2-dimension polar basis vectors

In summary, the conversation discusses the definition of unit polar basis vectors and how they relate to Cartesian coordinates. It is explained that the basis vectors should be contravariant and can be obtained from the unit vectors. It is also noted that the coordinate basis vectors are not necessarily unit vectors and should be considered covariant.
  • #1
epovo
114
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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?
 
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  • #2
epovo said:
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?
Yes. This is correct. The coordinate basis vectors are not (necessarily) unit vectors. Also, they should be considered covariant. That is,
[tex]d\vec{r}=\vec{e}_rdr+\vec{e}_θdθ[/tex]
where dr and dθ are considered contravariant.

Chet
 
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1. What are 2-dimension polar basis vectors?

2-dimension polar basis vectors are a set of two vectors that are used to describe the position of a point in a 2-dimensional space. They are typically represented as r and θ, where r represents the distance from the origin and θ represents the angle between the vector and a reference line.

2. How are 2-dimension polar basis vectors different from Cartesian coordinates?

Cartesian coordinates use two perpendicular axes, x and y, to describe the position of a point in a 2-dimensional plane. In contrast, 2-dimension polar basis vectors use a distance and angle to describe the position of a point from the origin.

3. How are 2-dimension polar basis vectors converted to Cartesian coordinates?

To convert 2-dimension polar basis vectors to Cartesian coordinates, we use the following formulas:
x = rcos(θ) and y = rsin(θ). These formulas use trigonometric functions to calculate the x and y coordinates of a point based on its distance and angle from the origin.

4. What is the purpose of using 2-dimension polar basis vectors?

2-dimension polar basis vectors are often used in physics and engineering to describe the position and movement of objects in circular or rotational motion. They are also useful for representing complex numbers in mathematics.

5. Can 2-dimension polar basis vectors be used in higher dimensions?

Yes, 2-dimension polar basis vectors can be extended to higher dimensions, such as 3-dimensional spherical coordinates. In these cases, additional variables such as φ are used to represent the angles in each dimension.

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