Given $\nabla\varphi = \mathbf{F}$ and $\mathbf{r}(t)$ from $a$ to $b$,
$$\int_c\mathbf{F}\cdot\mathrm{d}\mathbf{r} = \int_c\nabla\varphi\cdot\mathrm{d}\mathbf{r} = \int_a^b\nabla\varphi(\mathbf{r})\cdot\dfrac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}~\mathrm{d}t = \int_a^b\dfrac{\mathrm{d}}{\mathrm{d}t}(\varphi(\mathbf{r}))~\mathrm{d}t = \varphi(\mathbf{r}(b))-\varphi(\mathbf{r}(a)).$$