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How to show something is a tensor.

  1. Oct 12, 2005 #1
    I've got to show [tex] \nabla \vec{V} [/tex] is a (1 1) tensor with components [tex] V^\alpha{}_{;\beta} [/tex]. Do I need to show (a) that it is a multilinear map or (b) that the components transform tensorially? I don't know how to do it using method (a) and method (b) involves chrictoffel symbols and how they transform and it doesn't lool pretty. Any help?
    Last edited: Oct 12, 2005
  2. jcsd
  3. Oct 12, 2005 #2


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    Yes, the second method (behaving under general coordinate transformations) can't fail.

  4. Oct 12, 2005 #3


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    There is a general theorem that if [tex]V x^{\beta}y_{\alpha}[/tex] is a scalar for any vectors [tex]x^{\beta}[/tex] and [tex]y_{\alpha}[/tex] then V is a tensor of the correct form.
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