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I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ...

I need help in order to fully understand Tu's section on tangent vectors in \(\displaystyle \mathbb{R}^n\) as derivations... In his section on tangent vectors in \mathbb{R}^n as derivations, Tu writes the following:View attachment 8637

View attachment 8638In the above text from Tu we read the following:

" ... ... If \(\displaystyle f\) is \(\displaystyle C^{ \infty }\) in a neighborhood of \(\displaystyle p\) in \(\displaystyle \mathbb{R}^n\) and \(\displaystyle v\) is a tangent vector at \(\displaystyle p\), the directional derivative of \(\displaystyle f\) in the direction of \(\displaystyle p\) ... ... "

My questions are as follows:

What are these functions \(\displaystyle f\) that Tu is introducing ... and further, what is the point of them ... ?

The domain of \(\displaystyle f\) is clearly \(\displaystyle \mathbb{R}^n\) ... BUT ... what is the range of \(\displaystyle f\) ... I am guessing it is \(\displaystyle \mathbb{R}\) ... is that correct ... but why is \(\displaystyle f\) real-valued?

Hope that someone can clarify these issues ...

Peter

I need help in order to fully understand Tu's section on tangent vectors in \(\displaystyle \mathbb{R}^n\) as derivations... In his section on tangent vectors in \mathbb{R}^n as derivations, Tu writes the following:View attachment 8637

View attachment 8638In the above text from Tu we read the following:

" ... ... If \(\displaystyle f\) is \(\displaystyle C^{ \infty }\) in a neighborhood of \(\displaystyle p\) in \(\displaystyle \mathbb{R}^n\) and \(\displaystyle v\) is a tangent vector at \(\displaystyle p\), the directional derivative of \(\displaystyle f\) in the direction of \(\displaystyle p\) ... ... "

My questions are as follows:

**Question 1**What are these functions \(\displaystyle f\) that Tu is introducing ... and further, what is the point of them ... ?

**Question 2**The domain of \(\displaystyle f\) is clearly \(\displaystyle \mathbb{R}^n\) ... BUT ... what is the range of \(\displaystyle f\) ... I am guessing it is \(\displaystyle \mathbb{R}\) ... is that correct ... but why is \(\displaystyle f\) real-valued?

Hope that someone can clarify these issues ...

Peter