- #1
epovo
- 114
- 21
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 [itex]\hat{e_{r}}[/itex], [itex]\hat{e_{\vartheta}}[/itex] as the polar basis vectors, then they should be contravariant, meaning that they can be obtained from [itex]\hat{u_{x}}[/itex], [itex]\hat{u_{y}}[/itex] as:
[itex]\hat{e_{r}}[/itex] = [itex]\delta x/ \delta r\ \hat{u_{x}} + \delta y / \delta r \ \hat{u_{y}}[/itex] = cosθ [itex]\hat{u_{x}}[/itex] + sin θ [itex]\hat{u_{y}}[/itex]
and
[itex]\hat{e_{\vartheta}}[/itex] = [itex]\delta x/ \delta \vartheta \ \hat{u_{x}} + \delta y / \delta \vartheta \ \hat{u_{y}} [/itex] = -r sinθ [itex]\hat{u_{x}}[/itex] + r cosθ [itex]\hat{u_{y}}[/itex]
which implies that |[itex]\hat{e_{\vartheta}}[/itex]| = r, rather than being a unit vector as usually considered.
Is this right?
We have our usual polar coordinates relation to Cartesian:
x = r cosθ ; y = r sinθ
if I define [itex]\hat{e_{r}}[/itex], [itex]\hat{e_{\vartheta}}[/itex] as the polar basis vectors, then they should be contravariant, meaning that they can be obtained from [itex]\hat{u_{x}}[/itex], [itex]\hat{u_{y}}[/itex] as:
[itex]\hat{e_{r}}[/itex] = [itex]\delta x/ \delta r\ \hat{u_{x}} + \delta y / \delta r \ \hat{u_{y}}[/itex] = cosθ [itex]\hat{u_{x}}[/itex] + sin θ [itex]\hat{u_{y}}[/itex]
and
[itex]\hat{e_{\vartheta}}[/itex] = [itex]\delta x/ \delta \vartheta \ \hat{u_{x}} + \delta y / \delta \vartheta \ \hat{u_{y}} [/itex] = -r sinθ [itex]\hat{u_{x}}[/itex] + r cosθ [itex]\hat{u_{y}}[/itex]
which implies that |[itex]\hat{e_{\vartheta}}[/itex]| = r, rather than being a unit vector as usually considered.
Is this right?