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

  is a conservative force field then there exists a function  
  such that  
\[\mathbf{F}= - \mathbf{\nabla} \phi\]
.br> Take two points  

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  
\[\phi_A + \frac{1}{2}mv_A^2 = \phi_B + \frac{1}{2}mv_B^2\]