Deriving cross product and dot product, stuck at beginning.
1. The problem statement, all variables and given/known data
Assuming that ∅ is a differentiable scalar valued function and F a differentiable vector field, derive the following identities. a)∇(dotted with)(∅F) = ∇∅(dotted with)F + ∅∇(dotted with)F b)∇(crossed with)(∅F) = ∇∅(crossed with)F + ∅∇(crossed with)F 2. Relevant equations 3. The attempt at a solution Honestly don't know where to start. 
Re: Deriving cross product and dot product, stuck at beginning.
Nevermind, delete this, I've got it, just didn't put the initial effort into it.

Re: Deriving cross product and dot product, stuck at beginning.
Would this be the correct derivation for part a)
So far all I see is: ∅F is the vector field ∅ = ∅(x,y,z) F = <P,Q,R> ∇(dotted with)F = x partial P + y partial Q + z partial R ∇∅ = <x partial ∅, y partial ∅, z partial ∅> ∅F = <∅P, ∅Q, ∅R> a)∇(dotted with)(∅F) = [∇∅](dotted with)F + ∅∇(dotted with)F ∇(dotted with)(∅F) = [∇∅](dotted with)F + ∅[∇(dotted with)F] ∇(dotted with)(∅F) = [∇∅](dotted with)F + ∅(∂/∂xP + ∂/∂yQ + ∂/∂zR) ∇(dotted with)(∅F) = [∇∅](dotted with)F + ∅(∂/∂xP + ∂/∂yQ + ∂/∂zR) ∇(dotted with)(∅F) = <∂/∂x∅, ∂/∂y∅, ∂/∂z∅>(dotted with)<P,Q,R> + ∅(∂/∂xP + ∂/∂yQ + ∂/∂zR) ∇(dotted with)(∅F) = <0,0,0>(dotted with)<P,Q,R> + ∅(∂/∂xP + ∂/∂yQ + ∂/∂zR) ∇(dotted with)[∅F] = ∅(∂/∂xP + ∂/∂yQ + ∂/∂zR) <∂/∂x,∂/∂y,∂/∂z>(dotted with)<∅P, ∅Q, ∅R> = ∅(∂/∂xP + ∂/∂yQ + ∂/∂zR) ∅(∂/∂xP + ∂/∂yQ + ∂/∂zR) = ∅(∂/∂xP + ∂/∂yQ + ∂/∂zR) ∇(dotted with)(∅F) = [∇∅](dotted with)F + ∅∇(dotted with)F 
Re: Deriving cross product and dot product, stuck at beginning.
Hello? I did the work out and nobody can spot anything I did wrong, or if i did it right?

Re: Deriving cross product and dot product, stuck at beginning.
No, that isn't correct. You can't treat [itex]\phi[/itex] like a constant.

Re: Deriving cross product and dot product, stuck at beginning.
Thank you for the reply.
My professor wrote all of those on the board: ϕ = ϕ(x,y,z) ϕF is the vector field ϕ = ϕ(x,y,z) F = <P,Q,R> ∇(dotted with)F = x partial P + y partial Q + z partial R ∇ϕ = <x partial ϕ, y partial ϕ, z partial ϕ> ϕF = <ϕP, ϕQ, ϕR> ϕ is a function of x,y, and z. If I can't treat it as a constant in this situation, what can I do with it? 
Re: Deriving cross product and dot product, stuck at beginning.
Just start with the definition of the divergence and apply it to ϕF = (ϕP, ϕQ, ϕR):
[tex]\nabla\cdot(\phi \mathbf{F}) = \frac{\partial}{\partial x} (\phi P) + \frac{\partial}{\partial y} (\phi Q) + \frac{\partial}{\partial z} (\phi R)[/tex]Now use the product rule on each of the three terms. 
Re: Deriving cross product and dot product, stuck at beginning.
∇⋅(ϕF)=∂/∂x(ϕP)+∂/∂y(ϕQ)+∂/∂z(ϕR)
so =(ϕ'P + P'ϕ) + (ϕQ' + ϕ'Q) + (ϕR' + ϕ'R) = ϕ'(P+Q+R) + ϕ(P'+Q'+R') So it looks like ϕ'(P+Q+R) = (∇ϕ)⋅F and ϕ(P'+Q'+R') = ϕ(∇⋅F) and that is the end of the proof? 
Re: Deriving cross product and dot product, stuck at beginning.
You're on the right track, but you need to keep track of the fact that the derivatives are with respect to different variables so you can't, for example, simply collect terms and factor ϕ' out to get the first term.

All times are GMT 5. The time now is 08:08 AM. 
Powered by vBulletin Copyright ©2000  2014, Jelsoft Enterprises Ltd.
© 2014 Physics Forums