Need help proving vector identities

In summary, the conversation discusses proving the identity ∇x(FxG)=(G⋅∇)F-(F⋅∇)G+F(∇⋅G)-G(∇⋅F) for vector fields F and G using scalar multiplication and the cross product. The use of the completely antisymmetric or Levi-Civita symbol \epsilon_{ijk} is recommended as an easier method. The question also arises if (G⋅∇)F is equal to F(∇⋅G) when G is a constant vector and F is not.
  • #1
dooogle
21
0

Homework Statement



i have to prove that

∇x(FxG)=(G⋅∇)F-(F⋅∇)G+F(∇⋅G)-G(∇⋅F)

where F and G are vector fields with F=F1,F2,F3 and G=G1,G2,G3 ∇=d/dx,d/dy/d/dz

Homework Equations



The Attempt at a Solution



i have tried applying scalar multiplication and the cross product to multiply out the terms
and have got the right hand side as twice the left hand side.

is (G⋅∇)F equal to F(∇⋅G)?

thank you for your time

dooogle
 
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  • #2
Do you know about the completely antisymmetric or Levi-Civita symbol [tex]\epsilon_{ijk}[/tex]? This is by far the easiest way I know to prove this sort of identity.

dooogle said:
is (G⋅∇)F equal to F(∇⋅G)?

What if G is a constant vector and F is not?
 

1. How do I prove vector identities?

To prove vector identities, you can use various mathematical techniques such as substitution, simplification, and manipulation of vector equations. You can also use geometric interpretations of vectors to aid in understanding and proving identities.

2. What are some common vector identities?

Some common vector identities include the commutative and associative properties, the distributive property, and the cross product identity. Other commonly used identities involve the dot product and vector components.

3. Why is it important to prove vector identities?

Proving vector identities is important because it allows for a deeper understanding of vector operations and their properties. It also helps in solving more complex problems involving vectors and in developing new mathematical techniques.

4. Can I use software to help prove vector identities?

Yes, there are various software programs and online tools available that can assist in proving vector identities. However, it is important to understand the concepts and techniques behind the identities rather than relying solely on software.

5. How can I check if my proof for a vector identity is correct?

You can check your proof by verifying that both sides of the identity are equal using algebraic manipulation and substitution. You can also check for any errors in logic or calculation by going through each step of your proof carefully.

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