Problem $3.$ In this problem, we aim to justify the term conservative for gradient fields. A gradient field F is called conservative because it conserves the total energy of a mass mas it moves within the field. Recall that the total energy is defined as: E $=$ K $+$ U $=12$m jvj$2+$ U; where K represents the kinetic energy, v the velocity of m$,$ and U the potential energy associated with F$.$ a$)$ Use the relation: F $=$ mr $00($t$)$; and show that W$,$ the work done by a force field F to move a mass from point x$0$ to x$1$ is equal to: W $=$ K$($x$1)$ K$($x$0)$: b$)$ Show that if F is conservative, that is$,$ if F $=$rU$,$ where U is the potential energy, then the work W is equal to: W $=$ U$($x$0)$ U$($x$1)$: and conclude that if F is conservative, then total energy E is constant.