# Can a flux that varies only in $x$ through surface $ds$ be separated into $dx$ and $dy$ terms

For a vector such as this, if $$ds$$ was vertical, it could be written as $$F(x)dy$$. However, as $$ds$$ varies with $$x$$, can it be written as a with respect to only $$F(x)$$, $$dx$$ and $$dy$$ terms? The general term is written as

$$\int^{ds}_0 F(x) dS$$ but that is where I'm stuck

• That's not exactly the general term. For instance, usually $F$ would depend on both $x$ and $y$. And also, you wouldn't integrate from $0$ to $ds$, that's too short to be an interesting integral. Finally, I think you want to integrate $F(x, y)\cdot \vec n$, where $\vec n$ is a normal vector to $S$, because that's what the flux density is at a point on $S$. So, assuming $S$ is a curve parametrised by $t\in [0, 1]$, the general form would look something like $$\int_0^1F(S(t))\cdot \vec n(t)\,dt$$ Jun 11, 2020 at 11:31
• If F varies in both directions, it can't be simplified. But if it only varies in the y direction, we can integrate in y from 0 to dy. I just wonder if it can be integrated if it varies only in the streamwise direction of the flux Jun 11, 2020 at 11:43