# Axis Through x in the direction u .... in R^k .... ....

• MHB
• Math Amateur
In summary, the conversation discusses the concept of an axis in relation to an equation in the book "Several Real Variables" by Shmuel Kantorovitz. It is mentioned that the line through a given point in a certain direction can be considered as an axis, which is important for understanding the directional derivative of a function on that line. The conversation also explores the idea of plotting a graph in a different coordinate system and the use of partial derivatives in understanding this concept.
Math Amateur
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MHB
I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...

I am currently focused on Chapter 2: Derivation ... ...

I need help with an interpretation of a statement by Kantorovitz near to the start of Chapter 2 ...

The start to Chapter 2 in Kantorovitz reads as folows:
View attachment 7800
https://www.physicsforums.com/attachments/7801
At the end of the above quoted text we read the following:

" ... ... In general, for any unit vector $$\displaystyle u \in \mathbb{R}^k$$ and $$\displaystyle x \in \mathbb{R}^k$$, the axis through $$\displaystyle x$$ in the direction $$\displaystyle u$$ is the directed line with the parametric representation

$$\displaystyle \gamma \ : \ t \in \mathbb{R} \rightarrow \gamma (t) := x + tu \in \mathbb{R}^k$$ ... ... "
My question is as follows:

Why does Kantorovitz refer to the above line as "the axis through $$\displaystyle x$$ in the direction $$\displaystyle u$$" ... surely it is just a line as in my diagram below showing the line through $$\displaystyle x$$ in the direction $$\displaystyle u$$ in $$\displaystyle \mathbb{R}^3$$ ... it is not an axis but simply a line ..https://www.physicsforums.com/attachments/7802The required equation of the line, I think, arises as follows:Consider an arbitrary point, $$\displaystyle P$$, on the line given by $$\displaystyle \gamma (t)$$ where $$\displaystyle t \in \mathbb{R}$$.$$\displaystyle u \in \mathbb{R}^k$$ is a vector parallel to the direction of the line ...... we have that $$\displaystyle \gamma (t) = OP$$$$\displaystyle \Longrightarrow \gamma (t) = OP_0 + tu$$$$\displaystyle \Longrightarrow \gamma (t) = x + tu$$
Is that a correct interpretation of the line/axis through x in the direction u ... ?

Peter

Hi Peter,

As I understand, you are wondering why the line through $P_0$ is called an axis. The idea is that we will consider the restriction of $f$ on that line; in this case, $f$ is a function of $t$ only, and you could imagine plotting a graph of $f$ in a coordinate system where $t$ would be the axis of the independent variable and you would plot the value of $f$ on another axis (outside $\mathbb{R}^k$). The directional derivative of $f$ is simply the derivative of $f(t)$ on that graph.

You could also imagine setting up a new coordinate system centered at $P_0$ with one axis $t$ in the direction $\mathbf{u}$. The directional derivative of $f$ is simply the partial derivative $\partial f/\partial t$.

castor28 said:
Hi Peter,

As I understand, you are wondering why the line through $P_0$ is called an axis. The idea is that we will consider the restriction of $f$ on that line; in this case, $f$ is a function of $t$ only, and you could imagine plotting a graph of $f$ in a coordinate system where $t$ would be the axis of the independent variable and you would plot the value of $f$ on another axis (outside $\mathbb{R}^k$). The directional derivative of $f$ is simply the derivative of $f(t)$ on that graph.

You could also imagine setting up a new coordinate system centered at $P_0$ with one axis $t$ in the direction $\mathbf{u}$. The directional derivative of $f$ is simply the partial derivative $\partial f/\partial t$.

Thanks castor28 ... well ... that cleared up that matter ... ! ... appreciate the assistance ...

Thanks again ...

Peter

## 1. What is an axis through x in the direction u in R^k?

An axis through x in the direction u in R^k refers to a straight line passing through a specific point (x) and extending infinitely in the direction of a vector (u) in a k-dimensional space. This line is often used as a reference point for measuring distances and angles in the given space.

## 2. How is an axis through x in the direction u represented mathematically?

In mathematics, an axis through x in the direction u in R^k is typically represented using vector notation. The line can be expressed as x + tu, where x is the starting point, u is the direction vector, and t is a scalar that determines the position of points on the line.

## 3. What is the purpose of defining an axis through x in the direction u?

The purpose of defining an axis through x in the direction u is to provide a reference point for analyzing and understanding geometric relationships in a k-dimensional space. By establishing this line, we can measure the distance between points, determine the angle between vectors, and perform other calculations that help us understand the geometry of the space.

## 4. Can an axis through x in the direction u exist in spaces other than R^k?

Yes, an axis through x in the direction u can exist in any space that is defined by a set of coordinates and a direction vector. This includes spaces other than R^k, such as Euclidean spaces, affine spaces, and projective spaces.

## 5. How is an axis through x in the direction u used in real-world applications?

An axis through x in the direction u is used in many real-world applications, particularly in fields such as physics, engineering, and computer graphics. It is used to represent and analyze various physical phenomena, such as the motion of objects, the orientation of structures, and the flow of fluids. Additionally, it is used in computer graphics to create 3D models and animations by defining the position and orientation of objects in a virtual space.

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