Pseudo-Vector and Pseudo-Scalar
Excuse me ... .
A vector or polar-vector
can denote as
in a tangent space on point
at an
-dimensional manifold
, where
is a basis in
... .
The dot product
is a component of metric tensor on the manifold ... .
For each
, the dot product

,
is a scalar,
and the cross product
is a pseudo-vector alias axial-vector, as I have known ... .
For each
, the triple scalar product is defined as
which totally antisymmetric by a permutation ... .
is a pseudo-scalar, as I have known ... ,
and the triple vector product
is a vector alias polar-vector ... .
The other combinations are
is a scalar ... , and
is a pseudo-vector alias axial-vector, as I have known ... ,
for each
... .
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 ... . :)