When describing surface integrals in vector fields, it's common to use the notation $$\iint_S \vec{F} \cdot \text{d} \vec{S}$$ as a shorthand for $$\iint_S \vec{F} \cdot \vec{n}\, \text{d}S$$ This seems to be analogous to the notation for line integrals: $$\int_C \vec{F} \cdot \text{d}\vec{r} = \int_C \vec{F} \cdot \vec{T} \,\text{d}s$$ However, I don't understand why it makes sense to shorten $\vec{n} \,\text{d}S$ to $\text{d}\vec{S}$.
With line integrals, it makes sense to have $\vec{T} \,\text{d}s = \text{d}\vec{r}$ because $\text{d}\vec{r}$ notationally represents an infinitesimal movement along the curve described by $\vec{r}$. But with surface integrals, it doesn't seem like $\text{d}\vec{S}$ represents movement along the surface because the vector is normal to the surface.
At first I assumed that it was just a weird notation that resulted from the analogy to line integrals. However, I found a resource from MIT (http://math.mit.edu/~jorloff/suppnotes/suppnotes02/v9.pdf) that called the $\text{d}\vec{S}$ notation "suggestive". Of what is that notation suggestive?
Generally a surface integral is answering the informal question of, "How much "stuff" is flowing through the surface?" That's why it makes sense to dot with the normal vector to the surface, because just as $\text{d}\vec{r}$ represents movement along the curve, $\text{d}\vec{S}$ represents movement through the surface. See flux, or the divergence theorem.