How to show something is a tensor.

[Tzar]

Hey,
I've got to show \nabla \vec{V} is a (1 1) tensor with components V^\alpha{}_{;\beta}. 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?

 

[dextercioby]

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

Daniel.

 

[HallsofIvy]

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