Help with Newtonian Potentials for Helmholtz Decomposition

Click For Summary
SUMMARY

The discussion focuses on the application of Newtonian potentials within the context of Helmholtz decomposition to derive a divergenceless vector field from its curl. Participants confirm that the dimension referred to in the Newtonian potential article indeed corresponds to the number of spatial dimensions. The formula for the Newtonian potential in three dimensions is clarified as 1/(4π|x|), addressing concerns about the negative exponent and its implications for the problem at hand.

PREREQUISITES
  • Understanding of Helmholtz decomposition
  • Familiarity with Newtonian potentials
  • Knowledge of vector calculus
  • Basic concepts of divergenceless vector fields
NEXT STEPS
  • Study the mathematical derivation of Helmholtz decomposition
  • Explore the properties of Newtonian potentials in various dimensions
  • Learn about divergenceless vector fields and their applications
  • Investigate the implications of negative exponents in physical equations
USEFUL FOR

Physicists, mathematicians, and engineering students interested in vector calculus and potential theory, particularly those working on fluid dynamics or electromagnetic fields.

Savant13
Messages
85
Reaction score
1
I'm trying to find a divergenceless vector field based on its curl, and discovered that I could use a http://en.wikipedia.org/wiki/Helmholtz_decomposition" , and the article I found on this didn't make much sense to me.

First, can someone confirm that the dimension referred to in the Newtonian potential article is the number of spatial dimensions in which the field exists? If not, what is it?

Next, the exponent on what appears to be the absolute value of x for more than two dimensions can only be negative, which doesn't make sense to me in the context of the problem I am working on. Also, is that an absolute value, or some other notation?
 
Last edited by a moderator:
Physics news on Phys.org
Yes, d is the number of spatial dimensions. If d= 3 then the "Newtonian potential" is (-1/(3(2-3)(4pi/3))|x|-1= 1/(4pi|x|). Why would that "not make sense"? What problem are you working on?
 
See the thread in the classical physics forum. I started this one because no one was responding to the other one.
 

Similar threads

Potential flows and Helmholtz decomposition
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Newtonian potential in Helmholtz decomposition
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Pressure and Newtonian potential energy
  • · Replies 37 ·
2
Replies
37
Views
4K
Graduate Explore Potential Energy, Pressure & Newtonian Gravity
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Kinematic Decomposition for "Rod and Hole" Relativity Paradox
  • · Replies 78 ·
3
Replies
78
Views
6K
Undergrad Does potential energy curve spacetime?
  • · Replies 28 ·
Replies
28
Views
2K
Helmholtz potential for mixture of simple ideal gases
  • · Replies 9 ·
Replies
9
Views
3K
Graduate How Does the Four-Way Vector Operation Relate to Potential Energy in Gravity?
  • · Replies 5 ·
Replies
5
Views
2K
Calculating eletric potential using line integral of electric field
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Accelerations: Transformation of Newton potential
  • · Replies 21 ·
Replies
21
Views
2K