Htykuut

Basis vectors are defined as $vec {E_i}=vec{E_i}(x_1,x_2,x_3)=frac{partialvec{r}}{partial x^i}$ i=1,2,3.
In spherical coordinate system $x^1=r, x^2=theta, x^3=phi$

position vector $vec {R}=r hat{r}$.

That means $hat {r}=frac{partialvec{R}}{partial r}=hat {r}$

But $hat {theta}=frac{partialvec{R}}{partial theta}=rhat {theta}$

Similarly $hat {phi}=frac{partialvec{R}}{partial phi}=rhat {phi}$

Where am I wrong? Why is r appearing?

You're missing the scale factors

$$
hat{e}_alpha = frac{1}{h^alpha}frac{partial {bf x}}{partial u^alpha}
$$

As an example

$$
{bf x} = pmatrix{rsinthetacosphi \ rsinthetasinphi \ r costheta}
$$

So

$$
frac{partial {bf x}}{partial theta} = frac{partial}{partial theta}pmatrix{rsinthetacosphi \ rsinthetasinphi \ r costheta} = pmatrix{rcosthetasinphi \ rcosthetacosphi \ -rsintheta}
$$

The scale factor is just

$$
h^2 = [r^2cos^2theta(sin^2phi + cos^2phi) + r^2sin^2theta]^{1/2} = r
$$

So the unit vector is

$$
hat{e}_2 = pmatrix{costhetasinphi \ costhetacosphi \ -sintheta}
$$

In Cartesian co-ordinates the unit vectors parallel to the co-ordinate axes $hat{x}, hat{y}, hat{z}$ do not change with position. So if we have $vec{R}=xhat{x}+yhat{y}+zhat{z}$ then we are used to seeing

$frac{partialvec{R}}{partial x}=hat{x}$ etc.

However, in other co-ordinate systems this is not true. In particular, in spherical co-ordinates the direction of the $hat{r}$ vector changes depending on $theta$ and $phi$ - so we should more properly write $hat{r}$ as $hat{r}(theta, phi)$. So

$frac{partialvec{R}}{partial theta}=rfrac{partialvec{r}}{partial theta}=r hat{theta}$

$Rightarrow hat{theta} =frac{1}{r}frac{partialvec{R}}{partial theta}$

and similarly for $phi$:

$frac{partialvec{R}}{partial phi}=rfrac{partialvec{r}}{partial phi}=rsin theta hat{phi}$

$Rightarrow hat{phi} =frac{1}{r sin theta}frac{partialvec{R}}{partial phi}$