Definition:Differential Form

Let $$X \ $$ be a smooth manifold.

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

Here $$T_x(X) \ $$ is the tangent space.