Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Tensor of type (k,l)

  1. Feb 9, 2010 #1


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    In John Lee's books "Introduction to smooth manifolds" and "Riemannian manifolds", a tensor of type [tex]\begin{pmatrix}k\\ l\end{pmatrix}[/tex] on a vector space V is defined as a member of

    [tex]\underbrace{V^*\otimes\cdots\otimes V^*}_{k}\otimes\underbrace{V\otimes\cdots\otimes V}_{l}[/tex]

    or as a multilinear function

    [tex]\underbrace{V^*\times\cdots\times V^*}_{l}\times\underbrace{V\times\cdots\times V}_{k}\rightarrow\mathbb R[/tex]

    or, when [itex]l>0[/itex], as a multilinear function

    [tex]\underbrace{V^*\times\cdots\times V^*}_{l-1}\times\underbrace{V\times\cdots\times V}_{k}\rightarrow V[/tex]

    (These three vector spaces are isomorphic). But in Wald's "General relativity", this is called a tensor of type [itex](l,k)[/itex]. I just want to ask, is this a "math vs. physics" thing, like when physicsts make their inner products antilinear in the first variable and mathematicians make theirs antilinear in the second? Or is there a standard convention that one of these guys is ignoring?
  2. jcsd
  3. Feb 18, 2010 #2
    i saw the type of (l,k) for both of math and physics.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook