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