Horas.
Excuse me ... .
A vector or polar-vector $\mathbf{A}\in{T_pM}$ can denote as $\mathbf{A}:=\sum_{j=1}^nA^j\mathbf{e}_j$ in a tangent space on point $p\in{M}$ at an $n$-dimensional manifold $M$, where $\{\mathbf{e}_1,\dots,\mathbf{e}_n\}$ is a basis in $T_pM$ ... .
The dot product $\mathbf{e}_j\cdot\mathbf{e}_k=g_{jk}$ is a component of metric tensor on the manifold ... .
For each $\mathbf{A},\mathbf{B}\in{T_pM}$, the dot product
$\displaystyle\mathbf{A}\cdot\mathbf{B}=\sum_{j,k=1}^ng_{jk}A^jB^k$ ,
is a scalar,
and the cross product
$\displaystyle\mathbf{A}\times\mathbf{B}=\sum_{j,k=1}^nA^jB^k\mathbf{e}_j\times\mathbf{e}_k=\frac{1}{2}\sum_{j,k=1}^n\begin{vmatrix}A^j&A^k\\B^j&B^k\end{vmatrix}\mathbf{e}_j\times\mathbf{e}_k$
is a pseudo-vector alias axial-vector, as I have known ... .
For each $\mathbf{A},\mathbf{B},\mathbf{C}\in{T_pM}$, the triple scalar product is defined as $[\mathbf{A},\mathbf{B},\mathbf{C}]:=(\mathbf{A}\times\mathbf{B})\cdot\mathbf{C}$ which totally antisymmetric by a permutation ... .
$\displaystyle[\mathbf{A},\mathbf{B},\mathbf{C}]=\sum_{j,k,l=1}^nA^jB^kC^l[\mathbf{e}_j,\mathbf{e}_k,\mathbf{e}_l]=\frac{1}{6}\sum_{j,k,l=1}^n\begin{vmatrix}A^j&A^k&A^l\\B^j&B^k&B^l\\C^j&C^k&C^l\end{vmatrix}[\mathbf{e}_j,\mathbf{e}_k,\mathbf{e}_l]$
is a pseudo-scalar, as I have known ... ,
and the triple vector product
$(\mathbf{A}\times\mathbf{B})\times\mathbf{C}=(\mathbf{A}\cdot\mathbf{C})\mathbf{B}-(\mathbf{B}\cdot\mathbf{C})\mathbf{A}$
is a vector alias polar-vector ... .
The other combinations are
$(\mathbf{A}\times\mathbf{B})\cdot(\mathbf{C}\times\mathbf{D})=(\mathbf{A}\cdot\mathbf{C})(\mathbf{B}\cdot\mathbf{D})-(\mathbf{A}\cdot\mathbf{D})(\mathbf{B}\cdot\mathbf{C})$ is a scalar ... , and
$(\mathbf{A}\times\mathbf{B})\times(\mathbf{C}\times\mathbf{D})=[\mathbf{A},\mathbf{B},\mathbf{D}]\mathbf{C}-[\mathbf{A},\mathbf{B},\mathbf{C}]\mathbf{D}$ is a pseudo-vector alias axial-vector, as I have known ... ,
for each $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}\in{T_pM}$ ... .
U V U•V U×V
[vector] [vector] [scalar] [pseudo-vector]
[vector] [pseudo-vector] [pseudo-scalar] [vector]
[pseudo-vector] [vector] [pseudo-scalar] [vector]
[pseudo-vector] [pseudo-vector] [scalar] [pseudo-vector]
What do multiplication scalar with scalar, scalar with pseudo-scalar, pseudo-scalar with pseudo-scalar, scalar with vector, scalar with pseudo-vector, pseudo-scalar with vector, and pseudo-scalar with pseudo-vector yield ... ?
u v uv
[scalar] [scalar] [scalar]
[scalar] [pseudo-scalar] [pseudo-scalar]
[pseudo-scalar] [pseudo-scalar] [??]
u V uV
[scalar] [vector] [vector]
[scalar] [pseudo-vector] [pseudo-vector]
[pseudo-scalar] [vector] [pseudo-vector]
[pseudo-scalar] [pseudo-vector] [??]
What are the two [??] ‘s in the last two tables ... ?
Are the last two tables right ... ?
Thank you very much for the answer ... .
http://www.thescienceforum.com/mathematics/35117-pseudo-vector-pseudo-scalar.htmlDalam Nama Bapa dan Putera dan Roh Kudus. Amin.
