Prove the following identity [Einstein notation]

[Mulz]

Homework Statement

[/B]
Prove the following identity:

\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})

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 \vec{A} = A_{1}A_{\hat{x}} + A_{2}A_{\hat{y}} + A_{3}A_{\hat{z}} 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 \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}}} but then I got stuck. How to continiue? I'm not sure how all those \times appear using einstein notation.

 

[nrqed]

Homework Statement

[/B]
Prove the following identity:

\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})

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 \vec{A} = A_{1}A_{\hat{x}} + A_{2}A_{\hat{y}} + A_{3}A_{\hat{z}} 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 \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}}} but then I got stuck. How to continiue? I'm not sure how all those \times 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
$$ \partial_j \equiv \frac{\partial}{\partial x_j} $$.

 

[BvU]

Einstein notation is just a shorthand -- doesn't help you derive something. Might even confuse some folks ...

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

 

[nrqed]

Einstein notation is just a shorthand -- doesn't help you derive something. Might even confuse some folks ...

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

Just to not confuse the OP, we cannot write $(a\cdot\nabla)$ as $(\nabla\cdot a)$ when $a$ is a function of the coordinates.