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I have a question regarding the boundries in stoke's theorem. Stokes theorem states:

$$\oint_C \vec{F}\cdot d\vec{r}={\int\int}_S (\nabla \times \vec{F}) d\vec{S}$$

As far as I understand it relates the flux of the curl of a vector field $\vec{F}$ through a surface $S$ to a closed line integral along the boundry curve of $S$ called $C$. What I don't understand is: How do you choose a suitable path $C$ to integrate along? For example, consider the following surface:

enter image description here

Which path encloses the surface? Would it be the bottom "circle" or would it be some other curve?

Or suppose we have a cylinder with no top or bottom: enter image description here

Does it matter if I integrate along the top or the bottom "circle"?

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The curve needs to be the entire boundary. If there is more than one component to that boundary, you need to integrate over all of the pieces, each one suitably oriented. (In the case of the cylinder, oriented with its normal pointing outward, you orient the bottom circle counterclockwise and the top circle clockwise.)

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  • $\begingroup$ Ah I see. So in the case of the open cone (example 1) the boundry consists of the bottom circle and the top circe. Suppose I had some distorted cube with holes on each side, I would have to integrate along the closed path of all the holes right? $\endgroup$
    – qmd
    Dec 16 '19 at 22:29
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    $\begingroup$ Yes, you have to integrate over all the curves forming the boundary. Alternatively, you can modify the surface by filling in some of them and then adjust the flux integral accordingly. $\endgroup$ Dec 16 '19 at 22:32
  • $\begingroup$ Makes perfect sense now. Thank you very much for your help! $\endgroup$
    – qmd
    Dec 16 '19 at 23:23
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    $\begingroup$ You're welcome. At some point, you might find some of the YouTube lectures linked in my profile of interest. $\endgroup$ Dec 16 '19 at 23:27

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