For example, if I use spherical coordinates, r is X1, theta is X2 and phi is X3. But could I use, for example, phi as X1, theta as X2 and r as X3 in the calculus of tensors?
Sure you can but you have to stay consistent in your transformations!
and definitely define your metric tensor accordingly.
Hope that helps
Indeed, as Mr. Hady Joumaa said, the order does not matter as long as you make necessary changes in the
metric tensor g_{ij} etc
Order of coordinates is not necessary but you should annotate coordinates.
I also agree with Prof. Hady.
In fact, we must use oriented base vectors , because we can not always have coordinates (as in non-holonomic). The order is important because it determines an orientation of the axes and the signal of the Jacobian.
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