(Solved): 3. Find a potential for the conservative vector field: \[ \mathbf{F}(x, y, z)=\left\langle 2 x y+2 ...

3. Find a potential for the conservative vector field: \[ \mathbf{F}(x, y, z)=\left\langle 2 x y+2 x, x^{2}-2 y z^{3},-6 z-3 z^{2} y^{2}\right\rangle \] and use your answer in order to compute the vector line integral \[ \int_{C} \mathbf{F} \cdot d \mathbf{r} \] where \( C \) is the oriented broken line starting at \( (0,0,0) \), continuing to \( (1,1,1) \) and ending at \( (2,1,1) \)