 Fundamental theorem for line integrals

For a continuously differentiable function $\Phi$ and a piecewise smooth parametric curve $\gamma$ from $a$ to $b$,The chain rule converts the integrand into $d(\Phi (\gamma (t)))/dt$, then the fundamental theorem of calculus supplies the endpoint difference. This is why a conservative vector field has a path independent line integral. For $\Phi (x,y)=x^2+y^2$, every path from $(0,0)$ to $(1,2)$ gives $5$, provided it lies in the domain where the potential is defined.