Tangent Vectors in R^n as Derivations ....

[Math Amateur]

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:
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

 

[GJA]

Hi Peter,

Question 1

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

$f$ is a $C^{\infty}$ function on an open set containing the point $p$ (see page 4 of Tu). The point of introducing them is the following: Differential calculus in $\mathbb{R}^{n}$ boils down to basically one idea -- given a differentiable function $f$ at a point $p$ we calculate the linear mapping (i.e. derivative) from the tangent space at $p$ to the tangent space at $f(p)$. In $\mathbb{R}^{n}$ it is natural to define tangent vectors geometrically; i.e., as arrows emanating from a point.

When you are studying an abstract manifold, you want to do exactly the same thing. The problem is that you must now imagine the manifold as an object in its own right and not necessarily some subset of $\mathbb{R}^{n}.$ Because we lose the ambient $\mathbb{R}^{n}$ space, we now have nothing to naturally/geometrically guide our definition of tangent spaces and tangent vectors. Hence, we do what is always done in abstract mathematics: We look to the properties that a known object possesses and formulate a definition of the abstract object using said properties. In the case of tangent vectors, they can be thought of as linear functionals on the space of $C^{\infty}$ functions, as opposed to geometric arrows. How is this done? Through the directional derivative. Tu says this on page 11 when he says that the association of $v$ with $D_{v}$ gives us a way to think of vectors now as operators (instead of arrows). An operator needs an object onto which it acts, and this is the purpose of the $C^{\infty}$ functions $f$ at $p$.

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?

Tu defines $C^{\infty}$ functions to be real-valued. He uses the word "mapping" when talking about maps between manifolds.

 

[Math Amateur]

Hi Peter,
$f$ is a $C^{\infty}$ function on an open set containing the point $p$ (see page 4 of Tu). The point of introducing them is the following: Differential calculus in $\mathbb{R}^{n}$ boils down to basically one idea -- given a differentiable function $f$ at a point $p$ we calculate the linear mapping (i.e. derivative) from the tangent space at $p$ to the tangent space at $f(p)$. In $\mathbb{R}^{n}$ it is natural to define tangent vectors geometrically; i.e., as arrows emanating from a point.

When you are studying an abstract manifold, you want to do exactly the same thing. The problem is that you must now imagine the manifold as an object in its own right and not necessarily some subset of $\mathbb{R}^{n}.$ Because we lose the ambient $\mathbb{R}^{n}$ space, we now have nothing to naturally/geometrically guide our definition of tangent spaces and tangent vectors. Hence, we do what is always done in abstract mathematics: We look to the properties that a known object possesses and formulate a definition of the abstract object using said properties. In the case of tangent vectors, they can be thought of as linear functionals on the space of $C^{\infty}$ functions, as opposed to geometric arrows. How is this done? Through the directional derivative. Tu says this on page 11 when he says that the association of $v$ with $D_{v}$ gives us a way to think of vectors now as operators (instead of arrows). An operator needs an object onto which it acts, and this is the purpose of the $C^{\infty}$ functions $f$ at $p$.
Tu defines $C^{\infty}$ functions to be real-valued. He uses the word "mapping" when talking about maps between manifolds.

Well .. ! That made sense of things for me!

Thanks for a VERY helpful post, GJA ...

Very much appreciate your help ...

Peter