Laplacian of a vector function

[subzero0137]

Problem: The vector function A(r) is defined in spherical polar coordinates by A = (1/r) e_r

Evaluate ∇²A in spherical polar coordinates

Relevant equation: I'm assuming I have to use the equation 1671 on this website

But I haven't got a clue as to how I would apply it since, for example, I don't know what A_θ is. Any hint on how to get started would help.

 

[ShayanJ]

You're vector field has only a r component, other components are zero!
$\vec A=A_r \hat e_r +A_\theta \hat e_\theta+A_\varphi \hat e_\varphi$

 

[subzero0137]

You're vector field has only a r component, other components are zero!
$\vec A=A_r \hat e_r +A_\theta \hat e_\theta+A_\varphi \hat e_\varphi$

So it would just be ∇²A = ∇²A_r - 2(A_r/r²)?

 

[ShayanJ]

No. The Laplacian of a vector field, is a vector field. What you wrote, is only the r component of that vector field.

 

[subzero0137]

No. The Laplacian of a vector field, is a vector field. What you wrote, is only the r component of that vector field.

Right, so will it be ∇² A = (∇²A_r - 2(A₂/r²))e_r? Or am I missing the other components?

 

[ShayanJ]

That's correct.

 

[subzero0137]

That's correct.

EDIT: Oops, I meant the answer I got is (-2/r³)e_r