In the Arfken book, I came across an statement that has been bothering me for a long time. At the beginning I thought that I would not understand unless I became familiarized with tensors, but then I forced myself to learn the basics of that, and yet I am not at ease with the statement. Why in cartesian coordinate system, the following is true?
${\frac{\partial x'_i}{\partial x_j}=\frac{\partial x_j}{\partial x'_i}}$
Is there a formal way to demonstrate the equality above in the context of tensors, and an intuitive explanation where the following is assumed: the partial derivative on the left side is taken with the other x' coordinates fixed, while the partial derivative on the right side is taken with the other unprimed coordinates fixed. Maybe the arrangement below makes sense to someone else.
$\left(\frac{\partial x'_i}{\partial x_j}\right)_{x_k}=\left(\frac{\partial x_j}{\partial x'_i}\right)_{{x_k}^{'}}$
I understand the argument in terms of projections or dot products, but someone more experienced may see it immediately and have a better explanation.
Thank you for your time.