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Vector multiply that is NOT dot or cross?

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  1. Jan 21, 2015 #1
    Hi - just working through my text (studying by correspondence) on Del operator - so Curl, div etc. Came across some identities parts of which which have me confused. what does it mean when a vector is shown as multiplying something - but without dot or cross? For example F(∇.G) or ∇(F.G) or (G.∇)F ...

    I get that something like (G.∇) expands to each component of G times each component of ∇ - which is a scalar; also ∇.G is a normal dot product. So I understand f.(∇.G) and ∇.(F.G) and (G.∇).F and but am confused when the 'dot' outside the bracket is missing - how do we multiply those?

    Thanks
    Alan
     
  2. jcsd
  3. Jan 21, 2015 #2

    jedishrfu

    Staff: Mentor

    Are you seeing the gradient of a scalar function? That is represented by a del operator and a capital letter which you might think is a vector.
     
  4. Jan 21, 2015 #3
    These are all vectors, an example identity is
    ∇x(FXG) = F(∇.G) - G(∇.F) + (G.∇)F - (F.∇)G
     
  5. Jan 22, 2015 #4

    jedishrfu

    Staff: Mentor

  6. Jan 22, 2015 #5

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    ∇.G is a scalar function so F(∇.G) is "scalar multiplication"- each component of F multiplied by ∇.G

    F.G is a scalar function so ∇(F.G) is the gradient of F.G

    This is the same as G(∇.F)


     
  7. Jan 22, 2015 #6
    Nice explanation thanks hallsofivy, I could see dotting them was the only way to get anything done, but its nice to understand why.
     
  8. Feb 3, 2015 #7
    I'm not sure this is what you're looking for, but you might want to have a look at the definition of "geometric product" in Clifford algebras, plus the concept of "geometric derivative" proposed by D. Hestenes, which generalizes div,grad,curl operators.
     
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