Prove Radial Fields are conservative and at the origin.

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SUMMARY

The radial field F = r / |r|^p is proven to be conservative in regions that do not include the origin. The condition for F to remain conservative when the origin is included depends on the value of p. Specifically, the discussion indicates that for p = 1, where r is treated as a unit vector, the field maintains its conservative nature even at the origin. The key equation used in the analysis is v x F = 0, which confirms the conservation property.

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  • Understanding of vector calculus, specifically conservative fields
  • Familiarity with the concept of radial fields
  • Knowledge of the cross product and its implications in vector fields
  • Basic principles of mathematical proofs in physics
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  • Research the conditions under which vector fields are conservative
  • Study the implications of the cross product in vector calculus
  • Explore the concept of singularities in vector fields
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Prove Radial Fields are conservative... and at the origin.

1. Homework Statement

Prove the radial field F = r / |r|^p is conservative on any region not containing the origin. For what values of p is F conservative on a region that contains the origin?

2. Homework Equations

v x F = 0 (for conservation)

3. The Attempt at a Solution

I've proved the first part, but I don't know how to solve for p's such that F is conservative on a region that CONTAINS the origin.
 
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What about if r was a unit vector (p=1)
 

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