How to show something is a tensor.

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SUMMARY

The discussion centers on demonstrating that the expression \nabla \vec{V} is a (1 1) tensor with components V^\alpha{}_{;\beta}. Participants debate whether to prove this by showing it is a multilinear map or by verifying that its components transform tensorially under coordinate transformations. The consensus is that the second method, which involves the transformation properties of Christoffel symbols, is more straightforward and reliable. Additionally, a theorem is mentioned stating that if V x^{\beta}y_{\alpha} is a scalar for any vectors x^{\beta} and y_{\alpha}, then V qualifies as a tensor of the appropriate form.

PREREQUISITES
  • Understanding of tensor notation and components
  • Familiarity with Christoffel symbols and their transformation properties
  • Knowledge of multilinear maps in the context of tensors
  • Basic concepts of general coordinate transformations
NEXT STEPS
  • Study the properties of multilinear maps in tensor analysis
  • Learn about the transformation rules for Christoffel symbols
  • Explore the implications of the theorem regarding scalar products of vectors
  • Review general coordinate transformations and their effects on tensor components
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This discussion is beneficial for mathematicians, physicists, and students in fields involving differential geometry and tensor calculus, particularly those working with general relativity or advanced physics concepts.

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

Daniel.
 
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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