How Do You Derive the Vector Identity Involving Divergence and Curl?

In summary: F . ∇)G + (G . ∇)F + F x (∇ x G) + G x (∇ x F)In summary, the vectors F and G, which are arbitrary functions of position, can be used to obtain the identity ∇(F . G) = (F . ∇)G + (G . ∇)F + F x (∇ x G) + G x (∇ x F) by using the relations F x (∇ x G) and G x (∇ x F). However, the BAC-CAB rule cannot be applied in this case due to the difference in operators.
  • #1
bcjochim07
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Homework Statement


The vectors F and G are arbitrary functions of position. Starting w/ the relations F x (∇ x G) and G x (∇ x F), obtain the identity

∇(F . G) = (F . ∇)G + (G . ∇)F + F x (∇ x G) + G x (∇ x F)


Homework Equations





The Attempt at a Solution



I started off with the relation F x (∇ x G) and used the BAC-CAB rule:

F x (∇ x G) = ∇(F . G) - (G . ∇)F

so ∇(F . G = (G . ∇)F + F x (∇ x G) which seems to contradict the identity I am supposed to get. What am I doing wrong?
 
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  • #2
you can't apply the BAC-CAB rule here. That was derived for vectors by assuming that A X B = - B X A

However with the del operator del X A is a vector, but A X del is an operator...so there's a world of difference
 
  • #3
The trouble is commuting an opperator adds a commutator term
In single variable calculus
D(uv)=uDv+vDu not uDv
we can use partial opperators to avoid this
let a opperant in {} be fixed
D(uv)=D({u}v)+D(u{v})={u}Dv+{v}Du=uDv+vDu

∇(F . G )=∇({F} . G )+∇(F . {G} )
∇({F} . G )=Fx(∇xG)+(F.∇)G
∇(F . {G} )=Gx(∇xF)+(G.∇)F
∇(F . G )=∇({F} . G )+∇(F . {G} )=Fx(∇xG)+(F.∇)G+Gx(∇xF)+(G.∇)F
 

FAQ: How Do You Derive the Vector Identity Involving Divergence and Curl?

1. What are vector identities?

Vector identities are mathematical relationships or equations that involve vector quantities. They are used in physics, engineering, and other scientific fields to simplify calculations and solve problems.

2. Why are vector identities important?

Vector identities are important because they allow us to manipulate and transform vector equations in ways that make them easier to work with. They also help us to better understand the properties and behavior of vector quantities.

3. How are vector identities obtained?

Vector identities are obtained through mathematical derivation and manipulation. This involves using known equations and properties of vectors to derive new relationships between them.

4. What is the process for verifying a vector identity?

The process for verifying a vector identity involves substituting in values for the variables and simplifying both sides of the equation. If the simplified equations are equal, then the identity is verified. If they are not equal, then the identity is not valid.

5. Are there any common mistakes to avoid when obtaining vector identities?

Yes, some common mistakes to avoid when obtaining vector identities include not following the proper steps for derivation, not considering all possible cases, and making algebraic errors. It is important to be thorough and check your work carefully to ensure the validity of the identity.

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