= Solution
For a <smooth manifold> $M$ of dimension $n$, the <tangent space> $T_pM$ can be defined as the space of derivations at $p$: linear maps $v$ on germs of smooth real functions satisfying $v(fg)=f(p)v(g)+g(p)v(f)$. Equivalently, $v$ is the velocity of a <smooth curve> through $p$, with curves identified when their coordinate velocities agree. A chart $(x^1,\ldots,x^n)$ supplies the basis $\partial_i|_p$, so $v=v^i\partial_i|_p$.
The <tangent bundle> is the disjoint union $TM=\coprod_{p\in M}T_pM$, with projection $\pi(v)=p$. Its smooth structure is defined by the local <local trivializations>
$$
\pi^{-1}(U)\longrightarrow x(U)\times\mathbb R^n,\qquad (p,v)\longmapsto(x(p),v^1,\ldots,v^n).
$$
On overlapping charts, the change of trivialization is
$$
(x,v)\longmapsto\bigl(y(x),Dy(x)v\bigr).
$$
These are smooth changes of coordinates on a $2n$-dimensional <smooth manifold>, linear and invertible in each fibre. Thus $TM$ is a rank-$n$ <vector bundle>, and its projection is a <submersion>. Fibrewise addition, scalar multiplication and the zero section are smooth. A smooth <section of a vector bundle> $X:M\to TM$, satisfying $\pi\circ X=\operatorname{id}_M$, is exactly a <vector field>; locally $X=X^i\partial_i$, with components transforming by the displayed <Jacobian matrix>. Its action on functions is a derivation, and the commutator of two such derivations is their <Lie bracket>.
The <dual bundle> of $TM$ is the <cotangent bundle> $T^*M$. Its smooth <sections of a vector bundle> are <differential 1-forms>, locally $\alpha=\alpha_i\,dx^i$. The coefficient transformation is the inverse transpose of that for tangent vectors, ensuring that $\alpha(X)$ is a well-defined smooth function. Tensoring these two bundles gives the <tensor bundles>
$$
T^r_sM=(TM)^{\otimes r}\otimes(T^*M)^{\otimes s}.
$$
Their <sections of a vector bundle> are <tensor fields>; their <transition functions of a vector bundle> are the corresponding tensor products of the tangent and cotangent transformations. For example, the bundle $\operatorname{End}(TM)=TM\otimes T^*M$ contains fields of linear endomorphisms, while $S^2T^*M$ contains symmetric bilinear fields. Taking alternating covariant tensors gives $\Lambda^kT^*M$, whose sections are <differential forms>. The <wedge product>, <exterior derivative> and <pullback of a differential form> make these bundles central to integration, <Generalized Stokes theorem> and <de Rham cohomology>.
A nowhere-vanishing section of $\Lambda^nT^*M$ chooses an <orientation of a smooth manifold>; such a section exists exactly when $M$ is orientable. Without an <orientation of a smooth manifold>, one can still integrate sections of the <density bundle> $|\Lambda^nT^*M|$, whose coordinate changes use the absolute <Jacobian determinant>. A <Riemannian metric> is a positive-definite section $g$ of $S^2T^*M$. It identifies $TM$ with $T^*M$ by the <musical isomorphisms> $X\mapsto g(X,\cdot)$ and gives the <Riemannian volume density>
$$
\sqrt{\det(g_{ij})}\,|dx^1\cdots dx^n|.
$$
On an oriented <smooth manifold>, this density corresponds to a <volume form>.
Another organizing construction is the <frame bundle> $FM$. A frame over $p$ is a linear isomorphism $u:\mathbb R^n\to T_pM$, and the right action $u\cdot A=u\circ A$ makes $FM$ a <principal bundle> with <general linear group> $GL(n,\mathbb R)$ as its structure group. Given a <group representation> $\rho:GL(n,\mathbb R)\to GL(V)$, the <associated bundle> $FM\times_\rho V$ is obtained from pairs $(u,v)$ by the relation $(uA,v)\sim(u,\rho(A)v)$. The standard representation reconstructs $TM$; dual, tensor and exterior representations reconstruct the associated bundles described above. A <Riemannian metric> reduces the <frame bundle> to the <orthonormal frame bundle>, with <orthogonal group> $O(n)$ as structure group; an <orientation of a smooth manifold> reduces it further to the <special orthogonal group> $SO(n)$. A global section of $FM$ is a global frame, so it exists exactly when $M$ is a <parallelizable manifold>. Local triviality does not imply a global trivialization: the <Hairy ball theorem> prevents even one nowhere-zero <vector field> on $S^2$.
A <connection on a vector bundle> differentiates its sections in tangent directions and defines <parallel transport>. For $TM$, the <Levi-Civita connection> is distinguished by compatibility with a <Riemannian metric> and vanishing <torsion tensor>. Its coefficients are not tensor components, because a coordinate change introduces second derivatives; the difference of two connections is a section of $T^*M\otimes\operatorname{End}(TM)$. The <Riemann curvature tensor> is a genuine tensor measuring the failure of covariant derivatives to commute.
Finally, a smooth map $f:M\to N$ gives the <pullback tangent bundle> $f^*TN$. Its sections are <vector fields along a map>, including the velocity of a <curve> and <Jacobi fields> along a <geodesic>. For an embedded <submanifold> $i:M\hookrightarrow N$, the derivative identifies $TM$ with a subbundle of $i^*TN$, and the quotient is the <normal bundle>. With an ambient <Riemannian metric>, the <normal bundle> is identified with the orthogonal complement of $TM$ in $i^*TN$; the <second fundamental form> records the normal component of the ambient derivative of tangent fields. These constructions connect the bundle description to both intrinsic and extrinsic <differential geometry>.
\b[The tangent bundle is a rank-$n$ smooth vector bundle over an $n$-manifold; its dual, tensor, exterior, frame and pullback constructions encode the principal geometric fields and their natural operations.]
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