= Solution
Parametrize the smooth path by $\gamma:[0,1]\to G$. The <chain rule> turns the <line integral> of the <gradient> into a one-variable derivative:
$$
\int_\gamma\mathbf F\cdot d\mathbf x
=\int_0^1\nabla\phi(\gamma(t))\cdot\gamma'(t)\,dt
=\int_0^1\frac{d}{dt}\phi(\gamma(t))\,dt
=\boxed{\phi(\mathbf b)-\phi(\mathbf a)}.
$$
This is the <fundamental theorem for line integrals>; in particular the integral has <path independence>.
For a twice continuously differentiable <potential of a conservative vector field>, the <curl> of its <gradient> vanishes because mixed <partial derivatives> commute:
$$
(\nabla\times\nabla\phi)_i=\epsilon_{ijk}\partial_j\partial_k\phi=0.
$$
\b[A smooth <gradient> field is necessarily <curl>-free.]
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