PCM errata III

A quick note to say that I’ve just been asked by PUP to send all the corrections I have, so I have sent the corrections that are listed in comments on the post PCM errata II. Further errata should therefore ideally be pointed out here. I think this means that a reprint with corrections should appear in the next few months, but I’m not sure of the exact time scale.

Once again, many thanks to those who have notified me of errors. Particular thanks this time round go to Axel Boldt, who sent a very long list. He was to PCM errata II as Joseph Myers was to PCM errata I. But others too made observations that will lead to significant improvements when the reprint comes out. It has been disconcerting to see just how many mistakes remained after the effort we spent on eliminating them, but I suppose in a book this size it is inevitable, and I imagine that there are several more lurking there. So do please continue to let me know of any errors that you find. At some point in the not too distant future I’ll try to merge the second list with the first to make it easier for people to see which mistakes have already been spotted. For now, after the break, here is a copy of the instructions I have sent to PUP.

If you look carefully, you will see that I haven’t acted on every single comment, because I’m supposed to confine myself to easily implemented corrections that will have a minimal effect on page and line breaks. But I’ve taken account of almost all of them, and when I have not written about one, it is with regrets.

16.2 line 13

It’s been pointed out to me that I use the subset symbol without having
said what it is. That’s not ideal, but I think it’s not enough of a problem to
be worth the hassle it would take to fix it.

43.1 line 20

It would be better to say “a point on the line at infinity of $P’$” rather than
just “the point at infinity of $P’$”.

44.2 line 14

“ball” would be a better word than “sphere” here. In fact, this is an
essential change because of the wording earlier in the sentence.

126.1 Weierstrass quotation

Someone points out that it looks odd to date this 1988, but I think it’s
our convention that we refer to the date of the edition that is being
quoted and not the original book. Is that right? If so, then no need for
any action here.

149.2 line 17

Not sure how, but a “with with” slipped through the net.

163.2 line -3

Where it says SO(2m) it should say SO(m).

164.2 line 10

“motion through space-time” would be better replaced by “motion
in space-time”

180.2 lines 24-26

There are a couple of lower case phis that should be upper case.
(All phis round here should be upper case.)

186.9 line 9

In this heading, it should say $C_c^\infty[-1,1]^*$ rather than just
$C^\infty[-1,1]^*$. The font of the suffix c should be as it is in the
ensuing paragraph.

208.2 line -3

It should say “equal angles, just like the more familiar Euclidean rotations.”

218.2 line 6

In the definition of the ellipticity condition it should say \sum_{i,j}a_{ij}\xi^i\xi^j
In other words, the sum over i and j is missing.

265.1 line -6 and 265.2 line 2

In both places it should say \binom nk p^k(1-p)^{n-k}

That is, the power of 1-p should be n-k rather than just k.

287.1 line 13

The final psi in this definition of the Laplacian shouldn’t be there.
In other words, it should end with $\sum_{j=1}^n\frac{\partial^2\psi}{\partial q_j^2}$.

293.1 line 12

It should say \cos^{n-2r}\theta(1-\cos^2\theta)^r at the end. (In other words,
there is a theta missing.)

299.1 line -5

The words “a diffeomorphism” should be replaced by the single word
“symplectic”

303.2 line 19

The words “with a_r=b_r=0 for r>\frac 12 n” should be omitted.

313.2 line 9

Ideally, after the sentence that ends “vector space” I’d like to add
the following sentence:

Moreover, for every sufficiently small region $U$ of $X$, the inverse
image of $U$ is homeomorphic to $\R^n\times U$ (this property is
called \textit{local triviality}).

347.1 line -10

Where it says (b,q)=1 it should say (b,p)=1.

378.1 line -22

Am waiting for an email from the author before I know what to do here.

457.1 line -16

“derivatives of $f$” should be “derivatives of $u$”

470.1 line -3

It should say $B_{a+ct}(x_0)$ where it currently says $B_{a+t}(x_0)$.

478.2 line 11

The following rewriting is needed to remove an unfortunate ambiguity.

From this we wish to dedcue that the derivative of $S$ at $\phi$, in
some appropriate sense, is zero, and hence …

479.2 line 7

The sentence in parentheses needs to be rewritten as follows:

(For instance, for energy the one-parameter group of …

479.2 line 19

The displayed formula is missing two factors of \frac 12, which
should appear in the places where they appear in formula (58).
I think it would also be appropriate to have a dx at the end of
the integral, for consistency with (58).

490.1 line -14

\nabla_i A_0 should be \partial_i A_0

490.2 line 2

\partial_{x^i}^2 should be \partial_i^2

512.1 line 22 It would be slightly better to write

… a \textit{unitary} operator is an invertible …

but if this causes problems with line or page breaks then it is not essential.

520.1 line -16

The Z inside C^*(Z) should be blackboard bold, as it is four lines earlier.

521.2 middle of page

Where it says $T^*T-T^*T$ it should say $T^*T-TT^*$. (A mistake
that it doesn’t require any mathematical knowledge to spot …)

524.1 line -7

The word “period” should be replaced by the phrase “angular frequency”

527.2 line -4

There is a factor \frac 12 missing before the x^2.

543.1 formula (2)

There is a left square bracket missing at the beginning of the first line.

548.1 line 18

A valid-sounding objection has been raised to this formula. I’m not
sure what to do about it and may consult the author. But I think we’ll
have to leave it for the time being, unless I get back in touch about it.

639.2 line -10

It would be good, but not essential if it causes problems, to change
“all the true statements” to “all the true first-order statements”

645.1 line -1

Where it says R(x_i,z) it should say (x_i\sim z) (if \sim gives the
tilde symbol used just above). And where it says \neg R(y_i,z)
it should say \neg(y_i\sim z)

681.1 line -1

It should say “is at most $6x$” rather than “is $6x$”.

714.1 First sentence of V.25

Apparently I misstated the conjecture, which is embarrassing. Here
is an alternative first sentence.

The Poincar\’e conjecture is the statement that a COMPACT [III.9]
smooth $n$-dimensional manifold that is HOMOTOPY EQUIVALENT
[IV.6.2] to the $n$-sphere $S^n$ must in fact be homeomorphic to $S^n$.

This requires a further change at the end of the paragraph. Where it says,
“Thus, the Poincar\’e conjecture asks” it should be changed to “In three
dimensions, the Poincar\’e conjecture asks”

852.1 line -20

It should say

… which takes the value 1 between 0 and $\frac 12$ and $-1$ between
$\frac 12$ and 1.