Curvilinear Coordinates and Vector Calculus

[limddavid]

Homework Statement

With $\vec{L}$ = -i$\vec{r}$ x $\nabla$, verify the operator identities

$\nabla = \hat{r}\frac{\partial }{\partial \vec{r}}-i\frac{\vec{r}\times\vec{L}}{r^{2}}$
and
$\vec{r} \bigtriangledown ^2 - \bigtriangledown (1+\vec{r}\frac{\partial }{\partial \vec{r}})=i\bigtriangledown \times \vec{L}$

Homework Equations

The Attempt at a Solution

... I tried to expand it, and use some identities... But the equation becomes super complicated... Help!

 

[dextercioby]

I don't see other way to do it other than expanding and writing the nabla operator in spherical coordinates...The only identity you need is the one for double vector product

$$\vec{A}\times \left(\vec{B}\times\vec{C}\right) = \ ...$$