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

• MHB
• Math Amateur
In summary, Tu's book "An Introduction to Manifolds" (Second Edition) discusses the directional derivative of a function with respect to a point in a manifold, and how to think of vectors as operators. 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 studying
Math Amateur
Gold Member
MHB
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

#### Attachments

• Tu - 1 - Start of Section 2.1 ... PART 1 .png
3.7 KB · Views: 66
• Tu - 2 - Start of Section 2.1 ... PART 2 ... .png
33.6 KB · Views: 69
Hi Peter,

Peter said:
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 $p$.

Peter said:
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.

GJA said:
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

## 1. What are tangent vectors in R^n?

Tangent vectors in R^n are mathematical objects used to describe the direction and rate of change of a curve or surface at a given point in n-dimensional space. They are represented by arrows or vectors that are tangent to the curve or surface at that point.

## 2. How are tangent vectors related to derivatives?

Tangent vectors are closely related to derivatives, as they can be thought of as generalized derivatives. In fact, tangent vectors in R^n can be defined as derivations, which are linear operators that act on functions to produce directional derivatives.

## 3. What is the difference between a tangent vector and a normal vector?

A tangent vector is a vector that is tangent to a curve or surface at a given point, while a normal vector is perpendicular to the tangent vector and is used to describe the direction of the surface or curve's curvature at that point. In other words, a tangent vector lies on the surface or curve, while a normal vector points away from it.

## 4. How are tangent vectors used in real-world applications?

Tangent vectors have various real-world applications, such as in physics, engineering, and computer graphics. In physics, they are used to describe the motion and acceleration of objects, while in engineering, they are used to analyze the behavior of structures and materials. In computer graphics, they are used to model and render smooth surfaces and curves.

## 5. Can tangent vectors be defined in spaces other than R^n?

Yes, tangent vectors can be defined in other spaces, such as manifolds, which are generalizations of curves and surfaces in higher dimensions. In fact, tangent vectors can be defined in any space that has a smooth structure, where the concept of a curve or surface makes sense.

• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
4
Views
706
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
2
Views
2K
• Topology and Analysis
Replies
2
Views
1K
• Topology and Analysis
Replies
9
Views
2K
• Topology and Analysis
Replies
5
Views
2K
• Topology and Analysis
Replies
3
Views
2K
• Topology and Analysis
Replies
5
Views
866