Vector multiply that is NOT dot or cross?

[ognik]

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

 

[jedishrfu]

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.

 

[ognik]

These are all vectors, an example identity is
∇x(FXG) = F(∇.G) - G(∇.F) + (G.∇)F - (F.∇)G

 

[jedishrfu]

Just dot them together:

http://academics.smcvt.edu/jellis-monaghan/calc3/in%20class%20maple%20demos/graddivcurl1.pdf

Check out Example 6.6

 

[HallsofIvy]

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

 

[ognik]

Nice explanation thanks hallsofivy, I could see dotting them was the only way to get anything done, but its nice to understand why.

 

[mnb96]

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.