Let $Q$ be a smooth $n$-manifold, let $U\subset Q$ be a coordinate domain with local coordinates $(q_1,\dots,q_n)$, and let $\tau:T^*U\to U$ be the cotangent bundle projection. Define the tautological one-form $\lambda\in\Omega^1(T^*U)$ by $\lambda_\beta(v)=\beta(d\tau_\beta(v))$ for every $\beta\in T^*U$ and every $v\in T_\beta(T^*U)$. Define the induced fibre coordinate functions $p_i:T^*U\to\mathbb R$ by the identity $\beta=\sum_{i=1}^n p_i(\beta)(dq_i)_{\tau(\beta)}$ for every $\beta\in T^*U$. If $q_i$ also denotes the pulled-back coordinate function $q_i\circ\tau:T^*U\to\mathbb R$, then the tautological one-form is $\lambda=\sum_{i=1}^n p_i\,dq_i$ on $T^*U$.