How to prove that something transforms like a tensor?

[Chris B]

Homework Statement

I have several problems that ask me to prove that some quantity "transforms like a tensor"

For example:

"Suppose that for each choice of contravariant vector (a vector) A^nu(x), the quantities B_mu(x) are defined at teach point through a linear relationship of the form
B_mu(x) =T_mu_nu(x) A^nu(x)
transform like a covariant vector (1-form). Prove that the quantities T_mu_nu(x) transform like a tensor of type (0,2) at each point."

(Here an underscore followed by a letter is a lower index and a caret followed by a letter is an upper index).

Homework Equations

Transformation property of a tensor:
T'_mu_nu = dx^mu/dx'^mu dx^nu/dx'^nu T_mu_nu

(dx is a partial derivative and)

The Attempt at a Solution

My first guess is that I need to apply a coordinate transformation to both sides of the equation given in the problem, but I'm kind of stuck there. I don't know how to manipulate things to get T_mu_nu by itself and show it obeys the tensor transformation property.

 

[Diegor]

Homework Statement

I have several problems that ask me to prove that some quantity "transforms like a tensor"

For example:

"Suppose that for each choice of contravariant vector (a vector) A^nu(x), the quantities B_mu(x) are defined at teach point through a linear relationship of the form
B_mu(x) =T_mu_nu(x) A^nu(x)
transform like a covariant vector (1-form). Prove that the quantities T_mu_nu(x) transform like a tensor of type (0,2) at each point."

(Here an underscore followed by a letter is a lower index and a caret followed by a letter is an upper index).

Homework Equations

Transformation property of a tensor:
T'_mu_nu = dx^mu/dx'^mu dx^nu/dx'^nu T_mu_nu

(dx is a partial derivative and)

The Attempt at a Solution

My first guess is that I need to apply a coordinate transformation to both sides of the equation given in the problem, but I'm kind of stuck there. I don't know how to manipulate things to get T_mu_nu by itself and show it obeys the tensor transformation property.

I think you are correct. To get the partial derivative of x with respect to x prime you need x as a function x prime. And that is the coordinate transformation you mentioned. Probably in you problem or examples the tensor component is already given in one coordinaye system.