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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.

left-handed

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.

right-handed

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    $\begingroup$ By convention, the standard basis in $\mathbb R^n$ is viewed as right-handed. Any other basis $b_1,\dots,b_n$ is right-handed if $\det[b_1\dotsm b_n]>0$, where $b_j$ are expressed in the standard basis. $\endgroup$ Commented Oct 14 at 8:03
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    $\begingroup$ What is a "coordinate system" for you/the author? Is it not just the coordinates with respect to a chosen frame? $\endgroup$ Commented Oct 14 at 9:06
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    $\begingroup$ @tchappyha Maybe we use different terminology. To me a basis is an ordered sequence of vectors, so $(1,0),\,(0,1)$ is a right-handed basis and $(0,1),\,(1,0)$ a left-handed basis. Any basis gives a coordinate system, and I would call the coordinate system right-handed if the corresponding basis is. $\endgroup$ Commented Oct 14 at 9:10
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    $\begingroup$ You ask whether it is possible to define the concept of a right-handed coordinate system mathematically. Hasn't this been done in the definition on p.171? You should explain more precisely what is not clear to you. $\endgroup$ Commented Oct 14 at 9:20
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    $\begingroup$ You still did not define what you mean by a coordinate system. What you wrote in a comment is not a definition. A picture is not a substitute for a definition. $\endgroup$ Commented Oct 14 at 15:26

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In your question you make a nice observation. Certainly many readers (including myself) overlooked a subtlety in Example 2.

Munkres defines the concept of orientation for $n$-frames in $\mathbb R^n$. In Example 2 he wants to explain heuristically what it means that an $n$-frame with $n = 1,2, 3$ is right- or left-handed. Note that right-handed frames are often called positively oriented and left-handed frames negatively oriented.

The case $n=1$ is fairly trivial. A $1$-frame is nothing else than a non-zero number $a_1 \in \mathbb R$. But in $\mathbb R$ we have a natural concept of positive and negative numbers, thus it is clear that we want to define $a_1$ positively oriented if $a_1 > 0$. In this case right-handedness shows up in the fact that that $a_1$ lies on the right hand side of $0$ (provided we depict $\mathbb R$ as usual in form of a horizontal line).

The sublety shows up for $n =2$. To detect the right-handedness of a $2$-frame $(a_1,a_2)$ geometrically Munkres refers to a counterclockwise rotation of $a_1$ by an angle less than $\pi$ to make it point in the same direction as $a_2$. It is intuitevely clear what a counterclockwise/ clockwise rotation is, but indeed we need a single prototype of a $2$-frame to explain the precise meaning. You call this a coordinate system, but actually it is a frame. This prototype is the frame $(e_1, e_2)$. A rotation by $\pi/2$ moves $e_1$ onto $e_2$, and all rotations in this direction are called counterclockwise. What is the relation to right-handedness? If you put your right hand on the plane with index finger in the direction of $e_1$ (= direction of the positive $x$-axis), then your thumb can easily be arranged to point in the direction of $e_2$ (= direction of the positive $y$-axis). This method can then be applied to any frame $(a_1,a_2)$: Put your right hand on the plane with index finger in the direction of $a_1$ and try to move your thumb towards $a_2$. If you can reach $a_2$ without breaking your thumb, you call $(a_1,a_2)$ right-handed. If this is impossible, then it will work with your left hand. Another way to visualize it is this: Draw a line through the origin and $a_1$. This divides the plane in two open half-planes. Put your your right hand on the plane with index finger in the direction of $a_1$. If $a_2$ lies in the same half-plane as your thunb, then call $(a_1,a_2)$ right-handed.

Note that everything here is based on certain traditions (which arose "arbitrarily" in history).

  • The tens of a mechanical clock move clockwise.

  • The Euclidean plane $\mathbb R^2$ is usually depicted with the Cartesian coordinate system ($x, y$-axes) as in your second picture. This was introduced by Descartes in 1637.

All this may seem to be very natural because we are used to it. But actually is seems to me that these "standards" may be related the the "handedness" of the European writing systems (from the left to the right). This may also concern the movement of the tens of a watch starting at 12 o'clock.

There is no logical reason why one could not depict $\mathbb R^2$ with $x,y$-axes as in your first picture. The positive $y$-axis is the below the $x$-axis - but why not? If we would use this convention, than we would have to exchange the words "right-handed" and "left-handed".

The case $n=3$ does not refer to counterclockwise rotations (which do not exist in $3$-space), but to the standard $3$-frame $(e_1,e_2,e_3)$, or equivalently the $x,y,z$-axes. This gives the prototype of a "configuration" of middle finger, index finger and thumb of your right hand.

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