- Summary
- I'm attempting to understand how the definition of a tensor in terms of how it transforms arises from demanding invariance of the tensor when we define it as a multilinear map

Hi, I'm worried I've got a grave misunderstanding. Also, throughout this post, a prime mark (') will indicate the transformed versions of my tensor, coordinates, etc.

I'm going to define a tensor.

$$T^\mu_\nu \partial_\mu \otimes dx^\nu$$

Now I'd like to investigate how the tensor transforms under an arbitrary coordinate transformation where I require that the tensor is invariant under the coordinate transformation. In order to understand how partial-mu transforms, I'm going to apply the chain rule for partial derivatives.

$$\begin{equation} \partial'_\alpha = \frac{\partial}{\partial x'^\alpha} = \frac{\partial x^\mu}{\partial x'^\alpha}\partial_\mu \end{equation}$$

Similarly, I can write down how the basis one-forms transform.

$$\begin{equation} dx'^\alpha = \frac{\partial x'^\alpha}{\partial x^\mu} dx^\mu \end{equation}$$

Therefore, the basis of my tensor transforms like this:

$$\begin{equation} T'^\alpha_\beta \partial'_\alpha \otimes dx'^\beta = T^\mu_\nu \frac{\partial x^\mu}{\partial x'^\alpha} \partial_\mu \otimes \frac{\partial x'^\alpha}{\partial x^\mu} dx^\nu \end{equation}$$

Which mirrors my more familiar definition of a tensor in terms of how it's components transform:

$$\begin{equation} T'^\alpha_\beta = \frac{\partial x'^\alpha}{\partial x^\mu}\frac{\partial x^\nu}{\partial x'^\beta} T^\mu_\nu \end{equation}$$

Is it correct to write down the transformation of the tensor in the way I did in equation (3)? Is this equivalent to equation (4)?

Please help me if I've had a misunderstanding. Thank you very much for any help.

I'm going to define a tensor.

$$T^\mu_\nu \partial_\mu \otimes dx^\nu$$

Now I'd like to investigate how the tensor transforms under an arbitrary coordinate transformation where I require that the tensor is invariant under the coordinate transformation. In order to understand how partial-mu transforms, I'm going to apply the chain rule for partial derivatives.

$$\begin{equation} \partial'_\alpha = \frac{\partial}{\partial x'^\alpha} = \frac{\partial x^\mu}{\partial x'^\alpha}\partial_\mu \end{equation}$$

Similarly, I can write down how the basis one-forms transform.

$$\begin{equation} dx'^\alpha = \frac{\partial x'^\alpha}{\partial x^\mu} dx^\mu \end{equation}$$

Therefore, the basis of my tensor transforms like this:

$$\begin{equation} T'^\alpha_\beta \partial'_\alpha \otimes dx'^\beta = T^\mu_\nu \frac{\partial x^\mu}{\partial x'^\alpha} \partial_\mu \otimes \frac{\partial x'^\alpha}{\partial x^\mu} dx^\nu \end{equation}$$

Which mirrors my more familiar definition of a tensor in terms of how it's components transform:

$$\begin{equation} T'^\alpha_\beta = \frac{\partial x'^\alpha}{\partial x^\mu}\frac{\partial x^\nu}{\partial x'^\beta} T^\mu_\nu \end{equation}$$

Is it correct to write down the transformation of the tensor in the way I did in equation (3)? Is this equivalent to equation (4)?

Please help me if I've had a misunderstanding. Thank you very much for any help.