Proving vector identities with index notation (help with the del operator)

[ShearStress]

Homework Statement

Prove the vector identity: $$\left(a\times\nabla\right)\bullet\left(u \times v\right)=\left(a \bullet u \right)\left(\nabla \bullet v \right)+\left(v \bullet \nabla \right)\left(a \bullet u \right)-\left(a \bullet v \right)\left(\nabla \bullet u \right)-\left(u \bullet\nabla\right)\left(a \bullet v \right)$$
Where a, u, and v are vectors (and a is a "constant vector")

Homework Equations

N/A

The Attempt at a Solution

Okay, so in index notation I've gotten the left-hand side as...
$$LHS=a_{l}u^{l}\partial_{m}v^{m}-a_{m}v^{m}\partial_{l}u^{l}$$

Which, since the dot product on the RHS is commutative, it seems that the RHS is just twice the LHS I've come up with in index notation. What am I missing here? Is there some weird property of the del operator in index notation that I can just double the terms?

 

[Cyosis]

I haven't checked your calculation for the LHS, but $\nabla \cdot v \neq v \cdot \nabla$. So you can't just add them together,

 

[ShearStress]

Okay well then it seems I'm even more thoroughly confused than I originally thought. I still think I have the LHS correct but I'm not entirely sure about using the gradient/del in index notation. Does it matter if you move it around within the term? Such as, are the following terms equivalent:
$$a_{l}\partial_{m}u^{l}v^{m}=a_{l}u^{l}\partial_{m}v^{m}$$ ? Or does the order you write the stuff in matter when the del operator is involved?

I think what has really confused me is when you have a dot product with a del operator on the outside, that somehow results in one of the vectors times the partial derivative of the other plus the other vector times the partial derivative of the other. Help?

 

[gabbagabbahey]

Keep in mind that $\partial_m$ is a differential operator, and so order of operations and brackets are important. For example, $\partial_m\left[f(x_1,x_2,x_3)g(x_1,x_2,x_3)\right]=(\partial_m f)g+f(\partial_m g)$.

 

[Cyosis]

Don't let the indices get to you it is just the product rule. Also keep in mind that the exercise states that a is a constant vector. Without that fact you won't be able to prove the identity.

If you did everything correctly you get $a_m\partial_n(u_mv_n)-a_n \partial_m(u_mv_n)$ before using the product rule on the LHS.