Vector identity proof using index notation

In summary: That is, you should have\left(\vec{A}\left(\vec{\nabla}\bullet\vec{B}\right)\right) - \left(\vec{A}\left(\vec{\nabla}\bullet\vec{B}\right)\vec{A}\right) - \vec{B}\left(\vec{\nabla}\bullet\vec{A}\right)
  • #1
baffledboy
3
0

Homework Statement


Using index notation to prove

[tex]\vec{\nabla}\times\left(\vec{A}\times\vec{B}\right) = \left(\vec{B}\bullet\vec{\nabla}\right)\vec{A} - \left(\vec{A}\bullet\vec{\nabla}\right)\vec{B} + \vec{A}\left(\vec{\nabla}\bullet\vec{B}\right) - \vec{B}\left(\vec{\nabla}\bullet\vec{A}\right)[/tex]

Homework Equations


--?

The Attempt at a Solution


I tried converting the right side first...
[tex]B_{a}A_{k,a}-A_{a}B{k,a}+A_{a}\epsilon_{abc}B_{b,c}-[/tex][tex]B_{a}\epsilon_{abc} A_{b,c}[/tex]

but i don't really know if this is right... I'm very very new to the index notation system. i don't really understand the rules beyond the basics. will someone please give me some clue as to where to go from here?
 
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  • #2
Systems of index notation vary, so you may need to give a reference or a description of your system. Using the index notation I know, the del operator is written [tex]\partial_a[/tex], so the right side becomes [tex]B^b \partial_b A^a - A^b \partial_b B^a + A^a \partial_b B^b - B^a \partial_b A^b[/tex], and the left side becomes [tex]\mathbf{\nabla}\times \epsilon_{abc} A^b B^c = \epsilon^{ade} \partial_d \epsilon_{ebc} A^b B^c[/tex]. From there, try to figure out how to contract the epsilons on the left against each other. You definitely shouldn't have any epsilons on the right side, because all you have is dot products, they're not antisymmetric.
 
  • #3
Thanks for your response. I'm not sure what system the professor is using. He doesn't use superscripts at all and uses commas for derivatives. He didn't give a name for it other than "index notation" though.

is it okay to just change [tex]\epsilon_{ade}\partial_{d}\epsilon_{ebc}A_{b}B_{c}[/tex] to [tex]\epsilon_{ade}\epsilon_{ebc}\partial_{d}A_{b}B_{c}[/tex] or does this mess with the meaning?
 
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  • #4
baffledboy said:
is it okay to just change [tex]\epsilon_{ade}\partial_{d}\epsilon_{ebc}A_{b}B_{c}[/tex] to [tex]\epsilon_{ade}\epsilon_{ebc}\partial_{d}A_{b}B_{c}[/tex] or does this mess with the meaning?
That's fine. It's what makes index notation so useful in deriving identities like this.
 
  • #5
okay, so I've got

[tex]\epsilon_{ade}\epsilon_{ebc}\partial_{d}A_{b}B_{c} =
\left(\delta_{ab}\delta_{dc}-\delta_{ac}\delta_{db}\right)\partial_{d}A_{b}B_{c} =
\partial_{d}A_{a}B_{d}-\partial_{d}A_{d}B_{a} =
\vec{A}\left(\vec{\nabla}\bullet\vec{B}\right)-\left(\vec{\nabla}\bullet\vec{A}\right)\vec{B}[/tex]

but I can't figure out how to get that to have four terms, like on the right... It seems like I'm missing something here. Do I just keep adding onto terms in the second part with A, B, & del in different orders until I get something that looks right?
 
  • #6
You have to be a bit more careful. Remember the derivative acts on the product of Aa and Bc, so you need to use the product rule before you can simplify the indices using the Kronecker deltas.
 

1. What is vector identity proof using index notation?

Vector identity proof using index notation is a method used in vector calculus to prove mathematical equations involving vectors using indices instead of component form. This notation is useful in simplifying and generalizing vector operations and identities.

2. How is index notation used in vector identity proofs?

In index notation, vectors are represented by lowercase Latin letters with indices, and scalars are represented by lowercase Greek letters. Vector operations such as addition, subtraction, and scalar multiplication are performed by manipulating the indices according to specific rules. These rules allow for easier manipulation and proof of vector identities.

3. What are the advantages of using index notation for vector identity proofs?

Using index notation allows for a more concise and general representation of vector operations and identities. It also eliminates the need to work with individual vector components, which can be time-consuming and prone to errors. Additionally, index notation can be easily extended to higher dimensions, making it useful in more complex vector calculations.

4. Are there any limitations to using index notation for vector identity proofs?

One limitation of index notation is that it may not be as intuitive for beginners compared to working with vector components. It also requires a good understanding of vector operations and notational rules. Furthermore, index notation may not be suitable for visualizing vector quantities, as it primarily focuses on the algebraic manipulation of indices.

5. How can I learn more about vector identity proof using index notation?

If you are interested in learning more about vector identity proof using index notation, you can refer to textbooks or online resources on vector calculus. Practice problems and examples can also help in understanding and applying this notation. Seeking guidance from a tutor or instructor can also be beneficial in mastering this method.

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