I am reading Analysis on Manifolds by James R. Munkres.

I can’t clearly understand the following passage by the author. It leaves me with a vague, unsettled feeling.

On p.171 in this book:

Definition. Let $V$ be an $n$-dimensional vector space. An $n$-tuple $(a_1,\dots,a_n)$ of independent vectors in $V$ is called an $n$-frame in $V$. In $\mathbb{R}^n$, we call such a frame right-handed if $$\det [a_1\dots,a_n]>0;$$ we call it left-handed otherwise. The collection of all right-handed frames in $\mathbb{R}^n$ is called an orientation of $\mathbb{R}^n$; and so it the collection of all left-handed frames. More generally, choose a linear isomorphism $T:\mathbb{R}^n\to V$, and define one orientation of $V$ to consist of all frames of the form $\left(T(a_1),\dots,T(a_n)\right)$ for which $(a_1,\dots,a_n)$ is a right-handed frame in $\mathbb{R}^n$, and the other orientation of $V$ to consist of all such frames for which $(a_1,\dots,a_n)$ is left-handed. Thus $V$ has two orientations; each is called the reverse, or the opposite, of the other.

On p.172 in this book:

EXAMPLE 2. In $\mathbb{R}^1$, a frame consists of a single non-zero number; it is right-handed if it is positive, and left-handed if it is negative. In $\mathbb{R}^2$, a frame $(a_1,a_2)$ is right-handed if one must rotate $a_1$ in c counterclockwise direction throught an angle less than $\pi$ to make it point in the same direction as $a_2$. (See the exercises.) In $\mathbb{R}^3$, a frame $(a_1,a_2,a_3)$ is right-handed if curling the fingers of one's right hand in the direction from $a_1$ to $a_2$ makes one's thumb point in the direction of $a_3$. See Figure 20.2.

The author’s explanation in Example 2 clearly assumes the use of a right-handed coordinate system. However, for instance, $\mathbb{R}^2$ is nothing more than the set of all ordered pairs of real numbers. Is it possible to define the concept of a right-handed coordinate system mathematically?

Let $e_1:=(1,0)$ and $e_2:=(0,1)$.
Then, the frame $(e_1,e_2)$ is right-handed because $\det [e_1,e_2]=1>0$.

If we use a left-handed coordinate system, then we must rotate $e_1$ in a clockwise direction through an angle less than $\pi$ to make it point in the same direction as $e_2$. So, $(e_1,e_2)$ is not right-handed by the author's description in EXAMPLE 2.

So, the author assumes we use a right-handed coordinate system.

If we use a right-handed coordinate system, then we must rotate $e_1$ in a counterclockwise direction through an angle less than $\pi$ to make it point in the same direction as $e_2$. So, $(e_1,e_2)$ is right-handed by the author's description in EXAMPLE 2.

In this case, there is no contradiction between the definition of a right-handed frame given on p.171 and the author’s explanation in Example 2 on p.172 of what it means for the frame to be right-handed.

I am reading Analysis on Manifolds by James R. Munkres.

I can’t clearly understand the following passage by the author. It leaves me with a vague, unsettled feeling.

On p.171 in this book:

Definition. Let $V$ be an $n$-dimensional vector space. An $n$-tuple $(a_1,\dots,a_n)$ of independent vectors in $V$ is called an $n$-frame in $V$. In $\mathbb{R}^n$, we call such a frame right-handed if $$\det [a_1\dots,a_n]>0;$$ we call it left-handed otherwise. The collection of all right-handed frames in $\mathbb{R}^n$ is called an orientation of $\mathbb{R}^n$; and so it the collection of all left-handed frames. More generally, choose a linear isomorphism $T:\mathbb{R}^n\to V$, and define one orientation of $V$ to consist of all frames of the form $\left(T(a_1),\dots,T(a_n)\right)$ for which $(a_1,\dots,a_n)$ is a right-handed frame in $\mathbb{R}^n$, and the other orientation of $V$ to consist of all such frames for which $(a_1,\dots,a_n)$ is left-handed. Thus $V$ has two orientations; each is called the reverse, or the opposite, of the other.

