Vector multiply that is NOT dot or cross?

  • #1
174
2

Main Question or Discussion Point

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
 

Answers and Replies

  • #2
11,479
5,004
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.
 
  • #3
174
2
These are all vectors, an example identity is
∇x(FXG) = F(∇.G) - G(∇.F) + (G.∇)F - (F.∇)G
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
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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)
∇.G is a scalar function so F(∇.G) is "scalar multiplication"- each component of F multiplied by ∇.G

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

or (G.∇)F ...0
This is the same as 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
 
  • #6
174
2
Nice explanation thanks hallsofivy, I could see dotting them was the only way to get anything done, but its nice to understand why.
 
  • #7
713
5
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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