PX275 - B1b - cylindrical polar basis vectors

note: only ${\vec{e}}_{z}$ is always in the same direction, ${\vec{e}}_{ρ}$ and ${\vec{e}}_{ϕ}$ change directions wrt the axes
it is only in cartesian coordinates that the basis vectors hold their positions for all points in space
the position vectors, eg: $\vec{r}$ , should only be defined in the cartesian space
- eg: at a point, $P_{1} : \vec{r} = x \hat{i} + y \hat{j} + x \hat{k} = ρ \cos ϕ \hat{i} + ρ \sin ϕ \hat{j} + z \hat{k}$
the new basis vectors are related to $\hat{i}$ , $\hat{j}$ , $\hat{k}$ by taking the derivative of $\vec{r}$ wrt $ρ, ϕ, z :$

\begin{aligned} \frac{\partial \vec{r}}{\partial z} = {\vec{e}}_{z} & = \hat{k} \\ \frac{\partial \vec{r}}{\partial ρ} = {\vec{e}}_{ρ} & = \cos ϕ \hat{i} + \sin ϕ \hat{j} \\ \frac{\partial \vec{r}}{\partial ϕ} = {\vec{e}}_{ϕ} & = - ρ \sin ϕ \hat{i} + ρ \cos ϕ \hat{j} \end{aligned}

$| {\vec{e}}_{z |} = | {\vec{e}}_{ρ} | = 1 ⟹$ they are unit vectors, ie: ${\hat{e}}_{z} = {\vec{e}}_{z}$ and ${\hat{e}}_{ρ} = {\vec{e}}_{ρ}$
but, $| {\vec{e}}_{ϕ} | = ρ$ , so, ${\hat{e}}_{ϕ} = \frac{{\vec{e}}_{ϕ}}{ρ}$
note:
- $\vec{r} \neq ρ {\hat{e}}_{ρ} + ϕ {\hat{e}}_{ϕ} + z {\hat{e}}_{z}$ , as $ρ, ϕ$ are not fixed wrt the axes
- it can be written as: $\vec{r} = ρ {\hat{e}}_{ρ} (ϕ) + z {\hat{e}}_{z}$
- a vector field at $P$ will typically have components along the three directions: $\vec{F} = F_{x} \hat{i} + F_{y} \hat{j} + F_{z} \hat{k} = F_{ρ} {\hat{e}}_{ρ} + F_{ϕ} {\hat{e}}_{ϕ} + F_{z} {\hat{e}}_{z}$