Why is it that the surface integral of the flux of a vector field is the same as the surface integral of the vector field itself?

In other words, this:

Is this just a definition because what we really care about is how much the vectors are "pushing" through the surface? Or is it an actual equality?

1 Answer

This is just notation: $d\mathbf S = \mathbf n\,dS$. Both sides are computing the flux of $\mathbf F$, i.e., the net amount of $\mathbf F$ pushing outwards across the surface.