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Not really a homework problem. Need some help understanding tensors.

Ok, so the chapter in the book I am using, Vector Calculus by Paul C. Matthews introduces first the coordinate transformation and proceeds to say that a vector is anything which transforms according to the rule [tex]v_i'=L_{ij}v_j[/tex]. That's fair and I see why this is so. Later on, it introduces a tensor and says: A quantity is a

It will be nice if somebody could explain.

Thanks.

Ok, so the chapter in the book I am using, Vector Calculus by Paul C. Matthews introduces first the coordinate transformation and proceeds to say that a vector is anything which transforms according to the rule [tex]v_i'=L_{ij}v_j[/tex]. That's fair and I see why this is so. Later on, it introduces a tensor and says: A quantity is a

*tensor*if each of the free suffices transforms according to the rule [tex]x_i'=L_{ij}x_j[/tex]. "For example, consider a quantity that [tex]T_{ij}[/tex] that has two free suffices. This quantity is a tensor if its components in the dashed frame are related to those in the undashed frame by the equation: [tex]T_{ij}'=L_{ik}L_{jm}T_{km}[/tex]." I don't really understand what's in the quotations.It will be nice if somebody could explain.

Thanks.

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