Basis vectors are defined as $\vec {E_i}=\vec{E_i}(x_1,x_2,x_3)=\frac{\partial\vec{r}}{\partial x^i}$ 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 $\hat {r}=\frac{\partial\vec{R}}{\partial r}=\hat {r}$
But $\hat {\theta}=\frac{\partial\vec{R}}{\partial \theta}=r\hat {\theta}$
Similarly $\hat {\phi}=\frac{\partial\vec{R}}{\partial \phi}=r\hat {\phi}$
Where am I wrong? Why is r appearing?
On
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 $\vec{R}=x\hat{x}+y\hat{y}+z\hat{z}$ then we are used to seeing
$\frac{\partial\vec{R}}{\partial x}=\hat{x}$ 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
$\frac{\partial\vec{R}}{\partial \theta}=r\frac{\partial\vec{r}}{\partial \theta}=r \hat{\theta}$
$\Rightarrow \hat{\theta} =\frac{1}{r}\frac{\partial\vec{R}}{\partial \theta}$
and similarly for $\phi$:
$\frac{\partial\vec{R}}{\partial \phi}=r\frac{\partial\vec{r}}{\partial \phi}=r\sin \theta \hat{\phi}$
$\Rightarrow \hat{\phi} =\frac{1}{r \sin \theta}\frac{\partial\vec{R}}{\partial \phi}$
You're missing the scale factors
$$ \hat{e}_\alpha = \frac{1}{h^\alpha}\frac{\partial {\bf x}}{\partial u^\alpha} $$
As an example
$$ {\bf x} = \pmatrix{r\sin\theta\cos\phi \\ r\sin\theta\sin\phi \\ r \cos\theta} $$
So
$$ \frac{\partial {\bf x}}{\partial \theta} = \frac{\partial}{\partial \theta}\pmatrix{r\sin\theta\cos\phi \\ r\sin\theta\sin\phi \\ r \cos\theta} = \pmatrix{r\cos\theta\sin\phi \\ r\cos\theta\cos\phi \\ -r\sin\theta} $$
The scale factor is just
$$ h^2 = [r^2\cos^2\theta(\sin^2\phi + \cos^2\phi) + r^2\sin^2\theta]^{1/2} = r $$
So the unit vector is
$$ \hat{e}_2 = \pmatrix{\cos\theta\sin\phi \\ \cos\theta\cos\phi \\ -\sin\theta} $$