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Index conventions

  1. Jul 13, 2009 #1
    In Linear Algebra and its Applications, David Griffel writes, "The components of covectors are often denoted by superscripts, rather than subscripts." This differs from the usual convention, doesn't it? Unless I've completely misunderstood the concept (quite possible!), the introductory texts that I've seen so far have denoted the components of vectors with superscripts, and the components of covectors (one-forms, linear function(al)s) with subscripts). E.g.

    http://en.wikipedia.org/wiki/Covector#Bases_in_finite_dimensions

    For example, Griffel writes as follows:

    Let [tex]x_{r}[/tex] be the rth component of a vector x in V with respect to a basis [tex]\left\{e_{i},...,e_{n}\right\}[/tex], and [tex]g^{r}[/tex] the rth component of a covector g with respect to the dual basis. Then

    [tex](a)\; g^{r} = g\left(e_{r} \right)[/tex]
    [tex](b)\; g\left(x \right) = \sum_{}^{} g^{r} x_{r}[/tex]

    If this does differ from normal usage, as I suspect, how would it be rewritten according to the usual convention? Should I put the index on g down, and the index on x up, and leave the index down on e?

    And where he writes

    There is a basis [tex]\left\{f_{i},...,f_{n}\right\}[/tex] for [tex]V^{\ast}[/tex], called the dual basis, such that

    [tex]f_{r}\left(e_{s} \right) = \delta_{rs}[/tex], for r,s = 1,...,n

    would the normal convention be for the index written r to appear as a superscript on the basis of the dual space [tex]V^{\ast}[/tex], and for the Kronecker's delta here to have superscript r and subscript s?

    Any advice welcome!
     
    Last edited: Jul 13, 2009
  2. jcsd
  3. Jul 14, 2009 #2
    I thought he'd just reversed the conventional notation for subscripts and supetscripts that I've seen in various texts, including Geometrical Methods of Mathematical Physics by Bernard Schutz and Vector and Tensor Analysis by Borisenko and Taparov, the Wikipedia page I linked to, and others. But now I'm not sure what's going on.

    On p. 178, Griffel writes, "The components of a vector v in V are called covariant components. If V is a real inner product space, it is naturally isomorphic to V* (its dual), so each v in V corresponds to a certain v* in V*. Thus one thinks of v* as being another form of v. The components of v* are regarded as being another type of component of v; the contravariant components of a vector v with respect to a basis E are defined as the components of v* with respect to the dual basis F. The covariant and contravariant components of v with respect to a basis E are denoted [tex]v_{i}[/tex] and [tex]v^{i}[/tex] respectively.

    "[tex]v^{i} = v^{*}(e_{i})[/tex] = [tex]<e_{i},v> = <e_{i},\sum_{}^{}v_{j}e_{j}>[/tex]

    "Hence

    "[tex]v^{i} = \sum_{j}^{}v_{j}<e_{j},e_{i}>.[/tex] (3)

    "[...]If the basis vectors are normalised, the contravariant components have a simple geometrical interpretation. Equation (3) shows that [tex]v^{i}[/tex] is length of the orthogonal projection of v onto the direction of [tex]e_{i}[/tex]."

    But on p. 28 of Vector and Tensor Analysis, Borisenko and Taparov label the orthogonal projections of a vector they call A onto the unprimed basis vectors

    [tex]\frac{A_{i}}{||\mathbf{e_{i}}||}[/tex]

    and call [tex]A_{i}[/tex] the covariant components. So is Griffel reversing both the sub/superscript convention and the names "contravariant" and "covariant" components (with the exception of the practice of putting a subscript for the basis vectors [tex]\mathbf{e_{i}}[/tex] on which point he agrees with the others, or is there something deeper to it?
     
  4. Jul 16, 2009 #3
    By which I mean the basis vectors of the original basis, that Borisenko and Taparov with subscripts.

    [tex]\left\{ \mathbf{e_{1},...,\mathbf{e_{n}} }\right\}[/tex]

    (See attachment.)

    I wonder if the anomaly could be because Griffel really has used a different method of introducing these ideas which somehow allows him to use subscripted basis vectors (i.e. the kind of basis vectors which other authors denote with superscripts), where - if he'd been following the usual way of introducing it - he'd have denoted them with superscripts (to indicate the same sort of basis vectors that other authors denote with subscripts). Whatever that might mean...
     

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