Proving the Vector Calculus Identity: (1/g^2)(g∇f - f∇g)

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Discussion Overview

The discussion revolves around proving the vector calculus identity related to the gradient of a quotient of functions, specifically the expression (1/g^2)(g∇f - f∇g). The scope includes mathematical reasoning and exploration of vector calculus identities.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant seeks a proof for the identity involving the gradient of a quotient of functions.
  • Another participant suggests that the identity resembles a vector version of the derivative of a quotient and recommends examining each component separately.
  • A third participant references the quotient rule for ordinary derivatives, indicating that it applies to partial derivatives as well.
  • A later reply expresses gratitude and indicates that the participant has successfully understood the proof.

Areas of Agreement / Disagreement

The discussion does not present any explicit disagreements, but it does not establish a consensus on the proof itself, as the final understanding is subjective to the participant's acknowledgment.

Contextual Notes

There may be limitations in the assumptions made regarding the functions involved and the applicability of the quotient rule to vector calculus.

Rubik
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I am trying to figure out a proof for this identity

[itex]\nabla[/itex](f/g) = (1/g2) (g[itex]\nabla[/itex]f - f[itex]\nabla[/itex]g)

Any ideas?
 
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It looks like a vector version of the derivative of a quotient. Look at each component separately.
 
The quotient rule for ordinary derivatives:

(f/g)' = (gf' - fg')/g2

works for partial derivatives, too.
 
Thanks so much I got it :)
 

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