- #1
Marin
- 193
- 0
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 :)
[tex]A_i(\vec r)=\sum_{j=1}^3R_{ij}x_j[/tex], R constant 3x3 matrix
I have to calculate [tex]rot\vec A[/tex], [tex]rotrot\vec A[/tex]
[tex]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[/tex]
[tex]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[/tex]
and one more question: consider the scalar:
[tex]\phi(\vec r)=\sum_{i,j=0}^3Q_{ij}x_ix_j[/tex], Q 3x3 constant
is this the correct k-th component of the sum:
[tex]?(\sum_{i,j=0}^3Q_{ij}x_ix_j)_k=Q_{kj}x_kx_j=Q_{ik}x_ix_k ?[/tex]
thanks a lot in advance!
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 :)
[tex]A_i(\vec r)=\sum_{j=1}^3R_{ij}x_j[/tex], R constant 3x3 matrix
I have to calculate [tex]rot\vec A[/tex], [tex]rotrot\vec A[/tex]
[tex]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[/tex]
[tex]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[/tex]
and one more question: consider the scalar:
[tex]\phi(\vec r)=\sum_{i,j=0}^3Q_{ij}x_ix_j[/tex], Q 3x3 constant
is this the correct k-th component of the sum:
[tex]?(\sum_{i,j=0}^3Q_{ij}x_ix_j)_k=Q_{kj}x_kx_j=Q_{ik}x_ix_k ?[/tex]
thanks a lot in advance!