Prove the following identity [Einstein notation]

Mulz

1. Homework Statement

Prove the following identity:

$$\vec{\nabla}(\vec{A} \cdot \vec{B}) = (\vec{A} \cdot \vec{\nabla})\vec{B} + (\vec{B} \cdot \vec{\nabla})\vec{A} + \vec{A} \times (\vec{\nabla} \times \vec{B}) + \vec{B} \times (\vec{\nabla} \times \vec{A})$$

2. Homework Equations

Kronecker's delta, levi-civita tensor

3. The Attempt at a Solution

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.

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nrqed

Homework Helper
Gold Member
1. Homework Statement

Prove the following identity:

$$\vec{\nabla}(\vec{A} \cdot \vec{B}) = (\vec{A} \cdot \vec{\nabla})\vec{B} + (\vec{B} \cdot \vec{\nabla})\vec{A} + \vec{A} \times (\vec{\nabla} \times \vec{B}) + \vec{B} \times (\vec{\nabla} \times \vec{A})$$

2. Homework Equations

Kronecker's delta, levi-civita tensor

3. The Attempt at a Solution

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

BvU

Homework Helper
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

Homework Helper
Gold Member
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.

BvU

"Prove the following identity [Einstein notation]"

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