Proof of Principle of Conservation of Energy in a Conservative Vector Field

If
$\mathbf{F}$
is a conservative force field then there exists a function
$\phi(x,y,z)$
such that
$\mathbf{F}= - \mathbf{\nabla} \phi$
.br> Take two points
$A,B$

Newton's Second Law gives
$\mathbf{F} = m \frac{d^2 \mathbf{r}}{dt^2} \rightarrow \mathbf{F} \cdot \frac{d \mathbf{r}}{dt} =m \frac{d^2 \mathbf{r}}{dt^2} \cdot \frac{d \mathbf{r}}{dt}$

We can rewrite this as
$\mathbf{F} \cdot \frac{d \mathbf{r}}{dt} =m\frac{1}{2} \frac{d}{dt} ((\frac{d \mathbf{r}}{dt})^2)= \frac{1}{2} \frac{d (v^2)}{dt}$

Now integrate
$\int^B_A \mathbf{F} \cdot d \mathbf{r} = \int^B_A d(\frac{1}{2} mv^2 ) \rightarrow \phi_B - \phi_A =\frac{1}{2} m (V_B^2 - v_A^2 )$
.
This equation can be rearranged to give
$\phi_A + \frac{1}{2}mv_A^2 = \phi_B + \frac{1}{2}mv_B^2$