Why Can't Polar Basis Vectors Be Defined as Unit Vectors?

[epovo]

I'd like to understand why i cannot seem to be able to define unit polar basis vectors. Let me explain:

We have our usual polar coordinates relation to Cartesian:

x = r cosθ ; y = r sinθ

if I define $\hat{e_{r}}$, $\hat{e_{\vartheta}}$ as the polar basis vectors, then they should be contravariant, meaning that they can be obtained from $\hat{u_{x}}$, $\hat{u_{y}}$ as:

$\hat{e_{r}}$ = $\delta x/ \delta r\ \hat{u_{x}} + \delta y / \delta r \ \hat{u_{y}}$ = cosθ $\hat{u_{x}}$ + sin θ $\hat{u_{y}}$

and
$\hat{e_{\vartheta}}$ = $\delta x/ \delta \vartheta \ \hat{u_{x}} + \delta y / \delta \vartheta \ \hat{u_{y}}$ = -r sinθ $\hat{u_{x}}$ + r cosθ $\hat{u_{y}}$

which implies that |$\hat{e_{\vartheta}}$| = r, rather than being a unit vector as usually considered.

Is this right?

 

[Chestermiller]

I'd like to understand why i cannot seem to be able to define unit polar basis vectors. Let me explain:

We have our usual polar coordinates relation to Cartesian:

x = r cosθ ; y = r sinθ

if I define $\hat{e_{r}}$, $\hat{e_{\vartheta}}$ as the polar basis vectors, then they should be contravariant, meaning that they can be obtained from $\hat{u_{x}}$, $\hat{u_{y}}$ as:

$\hat{e_{r}}$ = $\delta x/ \delta r\ \hat{u_{x}} + \delta y / \delta r \ \hat{u_{y}}$ = cosθ $\hat{u_{x}}$ + sin θ $\hat{u_{y}}$

and
$\hat{e_{\vartheta}}$ = $\delta x/ \delta \vartheta \ \hat{u_{x}} + \delta y / \delta \vartheta \ \hat{u_{y}}$ = -r sinθ $\hat{u_{x}}$ + r cosθ $\hat{u_{y}}$

which implies that |$\hat{e_{\vartheta}}$| = r, rather than being a unit vector as usually considered.

Is this right?

Yes. This is correct. The coordinate basis vectors are not (necessarily) unit vectors. Also, they should be considered covariant. That is,
$$d\vec{r}=\vec{e}_rdr+\vec{e}_θdθ$$
where dr and dθ are considered contravariant.

Chet