Our physics prof wrote the following equation:
$\int\frac{\vec{r}}{r^3}d\vec{r} = \int\frac{1}{r^2}dr$
This is logical as long as I argue that $\vec{r}$ and $d\vec{r}$ are parallel, which is why the dot product evaluates as $|\vec{r}||d\vec{r}| = r dr$ However then i tried to do it by hand:
$\vec{r}d\vec{r} = \left(\begin{array}{c}x\\y\\z\\ \end{array}\right)\left(\begin{array}{c}dx\\dy\\dz\\ \end{array}\right) = xdx + ydy + zdz$
but this is nowhere near
$rdr = \sqrt{x^2+y^2+z^2}\sqrt{dx^2 + dy^2 + dz^2}$
which is why I would like to ask you what i am doing wrong.
Thanks in advance
ftiaronsem