Suppose $A^{\mu_1 \cdots \mu_n}_{\nu_{n+1}\cdots \nu_m}$ is a tensor. That means it transforms a tensor. How do I show that it transforms as a tensor? How do I see that $\cos (A^{\mu_1 \cdots \mu_n}_{\nu_{n+1}\cdots \nu_m})$ or some other non-tensor does not transform like a tensor?
Answer
A tensor (or more appropriately tensor field) $T^{i_1 \dots i_m}_{j_1 \dots j_n}$ of rank $(m,n)$ transforms as,
$$T'^{i'_1 \dots i'_m}_{j'_1 \dots j'_m} = \left( \frac{\partial\tilde{x}^{i'_1}}{\partial x^{i_1}} \dots \frac{\partial\tilde{x}^{i'_m}}{\partial x^{i_m}} \right) \left(\frac{\partial x^{j_1}}{\partial \tilde{x}^{j'_1}}\dots \frac{\partial x^{j_m}}{\partial \tilde{x}^{j'_m}} \right)T^{i_1 \dots i_m}_{j_1 \dots j_n} $$
under a coordinate transformation $x \to \tilde{x}$. Now, if we have some multi-dimensional array, we need to know specifically what it is to be able to check if it is a tensor. Without further details, the best we can say is it transforms as a tensor if it satisfies the above definition.
As an example, consider the Christoffel symbols,
$$\Gamma^i_{kl} = \frac12 g^{im} \left( \partial_l g_{mk} + \partial_k g_{ml} - \partial_m g_{kl} \right).$$
We know that the metric $g$ is a tensor and thus we know exactly how it transforms under a coordinate transformation. Thus, to figure out if the Christoffel symbols are tensors, we need only plug this in and compute. But we find that,
$$\Gamma'^{k}_{ij} = \frac{\partial x'^p}{\partial x^i}\frac{\partial x'^q}{\partial x^j} \Gamma^r_{pq} \frac{\partial x^k}{\partial x'^r} + \frac{\partial x^k}{\partial x'^m} \frac{\partial^2 x'^m}{\partial x^i \partial x^j}$$
and thus clearly does not transform in the manner defined for a tensor, which is why we call them the Christoffel symbols. As you can see, we need to know what the quantity in question is to be able to check if it transforms the right way.
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