Recall that a flow line, c(t), of a vector field F has c'(t)=F(c(t)) at all times t. Show all work below.
a.) Let c(t) be the flow line of a particle moving in a conservative force field F=-grad(f), where f:R^3->R, f(x,y,z) >=0 for all (x,y,z), represents the potential energy at each point in space. Prove that the particle will always move towards a point with lower potential energy. What is the limit of F(c(t)) as t goes to infinity?
The Attempt at a Solution
We are trying to show that f(c(t)) is a decreasing function of t. So far, I have that
grad[f(c(t))]=grad(f(c(t))*c'(t) thru chain rule
grad[f(c(t))]=grad(f(c(t))*F(c(t)) because c(t) is a flow line
grad[f(c(t))]=grad(f(c(t))*-grad(f(c(t)) by the definition of F
But I am not sure if this really helps at all. Is this the right direction to go in, or am I completely off base?