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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

  • $\begingroup$ 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 $$ \int_0^1F(S(t))\cdot \vec n(t)\,dt $$ $\endgroup$
    – Arthur
    Jun 11, 2020 at 11:31
  • $\begingroup$ 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 $\endgroup$ Jun 11, 2020 at 11:43


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