Definition of the differential of a function

Question

I have been reading some differential geometry from the book " An Introduction to Manifolds" by Tu and so far i like how he exposes the topic. Now i got to the part where he defines Differential forms in $\mathbb{R}^n$ and i understood the definition but then he defines the differential of a $C^{\infty}$ function, that its going to be a 1-form, and i am confused with is notation.

He writes $(df)_p(X_p)=X_pf$ and i dont quite understand what is going on in the last term , how is that going to be a tangent vector to the point $p$, i guess im having some trouble with the notation. Is the function calculated at some point?

Thanks.

Answer 1

Remember that $X_pf$ is the directional derivative of $f$ at $p$ in the direction of $X_p$. Remember from basic calculus that (for $C^1$ functions in Euclidean space) the directional derivative is the dot product $\nabla f(p)\cdot X_p$. The one-form $df$ gives exactly this computation: $df(p)(X_p)$ is likewise that directional derivative.