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Deriving cross product and dot product, stuck at beginning.

by JJRKnights
Tags: beginning, cross, deriving, product, stuck
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JJRKnights
#1
Nov24-11, 05:29 PM
P: 53
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.
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JJRKnights
#2
Nov24-11, 05:32 PM
P: 53
Nevermind, delete this, I've got it, just didn't put the initial effort into it.
JJRKnights
#3
Nov29-11, 07:14 PM
P: 53
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

JJRKnights
#4
Dec1-11, 02:32 PM
P: 53
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?
vela
#5
Dec1-11, 02:40 PM
Emeritus
Sci Advisor
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Thanks
PF Gold
P: 11,673
No, that isn't correct. You can't treat [itex]\phi[/itex] like a constant.
JJRKnights
#6
Dec1-11, 02:46 PM
P: 53
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?
vela
#7
Dec1-11, 03:47 PM
Emeritus
Sci Advisor
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PF Gold
P: 11,673
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.
JJRKnights
#8
Dec1-11, 08:55 PM
P: 53
∇⋅(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?
vela
#9
Dec1-11, 10:29 PM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,673
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.


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