Why is the dipole field formula coordinate-free if it uses r hat and theta hat?

[raddian]

I am "continuing this thread" in hopes of asking questions that deal with the meaning of the question. https://www.physicsforums.com/threa...dipole-moment-in-coordinate-free-form.359973/
1. Homework Statement
Show that the electric field of a "pure" dipole can be written in the coordinate-free form
$$E_{dip}(r)=\frac{1}{4\pi\epsilon_0}\frac{1}{r^3}[3(\vec p\cdot \hat r)\hat r-\vec p].$$

Homework Equations

$$E_{dip}(r)=\frac{p}{4\pi\epsilon_0r^3}(2\cos \hat r+\sin\theta \hat \theta)$$

The Attempt at a Solution

I am trying to understand what "coordinate free" means. If the answer is in terms of r hat and theta hat, doesn't that contradict "coordinate free"? AND i would get $$ p = pcos(\theta) \hat r - psin(\theta) \hat \theta $$. Why doesn't p depend on PHI? If it's coordinate free why are we restricting our coordinates to r and theta??

 

[blue_leaf77]

"Coordinate free" means you don't need to define the coordinate system to write your equation. $\hat{r}$ is a unit vector from the center of the dipole to the observation point, so given the orientation of $\mathbf{p}$ in space, the relative direction of $\hat{r}$ with respect to $\mathbf{p}$ will automatically follow.