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Suppose I am working in the polar coordinate $(r,\theta)$ system with unit vectors $\hat r,\hat \theta$ and wanted to evaluate the integral $$\int_C \theta \; \mathrm d\hat\theta$$ circular path $C$ (e.g. quarter circle).

I understand that, using coordinate transformations, I could express the d$\hat \theta $ in terms of the cartesian basis which are constant.

However, how could we perform this integral without changing coordinate systems.

My guess would be to express $\theta$ in terms of $\hat \theta$, but I do not now how to go about that without effectively ending up in cartesian coordinates again.

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  • $\begingroup$ What is the semicircular path? $\endgroup$ Jan 12, 2022 at 0:53
  • $\begingroup$ is $\hat{\theta}$ just $\theta/|\theta|$? It seems like your integral is $\int_0^\pi \theta/|\theta| d \theta$. Is that right? $\endgroup$ Jan 12, 2022 at 0:55
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    $\begingroup$ @AlfredYerger $\hat{\theta}$ is the unit vector $-\sin\theta \hat{x} + \cos\theta \hat{y}$ $\endgroup$ Jan 12, 2022 at 0:56
  • $\begingroup$ @AlfredYerger The unit vector is $\hat\theta = -\sin\theta \hat x + \cos\theta \hat y$. My problem is that the only way I can think of defining it is in terms of cartesian coordinates, and I want to perform this integral in cylindrical coordinates. $\endgroup$
    – user256872
    Jan 12, 2022 at 0:58
  • $\begingroup$ Typically the way people go about this sort of thing is they parametrize the curve and then do the integral over the parametrizing interval. $\endgroup$ Jan 12, 2022 at 1:02

1 Answer 1

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We have that

$$\int_C \theta \:d\hat{\theta} = \int_\alpha^\beta -\theta\:\hat{r}\:d\theta$$

Now we can use a justified integration by parts

$$= \theta \:\hat{\theta}\Bigr|_\alpha^\beta - \int_\alpha^\beta\hat{\theta}\:d\theta = \boxed{\beta\:\hat{\theta}(\beta)-\alpha\:\hat{\theta}(\alpha)-\hat{r}(\beta)+\hat{r}(\alpha)}$$

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