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In textbooks, a tensor is usually defined in terms of its transformation properties. But this definition actually seems vague when it comes to checking a set of quantities to see whether they form a tensor or not. Imagine I have four functions and want to see whether they form a 2d 2nd rank tensor together or not. Then if I want to use the transformation properties, I should check whether ## f_{ij}=J(i,k)J(j,l)f'_{kl} ##. But that's weird because that needs us to know what is ## f'_{kl} ## which means we should have an accepted method of transformation. But what is that method? Surely just replacing the coordinates doesn't work. Another way I thought of was through seeing tensors as operators. Then we would want the tranformed operator to do the same thing as the old operator, to the transformed vectors. But this method seems too subjective and doesn't seem general enough.
So I want to know, how can I check whether a bunch of functions form a tensor or not?
Thanks
So I want to know, how can I check whether a bunch of functions form a tensor or not?
Thanks