Corrections to Elementary Differential Geometry, Second Edition  (Apr '06)
       I am grateful to Allen B. Altman and Joseph E. Borzellino for most of these corrections.       Note: Since HTML cannot space mathematical italics properly, many ordinary letters below should be read as italic.
Ch.1
  Page/Line         Correction
  15/-6          Should read:  V[xi] Ui   (subscript "i").
  24/-8          Should read   R3, not   R2 , and fi= (Ui), with both subscript "i".
  26/-2         Should read "=xU1"   (subscript "1", not "i").
  26/-11        Formula should end: "...+2)v3".
  32/11         Should read r,,z   not  r, ,z.
  36/-1          Should read  ...=v[fi],  not ...=v[f].
  39/11         Should read: F*p  not  F*p.
Ch.2
  44/-1          The dot product should be:  ve i  (subscript "i", not "1").
  50/4          The upper limit in the integral should be "t", not "1".
  50/7          Replace   =0j by  =0; (semicolon).
  65/-15       Should be kappa tilde, not plain kappa.
  66/4          Should read =1/, not =1/r.
  66/7          Should read: = +N+ 'B.
  75/14        Should read: Example 4.2(2) of Chapter 1.
  83/-10       Should read ...increasing ,   not ... increasing .
  85/-9         Should read: ji= -ij.
  88/-2,-1     In the first matrix, replace "1" by "0". In the third matrix, both thetas should be "".
  89/4          Add a minus sign following "=", giving "...= - d
  90/6          In Ex.6, add the hypothesis V[r]=r.
  90/8          In Ex.7, replace the sentence "For simplicity, define..." by "(Hint: For Maple, use the differential operator d from the difforms package. For Mathematica, use the total differential operator Dt.)"
  94/-2         Delete the comma between and d.
Ch.3
   102/6       Should read:   F(p)=q.
  116/-6       Insert subscript V on the left side of the equation, between del and W.
  121/15      In (2), replace  E2  by  Ei.
  121/-1       Each  Ei  should have an overbar.
  123/14      Rename this exercise "Congruence of curves" and replace the second sentence of (b) by: "Find an isometry F=TC that carries to .
Ch.4
  125/7        Should read: R3 , not R2 .
  134/2        Replace the first sentence in Ex.12 by: Let C be a connected Curve in R2 (p.21) that is non-trivially symmetric about the x-axis. Show that (a) C crosses the x-axis, (b) all such crossings are orthogonal to the x-axis (Hint: A figure 8 is not a Curve.), (c) there is either one such crossing or two, giving Curves like those in Fig. 4.9.
  138/middle       The third equation should read:  r sin v = r sin v1
  142/9        Should read -1u1,   not -1v1.
  147/-1       Shrink "subspace" to "space".
  152/6        In the right side of the equation, the denominator should be ||xv||,  not ||xu||.
  172/-11    The first double integral in the Proof should be over R, not over x.
  172/-10     Should have g= (xv),   not xu.
  172/-9       This formula should be labeled (1), for reference on p.173.
  175/5        The integrand should be , not .
  176/13      Should read: ...and is a 2-form on N, then xF*= F(x) .
  177/9         (U,V) should map [cap script S] to [cap script R].
  177/14       In the second equation, change xU and xV to yU and yV
  177/16       Replace twice by .
  177/-7        Replace xV... by xV....
  179/-16      The inequality should be .
  180/17       In the formula, replace Z(p) by U(p).
  182/5         Should read:   x(u,v)= v(a) + (1-v)(u).
  187/-17      Replace "Exercise 9" by "Exercise 11".
  190/5         In the formula, add subscripts thus:   p3xu(p1, p2)+p4xv (p1, p2).
Ch.5
  202/7        Should read:  From Exercise 1 of Section 1...
  205/4        Should read:   normal sections of C all....
  217/13      Replace denominator du2 by u2.
  220/-1       Should read:  -1/b2K<0.
  221/1         Should read:   The minimum value K=-1/b2 occurs....
  223/3         Change + to -, yielding  ...-fuv 2.
  224/9         Delete the parens "(" following the first ± and following "± x".
  225/-4       Should read:  If is a unit speed curve....
  226/-9       In the formula for K(u,v), square the entire denominator.
