# Evaluate line integral in polar coordinates system with changing basis

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.

• What is the semicircular path? Jan 12, 2022 at 0:53
• is $\hat{\theta}$ just $\theta/|\theta|$? It seems like your integral is $\int_0^\pi \theta/|\theta| d \theta$. Is that right? Jan 12, 2022 at 0:55
• @AlfredYerger $\hat{\theta}$ is the unit vector $-\sin\theta \hat{x} + \cos\theta \hat{y}$ Jan 12, 2022 at 0:56
• @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. Jan 12, 2022 at 0:58
• Typically the way people go about this sort of thing is they parametrize the curve and then do the integral over the parametrizing interval. Jan 12, 2022 at 1:02

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

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