On p.172 in this book:

EXAMPLE 2. In $\mathbb{R}^1$, a frame consists of a single non-zero number; it is right-handed if it is positive, and left-handed if it is negative. In $\mathbb{R}^2$, a frame $(a_1,a_2)$ is right-handed if one must rotate $a_1$ in c counterclockwise direction throught an angle less than $\pi$ to make it point in the same direction as $a_2$. (See the exercises.) In $\mathbb{R}^3$, a frame $(a_1,a_2,a_3)$ is right-handed if curling the fingers of one's right hand in the direction from $a_1$ to $a_2$ makes one's thumb point in the direction of $a_3$. See Figure 20.2.

The author’s explanation in Example 2 clearly assumes the use of a right-handed coordinate system. However, for instance, $\mathbb{R}^2$ is nothing more than the set of all ordered pairs of real numbers. Is it possible to define the concept of a right-handed coordinate system mathematically?

Let $e_1:=(1,0)$ and $e_2:=(0,1)$.
Then, the frame $(e_1,e_2)$ is right-handed because $\det [e_1,e_2]=1>0$.

If we use a left-handed coordinate system, then we must rotate $e_1$ in a clockwise direction through an angle less than $\pi$ to make it point in the same direction as $e_2$. So, $(e_1,e_2)$ is not right-handed by the author's explanation in EXAMPLE 2.

So, the author assumes we use a right-handed coordinate system.

If we use a right-handed coordinate system, then we must rotate $e_1$ in a counterclockwise direction through an angle less than $\pi$ to make it point in the same direction as $e_2$. So, $(e_1,e_2)$ is right-handed by the author's explanation in EXAMPLE 2.

In this case, there is no contradiction between the definition of a right-handed frame given on p.171 and the author’s explanation in Example 2 on p.172 of what it means for the frame to be right-handed.

I am reading Analysis on Manifolds by James R. Munkres.

I can’t clearly understand the following passage by the author. It leaves me with a vague, unsettled feeling.

On p.171 in this book:

Definition. Let $V$ be an $n$-dimensional vector space. An $n$-tuple $(a_1,\dots,a_n)$ of independent vectors in $V$ is called an $n$-frame in $V$. In $\mathbb{R}^n$, we call such a frame right-handed if $$\det [a_1\dots,a_n]>0;$$ we call it left-handed otherwise. The collection of all right-handed frames in $\mathbb{R}^n$ is called an orientation of $\mathbb{R}^n$; and so it the collection of all left-handed frames. More generally, choose a linear isomorphism $T:\mathbb{R}^n\to V$, and define one orientation of $V$ to consist of all frames of the form $\left(T(a_1),\dots,T(a_n)\right)$ for which $(a_1,\dots,a_n)$ is a right-handed frame in $\mathbb{R}^n$, and the other orientation of $V$ to consist of all such frames for which $(a_1,\dots,a_n)$ is left-handed. Thus $V$ has two orientations; each is called the reverse, or the opposite, of the other.

On p.172 in this book:

EXAMPLE 2. In $\mathbb{R}^1$, a frame consists of a single non-zero number; it is right-handed if it is positive, and left-handed if it is negative. In $\mathbb{R}^2$, a frame $(a_1,a_2)$ is right-handed if one must rotate $a_1$ in c counterclockwise direction throught an angle less than $\pi$ to make it point in the same direction as $a_2$. (See the exercises.) In $\mathbb{R}^3$, a frame $(a_1,a_2,a_3)$ is right-handed if curling the fingers of one's right hand in the direction from $a_1$ to $a_2$ makes one's thumb point in the direction of $a_3$. See Figure 20.2.

The author’s explanation in Example 2 clearly assumes the use of a right-handed coordinate system. However, for instance, $\mathbb{R}^2$ is nothing more than the set of all ordered pairs of real numbers. Is it possible to define the concept of a right-handed coordinate system mathematically?

Let $e_1:=(1,0)$ and $e_2:=(0,1)$.
Then, the frame $(e_1,e_2)$ is right-handed because $\det [e_1,e_2]=1>0$.

