Vector Analysis: Seeking Guidance on Identity and Calculation

[Marin]

hi there!

I´m doing vector analysis the last two weeks and I feel unsure about this identity. Can anyone of you say if I´m on the right way, and if not where my mistakes lie :)

$$A_i(\vec r)=\sum_{j=1}^3R_{ij}x_j$$, R constant 3x3 matrix

I have to calculate $$rot\vec A$$, $$rotrot\vec A$$

$$rot\vec A=\epsilon_{jki}\partial_j(\sum_{j=1}^3R_{ij}x_j)_k=\epsilon_{jki}\partial_j(R_{ik}x_k)=R_{ik}\epsilon_{jki}\partial_jx_k$$

$$rotrot\vec A=\epsilon_{lkm}\partial_l(R_{ik}\epsilon_{jki}\partial_jx_k)_k=\epsilon_{lkm}\partial_lR_{ik}\epsilon_{jki}\partial_jx_k=\epsilon_{kml}\epsilon_{kij}R_{ik}\partial_l\partial_jx_k=(\delta_{mi}\delta_{lj}-\delta_{mj}\delta_{li})R_{ik}\partial_l\partial_jx_k=R_{mk}\partial_l^2x_k-R_{lk}\partial_l\partial_mx_k$$

and one more question: consider the scalar:

$$\phi(\vec r)=\sum_{i,j=0}^3Q_{ij}x_ix_j$$, Q 3x3 constant

is this the correct k-th component of the sum:

$$?(\sum_{i,j=0}^3Q_{ij}x_ix_j)_k=Q_{kj}x_kx_j=Q_{ik}x_ix_k ?$$


thanks a lot in advance!

 

[Marin]

any comments are welcome :)