Prove the following identity [Einstein notation]

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Homework Statement


[/B]
Prove the following identity:

[tex] \vec{\nabla}(\vec{A} \cdot \vec{B}) = (\vec{A} \cdot \vec{\nabla})\vec{B} + (\vec{B} \cdot \vec{\nabla})\vec{A} + \vec{A} \times (\vec{\nabla} \times \vec{B}) + \vec{B} \times (\vec{\nabla} \times \vec{A}) [/tex]

Homework Equations



Kronecker's delta, levi-civita tensor

The Attempt at a Solution


[/B]
My solution consisted of simply solving the RHS by decomposition. I wrote [tex] \vec{A} = A_{1}A_{\hat{x}} + A_{2}A_{\hat{y}} + A_{3}A_{\hat{z}} [/tex] likewise with B. I manually solved for each term in RHS and I did get the correct result which was LHS. My problem is that there is probably a more efficient way of doing this, perhaps by incorporating Einstein notation and solve LHS immediately without having to look at RHS.

How should I do this?

I tried [tex] \vec{\nabla}(\vec{A} \cdot \vec{B}) = \partial_{x_{i}}A_{i}B_{i} = \frac{\partial A_{i}}{\partial_{x_{i}}}B_{i} + A_{i} \frac{\partial B_{i}}{\partial_{x_{i}}}[/tex] but then I got stuck. How to continiue? I'm not sure how all those [tex] \times [/tex] appear using einstein notation.
 

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  • #2
nrqed
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Homework Statement


[/B]
Prove the following identity:

[tex] \vec{\nabla}(\vec{A} \cdot \vec{B}) = (\vec{A} \cdot \vec{\nabla})\vec{B} + (\vec{B} \cdot \vec{\nabla})\vec{A} + \vec{A} \times (\vec{\nabla} \times \vec{B}) + \vec{B} \times (\vec{\nabla} \times \vec{A}) [/tex]

Homework Equations



Kronecker's delta, levi-civita tensor

The Attempt at a Solution


[/B]
My solution consisted of simply solving the RHS by decomposition. I wrote [tex] \vec{A} = A_{1}A_{\hat{x}} + A_{2}A_{\hat{y}} + A_{3}A_{\hat{z}} [/tex] likewise with B. I manually solved for each term in RHS and I did get the correct result which was LHS. My problem is that there is probably a more efficient way of doing this, perhaps by incorporating Einstein notation and solve LHS immediately without having to look at RHS.

How should I do this?

I tried [tex] \vec{\nabla}(\vec{A} \cdot \vec{B}) = \partial_{x_{i}}A_{i}B_{i} = \frac{\partial A_{i}}{\partial_{x_{i}}}B_{i} + A_{i} \frac{\partial B_{i}}{\partial_{x_{i}}}[/tex] but then I got stuck. How to continiue? I'm not sure how all those [tex] \times [/tex] appear using einstein notation.
You should not have the same index (here the "i") used three times in an expression, this is a big no-no. Also notice that your left side is a vector an your right side is a scalar, so that cannot be right. What you need is

$$ \hat{e}_j ~\partial_j (A_i B_i ) $$
where by ##\partial_j## I mean
$$ \partial_j \equiv \frac{\partial}{\partial x_j} $$.
 
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  • #3
BvU
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Einstein notation is just a shorthand -- doesn't help you derive something. Might even confuse some folks ... :rolleyes:

This any use ? (where you may need the liberty to read ##(a\cdot\nabla)## as ##(\nabla\cdot a)## by virtue of ##a\cdot b = b \cdot a ## ... )

Must admit I got stuck with factors of 2, though :wideeyed:
 
  • #4
nrqed
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Einstein notation is just a shorthand -- doesn't help you derive something. Might even confuse some folks ... :rolleyes:

This any use ? (where you may need the liberty to read ##(a\cdot\nabla)## as ##(\nabla\cdot a)## by virtue of ##a\cdot b = b \cdot a ## ... )

Must admit I got stuck with factors of 2, though :wideeyed:
Just to not confuse the OP, we cannot write ##(a\cdot\nabla)## as ##(\nabla\cdot a)## when ##a## is a function of the coordinates.
 
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