Deriving cross product and dot product, stuck at beginning.

In summary: F is the vector fieldϕ = ϕ(x,y,z)F = <P,Q,R>∇ϕ = <x partial ϕ, y partial ϕ, z partial ϕ>ϕF = <ϕP, ϕQ, ϕR>In summary, the homework statement is that if ∅ is a differentiable scalar valued function and F a differentiable vector field, then the following identities hold: a) ∇(dotted with)(∅F) = ∇∅(dotted with)F + ∅∇(
  • #1
JJRKnights
53
0

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.
 
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  • #2
Nevermind, delete this, I've got it, just didn't put the initial effort into it.
 
  • #3
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
 
  • #4
Hello? I did the work out and nobody can spot anything I did wrong, or if i did it right?
 
  • #5
No, that isn't correct. You can't treat [itex]\phi[/itex] like a constant.
 
  • #6
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?
 
  • #7
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.
 
  • #8
∇⋅(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?
 
  • #9
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.
 

1. What is the difference between cross product and dot product?

The cross product is a vector operation that results in a vector perpendicular to the two given vectors, while the dot product is a scalar operation that results in a single number representing the magnitude of the projection of one vector onto the other.

2. How do you calculate the cross product and dot product?

The cross product is calculated by taking the determinant of a 3x3 matrix composed of the unit vectors and the two given vectors. The dot product is calculated by multiplying the corresponding components of the two vectors and then adding them together.

3. What are the applications of cross product and dot product?

The cross product is commonly used in physics and engineering to calculate torque, angular momentum, and magnetic fields. The dot product is used in physics and geometry to calculate the angle between two vectors, as well as in calculations involving work and energy.

4. How do you know when to use cross product or dot product in a problem?

The choice between cross product and dot product depends on the desired outcome of the problem. If the result should be a vector, then cross product should be used. If the result should be a scalar, then dot product should be used.

5. What are some tips for understanding and remembering how to derive cross product and dot product?

Practice is key for understanding and remembering how to derive cross product and dot product. Drawing diagrams and using mnemonic devices can also be helpful. It is also important to have a strong understanding of vectors and basic vector operations before attempting to derive cross product and dot product.

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