We are stuck with this question here because I cannot understand the following results. I find it hard to visualize this, let alone deduce from that. How to do it?
Objective to Attack The closely Related Problems with Orthogonal Basis and Dot Products In Polar-coordinates
$\left(\hat{e}_{r}\partial_{r}\right) \cdot \left(\frac{1}{r}\hat{e}_{\theta}\partial_{\theta}\right)= 0$
$\left(\frac{1}{r}\hat{e}_{\theta}\partial_{\theta}\right) \cdot \left(\hat{e}_{r}\partial_{r}\right) = \frac{1}{r} \partial_r$
$\partial_\theta \hat e_r = \hat e_\theta$
$\partial_\theta \hat e_\theta = -\hat e_r$
Trials
- I have some errors there, related to 3-4 apparently.
$$ \left(\hat{e}{r}\partial{r}\right) \cdot \left(\frac{1}{r}\hat{e}{\theta}\partial{\theta}\right) =
\left(\hat{e}{r}\partial{r}\right) \cdot \frac{1}{r}+ \left(\hat{e}{r}\partial{r}\right) \cdot \left( \hat{e}{\theta}\partial{\theta} \right)
\not
\frac{-\hat{e}_r}{r^2}+
\left(\hat{e}{r}\cdot\hat{e}\theta \right) \partial_{r} \partial_{\theta}
$$
Perhaps Related