Suppose one wants to take the surface integral of outward flux $\mathbf{F(x)}$ where $\mathbf{x}=[x_1,x_2,x_3]$ across a spherical surface.
$$\int\int_S \mathbf{F(x)}\cdot \mathbf{\hat{n}} \:\:dS$$
Say the spherical surface is given by $x_1^2+x_2^2+x_3^2=1$.
Since the surface is a sphere, then
$$\mathbf{\hat{n}}=[x_1,x_2,x_3]$$
(Right?)
So the surface integral can be rewritten
$$\int\int_S \mathbf{F(x)}\cdot \mathbf{x} \:\:dS$$
But say just before tackling this problem one learns that the quantity given by $\mathbf{F(x)}\cdot \mathbf{x}$ is known to be constant ($c$). So the surface integral can be rewritten again as
$$c\int\int_S \:dS$$
Since the surface was said to be a sphere at the outset, a novice in these matters might be tempted to say this evaluates to
$$c\int\int_S \:dS=c4\pi r^2$$
But another novice might say not so fast, since all information about the surface (in particular, the surface normal) has been factored out of the integral.
Which of the two novices would be right (or closer to being right)?
If the second novice was closer to the mark, then what does $\int\int_S \:dS$ evaluate to? Or is there something fundamentally ill posed about the notion of a flux which satisfies $\mathbf{F(x)}\cdot \mathbf{x}=c$?