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Could somebody please explain to me the bottom line here. I don't understand how dS becomes r dtheta. I thought dS was supposed to be an outward pointing normal which is surely just r? I'm guessing theres something obvious im missing here.

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user109331
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1 Answers1

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$dS$ is an infinitesimal surface element. In this case it pertains to the surface of the cylinder which has been reduced to a circle. In particular it points in the $\hat{r}$ direction but its magnitude must represent infinitesimal surface area. In this case its magnitude is $rd\theta$ which is exactly the infinitesimal arc length of a circle with radius $r$.

Alex R.
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