potential energy. Thus the trajectory of the particle is orthogonal to the level curves of V. Think about
this. Draw a picture. Savor the connection between mathematics and nature.
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