Proving B.(Gradient . B) - B X (Gradient X B)=Del{i}B{i}B{j}

[hellomynameisscottt]

Homework Statement

I need to prove B.(Gradient . B) - B X(Gradient X B)=Del{i} [B{i}B{j} -1/2 (kroneker delta {ij} B^2]

where I have used . as the dot product, {} as subscript. Thank you!

Homework Equations

The Attempt at a Solution

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I have attempted the solution but am not sure how it is possible to keep a kroneker delta on the right hand side.

 

[Charles Link]

I can not read your complete statement above, but this looks like it comes from a standard vector identity: (where $A$ and $B$ are both equal to $B$).
$\nabla (A \cdot B)=(A \cdot \nabla) B +(B \cdot \nabla) A+A \times \nabla \times B+B \times \nabla \times A$

 

[hellomynameisscottt]

I have uploaded the page from Jackson it is equation (6.119) I am trying to prove, however I must use Levi Cevita notation.

 

[Charles Link]

I have uploaded the page from Jackson it is equation (6.119) I am trying to prove, however I must use Levi Cevita notation.

On the left side of the equation in your OP, they are only taking one component. You may be able to use the identity I presented to simplify the $B \times \nabla \times B$ term.

 

[DrClaude]

The same question was asked in https://www.physicsforums.com/threads/can-you-help-me-with-levi-civita-notation-please.871728/

All replies should go there.

Thread closed.