Typo in spivak's calculus on manifolds?

In summary, in the first problem set of chapter 1, problem 1-8(b) deals with angle preserving transformations. In the newest edition of the book, the problem states that if there is a basis x_1, x_2, ..., x_n and numbers a_1, a_2, ..., a_n such that Tx_i = a_i x_i, then the transformation T is angle preserving if and only if all |a_i| are equal. However, in the first edition, the absolute value bars are omitted on the a_i. There are counterexamples to both formulations, as demonstrated by a matrix with eigenvectors and eigenvalues of different absolute values. It can be proved that equality of the |a
  • #1
Adeimantus
113
1
In the first problem set of chapter 1, problem 1-8(b) deals with angle preserving transformations. In the newest edition of the book the problem is stated

If there is a basis x_1, x_2, ..., x_n and numbers a_1, a_2, ..., a_n such that Tx_i = a_i x_i, then the transformation T is angle preserving if and only if all |a_i| are equal.

In the first edition (the one with the cool blue diagrams that you can actually see), the absolute value bars are omitted on the a_i. However, I'm pretty sure there are counterexamples to both formulations. If the basis happened to be orthogonal, then equal absolute values of the a_i would be enough, but if the basis were oblique, then you would need equality of the a_i. But the author doesn't mention orthogonality until a couple exercises later. Am I missing something?

thanks.
 
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  • #2
I found the same problem working through Spivak recently! I think it is a typo, and here's my counterexample:

Consider the matrix:
[tex]M=\left(\begin{matrix}1 & -2 \\ 0 & -1 \end{matrix}\right)[/tex]

This has eigenvector (1,0)' corresponding to eigenvalue 1, and eigenvector (1,1)' corresponding to eigenvalue -1. (this is how I constructed the matrix in the first place actually). Since the eigenvalues are the same in absolute value (+/-1) and the eigenvectors form a basis for R2, according to this problem M should be angle preserving.

On the other hand, compare the angle between (1,0)' and (1,2)', to the angle between M(1,0)'=(1,0)' and M(1,2)'=(-3,-2)'. These angles aren't the same.
 
  • #3
maze said:
I found the same problem working through Spivak recently! I think it is a typo, and here's my counterexample:

Consider the matrix:
[tex]M=\left(\begin{matrix}1 & -2 \\ 0 & -1 \end{matrix}\right)[/tex]

This has eigenvector (1,0)' corresponding to eigenvalue 1, and eigenvector (1,1)' corresponding to eigenvalue -1. (this is how I constructed the matrix in the first place actually). Since the eigenvalues are the same in absolute value (+/-1) and the eigenvectors form a basis for R2, according to this problem M should be angle preserving.

On the other hand, compare the angle between (1,0)' and (1,2)', to the angle between M(1,0)'=(1,0)' and M(1,2)'=(-3,-2)'. These angles aren't the same.

Exactly. I considered the same matrix for T for my counterexample. Thanks for confirming. I figured there must be some easy fix to the problem to make it make sense, which is why I called it a "typo" rather than a "mistake". But it must be something deeper than just leaving off the absolute value sign, because if the basis happened to be orthogonal you could have eigenvalues +1 and -1 and still preserve angles.


I think it can be proved that equality of the |a_i| is necessary (but not always sufficient) for T to be angle preserving. If for example |a_j| < |a_i|, the angle between x_i and x_i + x_j would, under the map T, become the angle between x_i and x_i + (a_j / a_i)x_j. Assuming these two angles are equal

angle(x_i, x_i + x_j) = angle(x_i, x_i + (a_j / a_i)x_j).

Applying this relation N times gives

angle(x_i, x_i + x_j) = angle(x_i, x_i + (a_j / a_i)^N x_j) ----> 0 as N ---> infinity. This contradicts the assumption that the x_k 's form a basis.
 
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What is a typo in Spivak's Calculus on Manifolds?

A typo in Spivak's Calculus on Manifolds refers to a mistake or error in the text, notation, or equations that were published in the book. These typos can lead to confusion for readers and may impact their understanding of the material.

How common are typos in Spivak's Calculus on Manifolds?

Typos in Spivak's Calculus on Manifolds are relatively common, as with any published work. However, they are not so frequent that they significantly diminish the value or accuracy of the text.

Can typos in Spivak's Calculus on Manifolds be corrected?

Yes, typos in Spivak's Calculus on Manifolds can be corrected. In fact, the author himself has released an errata document that lists and corrects the typos found in the first edition of the book. Subsequent editions may also include corrections to these typos.

Do typos in Spivak's Calculus on Manifolds affect the overall content and concepts presented in the book?

In most cases, typos in Spivak's Calculus on Manifolds do not significantly affect the overall content and concepts presented in the book. However, some typos may lead to confusion or incorrect solutions to problems, so it is important for readers to be aware of any known typos and refer to the errata document when necessary.

How can I report a typo I have found in Spivak's Calculus on Manifolds?

If you have found a typo in Spivak's Calculus on Manifolds, you can report it to the publisher or the author. You can also check the errata document to see if the typo has already been identified and corrected. Additionally, some online forums and communities dedicated to mathematics may have discussions about typos in the book.

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