# Definition:Differential Form

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## Definition

Let $X$ be a smooth manifold.

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

Here $T_x \left({X}\right)$ is the tangent space of $X$.