Exercise 11: Conservation of Energy
Suppose an object moves according to Newton's Law $F = m A$ where the force is given by the gradient of a scalar function $\phi(x, y, z)$, as
$m A[t] = -\nabla \phi[X[t]]$
Use the Dot Product Rule from Section 9.8 and the Chain Rule from Exercise 9 to prove that total energy is conserved, that is, if
$E[t] = \frac{1}{2} m |V[t]|^2 + \phi[X[t]]$, then $\frac{dE}{dt} = 0$
We use $X[t]$ for position, $V[t] = X'[t]$ for velocity, $v[t] = |V[t]|$ for speed, and $A[t] = V'[t]$ for acceleration.