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I am trying to learn how tensors transform under coordinate transformations. For an example, under a transformation from the coordinate system $x^\mu \longrightarrow x'^\mu$ a covariant tensor is defined by how it transforms, such as:

$$T'_{\mu \nu} = \frac{\partial x^\alpha}{\partial x'^\mu}\frac{\partial x^\beta}{\partial x'^\nu}T_{\alpha \beta}.$$

My question is then if the relation

$$\frac{\partial x'^\nu}{\partial x^\beta}T'_{\mu \nu} = \frac{\partial x^\alpha}{\partial x'^\mu}T_{\alpha \beta}$$

is true? Note that I have simply multiplied with $\frac{\partial x'^\nu}{\partial x^\beta}$ on both sides.

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    $\begingroup$ That equation is certainly true. But on its own it's not as meaningful as your first equation (tensor transformation law). What are you trying to show? $\endgroup$ Feb 8 at 22:27
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Based on your question it seems that just as myself you are at the early stages of trying to learn how tensors work and the rules which they follow. Therefore I will try and give you a brief overview of some of the key ideas that I have picked up that allow you to determine whether you're doing the correct thing.

If you keep in mind the following rules it would give you a rough idea whether the manipulations you're doing is correct:

  1. A particular index can only appear a maximum of twice in the equation, (this is the index that is summed over in the Einstein summation.)

  2. An index that is repeated should be written once as a covariant vector and once as a contravariant vector, that is to say that an index that is repeated should be written as once lowered and the other time raised, for example $v_iv^i$.

  3. Indices that are not repeated and only appear once in the equation are called dummy indices. If a dummy index appears on the left hand side of the equation the same dummy index should appear in the right hand side of the equation.

Keeping these rules in mind, it is easy to see the expression that you have written is consistent and is correct, although not as meaningful as

$$T'_{\mu \nu} = \frac{\partial x^\alpha}{\partial x'^\mu}\frac{\partial x^\beta}{\partial x'^\nu}T_{\alpha \beta}.$$ or alternatively, as $$ T_{\alpha \beta} = \frac{\partial x'^\mu}{\partial x^\alpha}\frac{\partial x'^\nu}{\partial x^\beta} T'_{\mu \nu}.$$ since these relationships show explicitly the ways in which a tensor will transform between coordinate systems. You can see how what I have written above is simply what you have written but 'rearranged' in a similar sense to what you have done to get your second expression.

We can write a general expression for the transformation of tensor of an arbitrary order as follows:

$$T'^{i_ii_2...i_p}_{j_1j_2...j_q} = \frac{\partial x'^{i_1}}{\partial x^{k_1}} ... \frac{\partial x'^{i_p}}{\partial x^{k_p}} \frac{\partial x^{m_1}}{\partial x'^{j_1}} ... \frac{\partial x^{m_q}}{\partial x'^{j_q}} T^{k_1k_2...k_p}_{m_1m_2...m_q}$$

You can see how this expression abides by the three rules I have stated earlier. I hope you find this to be of use.

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  • $\begingroup$ Yes I have just started to learn how tensors work and their rules. Thank you for your overview, it definitely helped! $\endgroup$
    – saundelin
    Feb 11 at 12:18

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