Thread: 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 ... .