I'm in $\mathbb{R^3}$ so $i=1,2,3$
- The star is dotproduct.
My goal is to simplify this: $$(\hat{e}_i \cdot \nabla)\vec{r} = \frac{\partial \vec{r}}{\partial x_i}$$
And this above I believe is equal to $\hat{e}_i$?
My attempt: $$(\hat{e}_i \cdot \nabla)\vec{r} = (\hat{e}_x \cdot \nabla)\vec{r} + (\hat{e}_y \cdot \nabla)\vec{r} + (\hat{e}_z \cdot \nabla)\vec{r} = (1,0,0)+(0,1,0)+(0,0,1) = \hat{e_x}+\hat{e_y}+\hat{e_z}= \hat{e_i}$$
Actually, the notation $\hat e_i$ denotes one of the Cartesian unit vectors and not the sum as expressed in the OP. So, we have
$$\begin{align} \hat e_i\cdot \left(\nabla \vec r\right)&=\hat e_i\cdot \left(\sum_{j=1}^3\hat e_j\frac{\partial }{\partial x_j} \right)\vec r\\\\ &=\left(\sum_{j=1}^3(\hat e_i\cdot \hat e_j)\frac{\partial }{\partial x_j} \right)\vec r\\\\ &=\left(\sum_{j=1}^3(\delta_{ij})\frac{\partial }{\partial x_j} \right)\vec r\\\\ &=\frac{\partial \vec r}{\partial x_i} \end{align}$$