1. Problem

What is $\displaystyle (\vec{r} \cdot \nabla) \vec{r}$ ?

where

$\displaystyle \vec{r} = r \hat{r} = x \hat{x} + y \hat{y} + z \hat{z}$

$\displaystyle \nabla = \hat{x} \frac{d}{dx} + \hat{y} \frac{d}{dy} + \hat{z} \frac{d}{dz}$

2. Attempt at a Solution

First,

$\displaystyle (\vec{r} \cdot \nabla) = (x \frac{d}{dx} + y \frac{d}{dy} + z \frac{d}{dz})$

Then,

$\displaystyle (\vec{r} \cdot \nabla) \vec{r} = x \frac{d}{dx} x \hat{x} + y \frac{d}{dy} y \hat{y} + z \frac{d}{dz} z \hat{z}$

...since all other terms would vanish. This would reduce to,

$\displaystyle = x \hat{x} + y \hat{y} + z \hat{z} = \vec{r}$

Wow! Could that really be correct?