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Spivak argues at page 577 in his book Physics for Mathematicians:

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What are the $2n$ relations he is talking about?

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In the very next line after OP's quote Spivak is considering the 1-form $$\vartheta~:=~\sum_{i=1}^n p_i \mathrm{d}q^i - \sum_{i=1}^n P_i \mathrm{d}Q^i~\in~\Gamma(T^{\ast}M) $$ in the cotangent bundle of the $2n$-dimensional phase space. He is effectively referring to the $2n$ component functions of the 1-form $\vartheta$. The fact that $\vartheta$ is closed $\mathrm{d}\vartheta=0$ means that it is locally exact $\vartheta=\mathrm{d}{\cal S}$ and specified by a single generating function ${\cal S}$, cf. the Poincare Lemma.

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