= Solution
A <tangent vector> at $p$ is a <derivation at a point>: a real-linear map $v:C_p^\infty(M)\to\mathbb R$ on <germs> of smooth functions satisfying
$$
v(fh)=f(p)v(h)+h(p)v(f).
$$
Addition and scalar multiplication preserve this identity, defining the <tangent space> $T_pM$ as a <vector space>. A derivation annihilates constants. In a <manifold chart> $x=(x^1,\ldots,x^n)$ centered near $p$, the local factorization $f-f(p)=\sum_i(x^i-x^i(p))f_i$, with $f_i(p)=\partial_i f(p)$, gives $v(f)=\sum_iv(x^i)\partial_i f(p)$. The factorization follows by integrating the first derivatives along a straight segment in a sufficiently small coordinate ball. Therefore
$$
\boxed{v=\sum_{i=1}^n v^i\partial_i|_p,\qquad v^i=v(x^i),\qquad\dim T_pM=n.}
$$
Conversely these coordinate derivatives satisfy the derivation identity, so they really give a basis rather than merely a spanning family.
The <cotangent space> is the <dual vector space> $T_p^*M=(T_pM)^*$. Its basis $dx^i|_p$ is characterized by $dx^i(\partial_j)=\delta^i_j$. If $\alpha=a_i dx^i=b_jdy^j$, the <chain rule> gives the <cotangent coordinate transition>
$$
\boxed{b_j=\sum_i a_i\frac{\partial x^i}{\partial y^j}(p).}
$$
The <cotangent bundle> is the disjoint union $T^*M=\coprod_{p\in M}T_p^*M$, with projection $\pi(p,\alpha)=p$. For each base <manifold chart> $(U,x)$, define
$$
\Phi_x:\pi^{-1}U\longrightarrow x(U)\times\mathbb R^n,\qquad(p,a_i dx^i|_p)\longmapsto(x(p),a_1,\ldots,a_n).
$$
Its inverse sends $(z,a)$ to $(x^{-1}(z),a_i dx^i|_{x^{-1}(z)})$. On overlaps the transition is $(x,a)\mapsto(y(x),b)$, with the displayed fibre-linear transformation. Its coefficients and inverse are smooth, so these maps give a <smooth atlas> in dimension $2n$. Give the total space the topology obtained by transporting the product topology through these charts. Different base points are separated by disjoint base neighborhoods; distinct points over the same base point are separated within one bundle chart. A countable base <smooth atlas> and countable product bases give second countability. Thus the total space is a Hausdorff, second-countable <smooth manifold>. The same charts are <vector bundle trivializations>, since $\pi$ is product projection and their fibre changes are invertible linear maps.
The intrinsic <canonical one-form on a cotangent bundle> is
$$
\lambda_{(p,\alpha)}(W)=\alpha(d\pi_{(p,\alpha)}W).
$$
In the bundle chart it is $\lambda=\sum_i a_i dx^i$. This definition is coordinate independent, hence its <exterior derivative> is the globally defined smooth <differential two-form>
$$
\boxed{\eta=d\lambda=\sum_i da_i\wedge dx^i.}
$$
This sign follows the fibre-first order in this problem; the equally common position-first <symplectic form> is $-d\lambda$. The <volume form>
$$
\eta^n=n!\,da_1\wedge dx^1\wedge\cdots\wedge da_n\wedge dx^n
$$
never vanishes. A smooth manifold is orientable exactly when it admits a nowhere-zero top-degree form; its positive ordered bases determine a consistent <orientation>. Consequently \b[$T^*M$ is an <orientable smooth manifold>, even when $M$ is not]. This is <cotangent bundle orientation>. In dimension zero the same conclusion uses the nowhere-zero zero-form $1$.
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