Spivak - Calculus on Manifolds, Comments and Errata

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• Firstly, check on page 145 in the book itself for some errata and comments.
• Petra Axolotl also put together another website for errata in Spivak, so also look there: http://www.petraaxolotl.com/mathematics/Calculus-on-Manifolds/
• In several places, e.g. pages 16, 21, 31, the book does something rather terrible. To prove that \(|A(x)|\) goes to 0, knowing that \(|B(x)|\) goes to zero it uses \[ \lim_{x\to 0} |A(x)| \leq \lim_{x\to 0} |B(x)| = 0. \] But we cannot take the limit of A before we know it exists! Really, what was being proved was \(|A(x)| \leq |B(x)|\), and only then can we take the limit because \(B(x)\) goes to zero. This would be enough to lose a point or two in a beginning real analysis class, and ought to at least earn a raised eybrow in a more advanced class.
• In a few places n is used as the dimension and also in a different context. Perhaps most confusingly on page 51 in the definition of content 0, n is both the dimension and the number of rectangles, but of course those numbers are different. Same problem appears on page 9 and 10 in the proofs of corollaries 1–5 and 1–7.
• Page 43) In problem 2–41. We want to assume that f is twice continuously differentiable.
• Page 61) Problem 3–27 is missing the limits of integration, both should be b.
• Page 62) Problem 3–32. The hint should really use a dummy variable for y in the integral, this is terrible form.
• Page 65) The integral \(\int_A \varphi |f|\) exists in the usual sense only if \(\varphi\) is compactly supported. The partition of unity constructed in the theorem is compactly supported, it should really be part of the conclusion of theorem 3–11. The book should really expand on this issue, it glosses over why this integral exists. Hint: The integral exists since \(\varphi |f|\) is nonzero on a compact set \(C \subset A\), so cover C by rectangles that are in A and as C is compact we only need finitely many. Union of finitely many rectangles is Jordan measurable, and we only need to integrate over that.
• Page 66) Problem 30–37 b) The continuous limit may not exist (so the problem is impossible as stated). For example, you could cook up a function which as \(\epsilon\) goes accross \(A_n\) becomes very positive and then very negative so that the continuous limit does not exist. Either you must take a limit of a certain sequence (say endpoints of \(A_n\)) or you have to give extra conditions on f. For example, the problem works if you specify that on each \(A_n\), f is either positive or negative but not both.
• Page 68) We really use the fact that g pulls back the partition of unity, but then we only get \(C^1\), which is sufficient. The definition of partition of unity and integrable in the extended sense should be done with continuous, and existence of smooth partition of unity is simply something extra that we sometimes do not need.
• Page 72) Missing parentheses in the first displayed equation. g(y)-g(x) should be in parentheses.
• Page 73) Problem 3–40. Firstly you should assume \(C^1\) (continuously differentiable) to be able to apply the inverse function theorem. Second, the second part of the problem (about \(g'(x)\) being diagonal) is not true, this can easily be checked by finding the derivative of the composition in two dimensions for \(g_1(x) = (f_1(x),x^2)\) and \(g_2(x) = (x^1,f_2(x))\). This is diagonal (and still invertible) only if \(f_1\) does not depend on \(x^2\) and \(f_2\) does not depend on \(x^1\).
• Pages 78, 80, 81) The notation for multiplication in the symmetric group is reversed. The multiplication is normally composition as far as I know; I have never seen it be the reverse as Spivak does it.
• Page 85) Problem 4–3, the hint is wrong. The matrix \(g_{ij}\) is not \(AA\) but \(A^TA\), where A is the matrix \(a_{ij}\). So the indices on one of the \(a_{??}\) in the formula for \(g_{ij}\) are flipped.
• Page 96) Problem 4–14. The definition of "the" tangent vector is odd nomenclature. It is one of the tangent vectors and it happens to span the tangent space if the derivative of c does not vanish.
• Problem 4–21 on page 97. The student should be referred to top of page 93 where the problem is mentioned, or the definition of \(\theta\) should be given here.
• Page 97) Definition of an n-cube should not be restricted to \({\mathbb{R}}^n\). It should read \(A \subset {\mathbb{R}}^m\).
• Page 97) n-cubes should really be defined to be smooth not just continuous. Otherwise the definition of integral on page 101 doesn't work.
• Page 101) On the bottom of the page in the computation of \(\int_{[0,1]^{k-1}} {I^k_{(j,\alpha)}}^* ( \dots )\) there is some detail left out. At this point the student is probably not completely proficient in these calculations and so it seems like there is quite a bit skipped over. Also the integral in the answer is strangely over \([0,1]^k\). Really there is no point in writing that integral here (or later in the proof) as integral over \([0,1]^k\), it is really an integral over \([0,1]^{k-1}\), but then you can add one more integration and as we are integrating a constant over the interval \([0,1]\) it works, but you really use Fubini in that argument which is not needed. So not an error, but there should be more detail than "Note that".
• Page 111) In theorem 5–1, it is not an immediate consequence of 2–13. Theorem 2–13 talks about continuously differentiable functions and even then only says that \(h\) is differentiable (though continuously differentiable is easy to get). Thoerem 5–1 talks about smooth (\(C^\infty\)) functions. What is missing is a smooth version of 2–13 that ends with a smooth \(h\). This follows from Addenda 1 in the back of the book.
• Page 118) In definition of orientation the W must be connected, otherwise no manifold would be orientable. Probably would be best to just define a coordinate system to automatically be connected on page 111, as there could be other places where this kind of subtle error lurks.
• Page 126) In problem 5–20, M should of course be orientable.

Some of these were found by my students, so thanks, though I didn't keep track of who told me about what. I also didn't manage to mark down some issues from the beginning of the book (I marked it down but can't find it), so don't think there are no comments on the first part, I just lost them or forgot them.