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Thread: Pseudo-Vector and Pseudo-Scalar

  1. #1 Pseudo-Vector and Pseudo-Scalar 
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    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 UV UV
    [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 ... .


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