= Fundamental theorem for line integrals
{wiki}
For a continuously differentiable <function> $\Phi$ and a piecewise smooth <parametric curve> $\gamma$ from $a$ to $b$,
$$
\int_\gamma\nabla\Phi\cdot d\mathbf x=\Phi(b)-\Phi(a).
$$
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.
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