Proving vector identities using Cartesian tensor notation

[QuanticEnigma]

Homework Statement

1. Establish the vector identity
<br /> <br /> \nabla . (B x A) = (\nabla x A).B - A.(\nabla x B)<br />

2. Calculate the partial derivative with respect to x_{k} of the quadratic form A_{rs}x_{r}x_{s} with the A_{rs} all constant, i.e. calculate A_{rs}x_{r}x_{s,k}

Homework Equations

The Attempt at a Solution

1.
<br /> <br /> \nabla . (B x A) = \epsilon_{ijk}A_{j}B_{k,i}<br />

Now I don't know what to do next.

2.

<br /> A_{rs}x_{r}x_{s,k} = A_{rs}\partial_{k}(x_{r}x_{s}) = A_{rs}(x_{r}\partial_k x_{s} + x_{s}\partial_k x_{r})

I have no idea if this is right or not.

I'm pretty good at proving vector identities (and Cartesian tensor notation in general), but I get lost when partial derivatives/nablas are involved. Any tips would be greatly appreciated!

Homework Statement

Homework Equations

The Attempt at a Solution

 

[vela]

Your notation is unclear and misleading. When you write \nabla\cdot(B\times A) = \epsilon_{ijk}A_j B_{k,i}, it looks like you're only differentiating B_k. Also, you got the indices wrong since you have BxA, not AxB. On the other hand, you should have \nabla\cdot(A \times B) on the LHS anyway.

Instead, you should write

\nabla\cdot(A \times B) = \partial_i (\epsilon_{ijk} A_j B_k) = \epsilon_{ijk} \partial_i (A_j B_k)

Use the product rule to differentiate and then convert back to vector notation.In the second problem, use the fact that the x's are independent, so \partial_i x_j = 0 if i \ne j and \partial_i x_j = 1 if i = j, i.e. \partial_i x_j = \delta_{ij}.