Laplacian of a vector function

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

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 Aθ is. Any hint on how to get started would help.
 

Answers and Replies

  • #2
2,797
602
You're vector field has only a r component, other components are zero!
[itex]
\vec A=A_r \hat e_r +A_\theta \hat e_\theta+A_\varphi \hat e_\varphi
[/itex]
 
  • #3
You're vector field has only a r component, other components are zero!
[itex]
\vec A=A_r \hat e_r +A_\theta \hat e_\theta+A_\varphi \hat e_\varphi
[/itex]

So it would just be ∇2A = ∇2Ar - 2(Ar/r2)?
 
  • #4
2,797
602
No. The Laplacian of a vector field, is a vector field. What you wrote, is only the r component of that vector field.
 
  • #5
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 = (∇2Ar - 2(A2/r2))er? Or am I missing the other components?
 
  • #7
That's correct.

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

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