Angular Components of Polar coordinate vectors

Question

I’m a third year physics student who was relaxing on vacation during spring break, and now am having a crisis because I realized I somehow never came across this problem before.

A vector in polar coordinates can be written as:

$$r \cos\theta \hat{r} + r \sin\theta \hat{\theta}.$$

Position vectors can be simplified to just

$$r \hat{r},$$

as there is no angular component.

My confusion is two-fold, firstly the term "angular component"—how does a vector which starts from the origin and points radially to a set point, have an angular component? What exactly is the formula $$r \cos\theta \hat{r} + r \sin \theta \hat{\theta}$$ getting at? When I convert this vector to Cartesian I get $$r \cos(2\theta)\hat{x} + r \sin(2\theta)\hat{y},$$ which, if you define a $$\theta' = 2\theta,$$ is just the standard $$x \hat{x} + y \hat{y}$$, which is just a position vector, which we already know is represented in polar coordinates as $$r \hat{r}$$. So then what was the point of defining the polar form with an angular component (whatever that means)?

Answer 1

A vector in polar coordinates is expressed as a linear combination of its basis: $a\mathbf e_r + b\mathbf e_{\theta}$. Of course, the basis is a function of the coordinates. For example, we can define: $\mathbf e_r = (cos(\theta), sin(\theta))$, and $\mathbf e_{\theta} = r(-sin(\theta), cos(\theta))$.

It is a convenient definition because the conversion from the Cartesian basis is: $$\mathbf e_r = \frac{\partial x}{\partial r}\mathbf e_x + \frac{\partial y}{\partial r}\mathbf e_y $$ $$\mathbf e_{\theta} = \frac{\partial x}{\partial \theta}\mathbf e_x + \frac{\partial y}{\partial \theta}\mathbf e_y $$

And the inverse conversion from polar to Cartesian basis is: $$\mathbf e_x = \frac{\partial r}{\partial x}\mathbf e_r + \frac{\partial \theta}{\partial x}\mathbf e_{\theta} $$ $$\mathbf e_y = \frac{\partial r}{\partial y}\mathbf e_r + \frac{\partial \theta}{\partial y}\mathbf e_{\theta} $$

Answer 2

Some confusion is understandable, considering the usual sloppiness in the introductions of such topics to physics students.

Polar coordinates are useful coordinates to represent points in the 2D plane. As well as other systems of coordinates, they correspond to a double set of parametrized curves with an orthogonal mutual intersection at each point. Cartesian coordinates use two orthogonal families of parallel lines. Polar coordinates instead use a family of concentric circles and a bundle of half-lines starting from the common center of the family of circles. Other systems of orthogonal coordinates exist based on different families of curves.

Therefore, a generic point of the plane has two coordinates: the two parameters selecting the two curves that intersect at that point. In cartesian coordinates, we choose the orthogonal lines ($x,y$); in polar coordinates, a circle ($r$) and a half-line ($\theta$).

There are two important things to notice:

1. there could be one or more points where the one-to-one association between parameters and points fails. In cartesian coordinates, it never happens. In polar coordinates, the origin corresponds to $r=0$ and any value of $\theta$ between $0$ and $2\pi$. Such a problem can be managed by introducing more than one set of polar coordinates with different origins. In the technical language of differential geometry, we can use more than one chart to ensure a one-to-one correspondence between points and two real parameters.
2. In general, the two coordinates are not the components of a vector even though the Euclidean plane can be equipped with a vector space structure with the choice of an origin.

Point (2) is worth some elaboration. Once we choose a point as the origin, we can associate the oriented segment starting from the origin and ending at that point with any point of the plane. It is easy to prove that the set of oriented segments is a real vector space of dimension two. As such, we can choose any two non-aligned vectors (i.e., oriented segments starting from the origin) as a vector basis. However, since both basis vectors are segments starting from the origin, which is a singular point for the polar coordinates, they are both radial vectors, and there is no tangential vector (a tangent to a circle of zero radius is an ill-defined concept).

In other words, the 2D position can be well described by polar coordinates, but the pair of radial and tangential unit vectors cannot be used as a basis because there is no tangential vector at the origin.

The situation is entirely different if we deal with vector fields in the plane. At variance with the positions that can be mapped to the vector space of oriented segments starting from the origin, vector fields imply the presence of a vector space in correspondence with each 2D point. Therefore, at each point but the origin, we have well-defined unit vectors tangent to the pair of coordinate curves crossing at that point. Consequently, we can introduce at each point a different basis (${\bf e}_r, {\bf e}_{\theta}$).

Summarizing,

• in general, positions in the plane can be described by polar coordinates;
• polar coordinates are not projections of a vector onto basis vectors
• polar coordinates introduce at each point, but the origin, two orthogonal unit vectors ${\bf e}_r$ and $ {\bf e}_{\theta}$, tangent to the coordinate curves;
• the vector space of positions referred to the origin of coordinates cannot have a basis containing a vector tangent to a degenerate coordinate curve (no $ {\bf e}_{\theta}$ for positions);
• Two-dimensional vector fields can be projected into a radial and tangential component at any point except the origin.