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Let $M$ be a smooth manifold with an atlas $\{(U_i,\psi_i)\}$. If $x\in U_i\cap U_j$, let $a_{ji}(x)$ denote the Jacobian matrix at $\psi_i(x)$ of the smooth map $\psi_j^{-1}\circ \psi_i:\psi_i(U_i\cap U_j)\to \psi_j(U_i\cap U_j)$. Then $a_{ji}$ is a smooth map $U_i\cap U_j\to GL(n,\Bbb R)$ and satisfies the cocycle condition $a_{ki}=a_{kj}a_{ji}$.

Now define $g_{ji}=a_{ji}^m$, where $m$ is a fixed positive integer, and let $E\to M$ be a smooth vector bundle that has the transition functions $\{g_{ji}\}$. Then $E$ is called the the bundle of relative scalars over $M$ of weight $m$, according to Steenrod's The Topology of Fibre Bundles, section 6.

I see that if $m=1$, then $E\to M$ is the tangent bundle of $M$. But for $m>1$, what is $E$? I mean, I haven't seen these kind of definitions in textbooks about smooth manifolds, for example, Lee's Introduction to Smooth Manifolds. But maybe these bundles have another familiar names nowadays, I hope.

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I think you're misinterpreting Steenrod's definition. (Admittedly, his exposition is hard to follow on this point.) What he calls the bundle of relative scalars of weight $\boldsymbol w$ is the rank-$1$ vector bundle with transition functions $\det(a_{ij})^w$, where $(a_{ij})$ are the transition functions for the tangent bundle as you defined them above.

Modern mathematicians usually call this the bundle of densities of weight $\boldsymbol w$. See my answer to this question for a detailed explanation.

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