Deriving cross product and dot product, stuck at beginning.

[JJRKnights]

Homework Statement

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

Homework Equations

The Attempt at a Solution

Honestly don't know where to start.

 

[JJRKnights]

Nevermind, delete this, I've got it, just didn't put the initial effort into it.

 

[JJRKnights]

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]

Hello? I did the work out and nobody can spot anything I did wrong, or if i did it right?

 

[vela]

No, that isn't correct. You can't treat $\phi$ like a constant.

 

[JJRKnights]

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]

Just start with the definition of the divergence and apply it to ϕF = (ϕP, ϕQ, ϕR):
$$\nabla\cdot(\phi \mathbf{F}) = \frac{\partial}{\partial x} (\phi P) + \frac{\partial}{\partial y} (\phi Q) + \frac{\partial}{\partial z} (\phi R)$$Now use the product rule on each of the three terms.

 

[JJRKnights]

∇⋅(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]

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.