Radial component of del^2 in spherical coordinates?

[philip041]

I'm doing a question on a 3D isotropic harmonic oscillator. At one point I need to find write the radial component of del^2.

The lecturer has written 1/r^2 * d/dr * (r^2 * d/dr)

I don't understand cause it looks like he hasn't actually changed anything, r^2 over r^2 ?

 

[Vid]

In Latex, with parenthesis, the expression might be more clear:

$$\frac{1}{r^{2}} \frac{d}{dr} \left( r^{2} \frac{d}{dr} \right)$$

You're taking the derivative of r^2*d/dr so the r^2 won't just cancel. Also, del^2 isn't a vector so there aren't components. I guess a more correct way would be the radial term.

 

[philip041]

Ok, I'm really confused.. why isn't the radial part of del^2 in spherical polar coords just dr^2?