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Ideal representation for vectors/covectors

  1. Mar 4, 2014 #1
    A vector can be expressed by a column matrix or by a row matrix, however is preferably use the column matrix for represent vectors and row matrix for covectors.

    So, analogously, we can express a vector v as v1 e1 + v2 e2 or as v1 e1 + v2 e2. But I think that a vector is represented more correctly with one of those two possibilities, whereas that a covector is better represented with the other. The question is, which representation is the ideal representation for a vector and for a covector?
     
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  3. Mar 4, 2014 #2

    micromass

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    The notation ##v_1 \mathbf{e}^1 + v_2 \mathbf{e}^2## is used for covectors. For example, in differential geometry, you have the covectors ##dx^i##.

    The notation ##v^1 \mathbf{e}_1 + v^2 \mathbf{e}_2## is used for vectors. For example, tangent vectors to a manifold are given by ##\frac{\partial}{\partial x^i}## (which is seen as a lower index).

    This is consistent with matrices. A matrix is written as ##(\alpha^i_j)_{i,j}##, where ##i## denotes the rows and ##j## the columns. So we have, for example

    [tex]\left(\begin{array}{cc} \alpha_1^1 & \alpha^1_2\\ \alpha^2_1 & \alpha_2^2 \end{array}\right)[/tex]

    In particular, in the case of one column, we get a vector:

    [tex]\left(\begin{array}{c} v^1 \\ v^2\end{array}\right)[/tex]

    And in the case of one row, we get a covector:

    [tex]\left(\begin{array}{cc} v_1 & v_2\end{array}\right)[/tex]

    Also see: http://en.wikipedia.org/wiki/Einstein_notation
     
    Last edited: Mar 4, 2014
  4. Mar 4, 2014 #3
    Thanks you!

    To use the indices mixed is a thing that irritate me a lot, the indices for scalar components is the contrary of the indices of the unit vectors. I don't understand the why this...
     
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