How to Derive Green's Function for the Laplacian in 3D?

Join the discussion
Ask a follow-up here, or get your own question answered by working scientists, mathematicians and engineers — people, not an autocomplete.
Real named experts · corrections over time · the nuance an AI answer skips
Sign up Log in
1 reply · 5K views
centry57
Messages
8
Reaction score
0
Here are some pages of Arfken's “Mathematical Methods for Physicists ”
I don't How to work out the Green's function!
attachment.php?attachmentid=21046&stc=1&d=1255190271.png

attachment.php?attachmentid=21049&stc=1&d=1255190488.png

attachment.php?attachmentid=21048&stc=1&d=1255190271.png

Can anyone explain (9.174)and(9.175) for me ?
I'm hoping for your help, Thank you !
 

Attachments

  • 1.png
    1.png
    36.4 KB · Views: 1,782
  • 3.png
    3.png
    45.1 KB · Views: 1,484
  • 2.png
    2.png
    48.3 KB · Views: 826
Physics news on Phys.org
This is about finding the Green's function for the Laplacian operator under infinite boundary condition. Basically, 9.173 gives the relation between the Green's function and the unit excitation. In 3D, the unit excitation is assumed located at the center of a sphere. If you write out the gradient operator in 9.173 in the spherical coordinate on the radial component, and perform surface integration, you will get 9.174. 9.175 is the indefinite integral of 9.174.

Haberman's book on PDE Section9.5.6 has the explicit derivation.
 

Similar threads

Undergrad Green's function for 2-D Laplacian within square/rectangular boundary
  • Swamp Thing
  • · Replies 5 ·
  • Differential Equations
Replies
5
Graduate Green's function for Stokes equation
  • steve1763
  • · Replies 1 ·
  • Differential Equations
Replies
1
Undergrad Green's function for the wave equation
  • dRic2
  • · Replies 4 ·
  • Differential Equations
Replies
4
Graduate How Do You Construct the Chain Rule for Green's Function in Half-Space?
  • Shackleford
  • · Replies 1 ·
  • Differential Equations
Replies
1
Graduate How Do You Derive Green's Function Using Vector Calculus?
  • yungman
  • · Replies 5 ·
  • Calculus
Replies
5
Calculating exciton Green's function on a tight binding chain
  • physicsxanime
  • · Replies 1 ·
  • Advanced Physics Homework Help
Replies
1
Graduate Understanding the method of Green's function
  • alivedude
  • · Replies 2 ·
  • Differential Equations
Replies
2
Graduate Assuming separability when solving for a Green's Function
  • Exp HP
  • · Replies 6 ·
  • Differential Equations
Replies
6
Graduate Normal derivative of Green's function on a disk.
  • yungman
  • · Replies 8 ·
  • Differential Equations
Replies
8
Graduate Green's function in electrostatics
  • amjad-sh
  • · Replies 6 ·
  • Electromagnetism
Replies
6