# Definition:Differential Form

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## Definition

Let $X$ be a smooth manifold.

Let $\map {T_x} X$ denotes the tangent space of $X$.

A **$p$-form** on $X$ is a function $\omega: \map {T_x} X^p \to \R$ defined at each point of $X$ which takes $p$ vectors as inputs, and outputs a real number.