# How to show something is a tensor.

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?

Last edited:

dextercioby
Homework Helper
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.