If we use a left-handed coordinate system, then we must rotate $e_1$ in a clockwise direction through an angle less than $\pi$ to make it point in the same direction as $e_2$. So, $(e_1,e_2)$ is not right-handed by the author's description in EXAMPLE 2.

So, the author assumes we use a right-handed coordinate system.

I am reading Analysis on Manifolds by James R. Munkres.

I can’t clearly understand the following passage by the author. It leaves me with a vague, unsettled feeling.

On p.171 in this book:

Definition. Let $V$ be an $n$-dimensional vector space. An $n$-tuple $(a_1,\dots,a_n)$ of independent vectors in $V$ is called an $n$-frame in $V$. In $\mathbb{R}^n$, we call such a frame right-handed if $$\det [a_1\dots,a_n]>0;$$ we call it left-handed otherwise. The collection of all right-handed frames in $\mathbb{R}^n$ is called an orientation of $\mathbb{R}^n$; and so it the collection of all left-handed frames. More generally, choose a linear isomorphism $T:\mathbb{R}^n\to V$, and define one orientation of $V$ to consist of all frames of the form $\left(T(a_1),\dots,T(a_n)\right)$ for which $(a_1,\dots,a_n)$ is a right-handed frame in $\mathbb{R}^n$, and the other orientation of $V$ to consist of all such frames for which $(a_1,\dots,a_n)$ is left-handed. Thus $V$ has two orientations; each is called the reverse, or the opposite, of the other.

On p.172 in this book:

EXAMPLE 2. In $\mathbb{R}^1$, a frame consists of a single non-zero number; it is right-handed if it is positive, and left-handed if it is negative. In $\mathbb{R}^2$, a frame $(a_1,a_2)$ is right-handed if one must rotate $a_1$ in c counterclockwise direction throught an angle less than $\pi$ to make it point in the same direction as $a_2$. (See the exercises.) In $\mathbb{R}^3$, a frame $(a_1,a_2,a_3)$ is right-handed if curling the fingers of one's right hand in the direction from $a_1$ to $a_2$ makes one's thumb point in the direction of $a_3$. See Figure 20.2.

The author’s explanation in Example 2 clearly assumes the use of a right-handed coordinate system. However, for instance, $\mathbb{R}^2$ is nothing more than the set of all ordered pairs of real numbers. Is it possible to define the concept of a right-handed coordinate system mathematically?

Let $e_1:=(1,0)$ and $e_2:=(0,1)$.
Then, the frame $(e_1,e_2)$ is right-handed because $\det [e_1,e_2]=1>0$.

If we use a left-handed coordinate system, then we must rotate $e_1$ in a clockwise direction through an angle less than $\pi$ to make it point in the same direction as $e_2$. So, $(e_1,e_2)$ is not right-handed by the author's description in EXAMPLE 2.

So, the author assumes we use a right-handed coordinate system.

If we use a right-handed coordinate system, then we must rotate $e_1$ in a counterclockwise direction through an angle less than $\pi$ to make it point in the same direction as $e_2$. So, $(e_1,e_2)$ is right-handed by the author's description in EXAMPLE 2.

In this case, there is no contradiction between the definition of a right-handed frame given on p.171 and the author’s explanation in Example 2 on p.172 of what it means for the frame to be right-handed.

I am reading Analysis on Manifolds by James R. Munkres.

I can’t clearly understand the following passage by the author. It leaves me with a vague, unsettled feeling.

On p.171 in this book:

Definition. Let $V$ be an $n$-dimensional vector space. An $n$-tuple $(a_1,\dots,a_n)$ of independent vectors in $V$ is called an $n$-frame in $V$. In $\mathbb{R}^n$, we call such a frame right-handed if $$\det [a_1\dots,a_n]>0;$$ we call it left-handed otherwise. The collection of all right-handed frames in $\mathbb{R}^n$ is called an orientation of $\mathbb{R}^n$; and so it the collection of all left-handed frames. More generally, choose a linear isomorphism $T:\mathbb{R}^n\to V$, and define one orientation of $V$ to consist of all frames of the form $\left(T(a_1),\dots,T(a_n)\right)$ for which $(a_1,\dots,a_n)$ is a right-handed frame in $\mathbb{R}^n$, and the other orientation of $V$ to consist of all such frames for which $(a_1,\dots,a_n)$ is left-handed. Thus $V$ has two orientations; each is called the reverse, or the opposite, of the other.

