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Components of Curl, Divergence

  1. Feb 2, 2013 #1
    1. The problem statement, all variables and given/known data
    I'm trying to understand where the Cartesian components of the rotor and the divergence of a vector field derived.

    I read that the divergence of a vector field is defined by:
    [itex]\vec { \nabla } \cdot \vec { F } =\lim _{ V\rightarrow \left\{ P \right\} }{ \frac { 1 }{ \left| V \right| } \iint _{ \partial V }^{ }{ \vec { F } \cdot \vec { dA } } } [/itex]

    Also, that components of the rotor are defined by:
    [itex]\vec { \nabla } \times \vec { F } \cdot \hat { u } =\lim _{ A(\Gamma )\rightarrow 0 }{ \frac { 1 }{ \left| A(\Gamma ) \right| } \int _{ \Gamma }^{ }{ \vec { F } \cdot \vec { dr } } } [/itex]

    I'm trying to understand where the standard "sum of partial derivatives", and the mnemonic determinant with a row of unit vectors come from. I don't see a correlation between the definitions and these simple representations. How are they derived from the definition?

    2. Relevant equations
    [itex]\vec { \nabla } \times \vec { F } \cdot \hat { u } =\lim _{ A(\Gamma )\rightarrow 0 }{ \frac { 1 }{ \left| A(\Gamma ) \right| } \int _{ \Gamma }^{ }{ \vec { F } \cdot \vec { dr } } } [/itex]

    [itex]\vec { \nabla } \cdot \vec { F } =\lim _{ V\rightarrow \left\{ P \right\} }{ \frac { 1 }{ \left| V \right| } \iint _{ \partial V }^{ }{ \vec { F } \cdot \vec { dA } } } [/itex]

    3. The attempt at a solution
    No clue
     
  2. jcsd
  3. Feb 2, 2013 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    You can choose a cube as volume (if the general limit exists, it is the same for a special type of volume), and get the coordinate representation in the limit.
    In the same way, you can use a square as path.
     
  4. Feb 2, 2013 #3

    Mark44

    Staff: Mentor

    This is obviously not a precalculus question, so I am moving it to the appropriate section.
     
  5. Feb 2, 2013 #4
    My bad, didn't double check. Sorry
     
  6. Feb 2, 2013 #5
    Can you please be a bit more specific?
    I divided my field into orthogonal component functions, and i'm trying to look at the circulation of the field over a square with sides parallel to the axis. In order to get my result i'd have to approximate the field as constant over each side.. Why can I do that?
     
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