The weight ($W$), determined by the determinant factor $\left(J^{-1}\right)^W$, dictates their algebraic properties. Addition of two tensor densities is only possible if they share the same type and the same weight, as the determinant factor must be identical to be factored out. Multiplication (outer product) results in a new tensor density whose weight is the sum of the individual weights ( $W_1+W_2$ ), a direct consequence of multiplying the two determinant factors. Finally, contraction (summing over a contravariant and a covariant index) uniquely preserves the weight of the original tensor density, because the transformation factors for the contracted indices cancel each other out, leaving the determinant factor $\left(J^{-1}\right)^W$ unchanged.
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%% Condensed Notes
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%% Proof and Derivation
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%% Proof and Derivation
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