How to show something is a tensor.

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In summary, the conversation discusses the proof of \nabla \vec{V} being a (1 1) tensor with components V^\alpha{}_{;\beta}. The speaker is unsure about which method to use for the proof and is seeking help. The other person mentions a general theorem that can be used to show that V is a tensor.
  • #1
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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  • #2
Yes, the second method (behaving under general coordinate transformations) can't fail.

Daniel.
 
  • #3
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.
 

FAQ: How to show something is a tensor.

1. What is a tensor?

A tensor is a mathematical object that is used to represent the relationships between different quantities in a system. It is a type of multidimensional array that can have any number of dimensions, and it is commonly used in physics and engineering.

2. How do you show that something is a tensor?

To show that something is a tensor, you need to demonstrate that it follows the rules of tensor algebra. This includes satisfying the properties of linearity, transformation under coordinate changes, and the use of tensor notation. Additionally, you can use mathematical proofs or examples to show that the object behaves like a tensor.

3. What are the properties of a tensor?

The properties of a tensor include linearity, meaning that it can be scaled and added together; transformation under coordinate changes, meaning that it changes in a predictable way when the coordinate system is changed; and the use of tensor notation, which allows for a concise and consistent way of representing tensors.

4. How can you tell if a tensor is symmetric or antisymmetric?

A symmetric tensor is one that remains the same after swapping any two indices. An antisymmetric tensor is one that changes sign after swapping any two indices. To determine if a tensor is symmetric or antisymmetric, you can use mathematical operations such as transposition or permutation to see if the tensor remains the same or changes sign, respectively.

5. Can a scalar be considered a tensor?

Yes, a scalar can be considered a tensor of rank 0, meaning it has no indices and is just a single value. In fact, all scalars are tensors, but not all tensors are scalars.

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