On p.172 in this book:

EXAMPLE 2. In $\mathbb{R}^1$, a frame consists of a single non-zero number; it is right-handed if it is positive, and left-handed if it is negative. In $\mathbb{R}^2$, a frame $(a_1,a_2)$ is right-handed if one must rotate $a_1$ in c counterclockwise direction throught an angle less than $\pi$ to make it point in the same direction as $a_2$. (See the exercises.) In $\mathbb{R}^3$, a frame $(a_1,a_2,a_3)$ is right-handed if curling the fingers of one's right hand in the direction from $a_1$ to $a_2$ makes one's thumb point in the direction of $a_3$. See Figure 20.2.

The author’s explanation in Example 2 clearly assumes the use of a right-handed coordinate system. However, for instance, $\mathbb{R}^2$ is nothing more than the set of all ordered pairs of real numbers. Is it possible to define the concept of a right-handed coordinate system mathematically?

Let $e_1:=(1,0)$ and $e_2:=(0,1)$.
Then, the frame $(e_1,e_2)$ is right-handed because $\det [e_1,e_2]=1>0$.

If we use a left-handed coordinate system, then we must rotate $e_1$ in a clockwise direction through an angle less than $\pi$ to make it point in the same direction as $e_2$. So, $(e_1,e_2)$ is not right-handed by the author's description in EXAMPLE 2.

I am reading Analysis on Manifolds by James R. Munkres.

I can’t clearly understand the following passage by the author. It leaves me with a vague, unsettled feeling.

On p.171 in this book:

Definition. Let $V$ be an $n$-dimensional vector space. An $n$-tuple $(a_1,\dots,a_n)$ of independent vectors in $V$ is called an $n$-frame in $V$. In $\mathbb{R}^n$, we call such a frame right-handed if $$\det [a_1\dots,a_n]>0;$$ we call it left-handed otherwise. The collection of all right-handed frames in $\mathbb{R}^n$ is called an orientation of $\mathbb{R}^n$; and so it the collection of all left-handed frames. More generally, choose a linear isomorphism $T:\mathbb{R}^n\to V$, and define one orientation of $V$ to consist of all frames of the form $\left(T(a_1),\dots,T(a_n)\right)$ for which $(a_1,\dots,a_n)$ is a right-handed frame in $\mathbb{R}^n$, and the other orientation of $V$ to consist of all such frames for which $(a_1,\dots,a_n)$ is left-handed. Thus $V$ has two orientations; each is called the reverse, or the opposite, of the other.

On p.172 in this book:

EXAMPLE 2. In $\mathbb{R}^1$, a frame consists of a single non-zero number; it is right-handed if it is positive, and left-handed if it is negative. In $\mathbb{R}^2$, a frame $(a_1,a_2)$ is right-handed if one must rotate $a_1$ in c counterclockwise direction throught an angle less than $\pi$ to make it point in the same direction as $a_2$. (See the exercises.) In $\mathbb{R}^3$, a frame $(a_1,a_2,a_3)$ is right-handed if curling the fingers of one's right hand in the direction from $a_1$ to $a_2$ makes one's thumb point in the direction of $a_3$. See Figure 20.2.

The author’s explanation in Example 2 clearly assumes the use of a right-handed coordinate system. However, for instance, $\mathbb{R}^2$ is nothing more than the set of all ordered pairs of real numbers. Is it possible to define the concept of a right-handed coordinate system mathematically?

Let $e_1:=(1,0)$ and $e_2:=(0,1)$.
Then, the frame $(e_1,e_2)$ is right-handed because $\det [e_1,e_2]=1>0$.

If we use a left-handed coordinate system, then we must rotate $e_1$ in a clockwise direction through an angle less than $\pi$ to make it point in the same direction as $e_2$. So, $(e_1,e_2)$ is not right-handed by the author's description in EXAMPLE 2.

So, the author assumes we use a right-handed coordinate system.