  232/20      Should read:   ...principal curvature ki   (not i).
  238/-16     The first row of the determinant should read:  a2'2    -a1' a2'     a1'2.
  240/middle        In Ex.14, add primes to in (a) and to in (b).
  241/-8        Delete the very difficult Ex.19.
  242/-4        The denominator for N should be the same as the denominator for L.
  244/11       Should read:   The curve y=c cosh(x/c)   (not v/c).
  245/11       Should read:   By Exercise 2.8   (not 2.10).
  245/-3       Should read:    ...are given in Exs. 5.5 and 6.15.
  248/12      Should read: "As a shrinks down..." not "As c shrinks down...".
  248/-6       Should read:    sin(a*/c),   not sin(u*/c).
  248/-5       Should read:   h(a*)= a cos(a*/c)=....
Ch.6
  269/5        Should read: Ex. 2.2(a), not Corollary 2.4.
  270/9        Should read: Ex. 8.14, not Ex. 8.7.
  270/-15     Should read: M R3    (not R2 ).
  286/12      Should read: Ex. 4.8   (not Ex. 4.7).
  287/3        Should read:  x(u,v) to x(u,v+v)  is approximated by v xv.
  289/9        Should read: ...one set xi(Rio).
  290/5        Should read:   ...on any pair of tangent vectors to M is ±||v × w||.
  291/7        Should read:   area(xi(Ri).
  294/-7       Should read:  Example 7.1 of Chapter 5  (not Example 6.1).
  294/-2       Delete du dv.
  297/-12     Should read:  G*(d)(v,w)=....
  301/13       Change "Extend (b)" to "Extend (a)".
  302/4         Should read:  Ex. 7.6, not Ex. 7.8.
  303/-13      Replace x2+y2 in the numerator by  x2-y2.
Ch.7
  313/2         The first subscript on this line should be "2", not "1".
  314/11       The expression beneath the square root sign should be:  a1'2 + a2'2.
  314/middle          The speed in Ex. 2(a) is csc t, not r csc t.
  315/11       Delete the "2" from the formula.
  320/-1        In the integral, replace the denominator by:  1-(u/2)2.
  326/10       Delete parens "(" before the first E1 and before the first E2.
  328/19       Here J is the rotation operator defined in Ex. 1.3 of this chapter.
   332/-9        Should read: ...where denotes....
   334/-1       Change sign ...-Gva2'2  to  +Gva2'2.
  345/1         Should read:   ...=±cos-1(c/u).
  345/7         Replace v by x.
  347/19       Should read  "Poincaré."
  351/-11      Should read:  ...but g< 0 for a (right-turning) clockwise trip.
  351/-9        Replace "Lemma 4.6" by "Corollary 4.6".
  354/5         Replace "of xu" by "of x".
  364/-10      In (1), delete "no".
  368/11       The first integral is over all of M, not just the polygonal region.
  372/1         Should read:  Figure 7.26, not Example 7.26.
Ch.8
  375/14,17   Should read:  arc length.
  377/5          Should read: ...v'(0) =v   (not v(0)).
   379/-15     Should read expp,  not expv.
  380/-3        The expression under the square root should read: a1'2+Ga2'2.
  384/4,5       Reletter (b)...(c) as (c)...(d).
  385/2,3       (a) should read:  "...to these geodesics." (b) should read: "...curves in (a)."
  389/11       Delete the sentence: "Similarly,...."
  389/-2        Should read: Section 7, not Section 6.
  392/4         Should read:  (u cos v, u sin v).
  392/15       Should be the square root of G(,v).
  396/10       Should read:   g'(s)1.
  398/8         Should read:   ...+o(3).
  400/8         Should read:  (Ch.4, Sec.2).
  402/-2        The N should be an M tilde, as in the Proposition.
  410/14       Should read:  Hopf-Rinow theorem (2.1)  (not (2.2)).
  416/9         Should read: ...x2/2, y2/2, z2/2....
  426/-14,-13   Replace larger by smaller and longer by shorter.
  426/-6        Should be:  4/k.
Answers
  451/1         Should read:    3.  (a) y3.
  451/6         Use:  '(/2)= (-1,0,1/2) at p=(1,1,2).
  453/4          Replace "...2f 2" by "...2/f 2".
  458/5          Replace V=(x,0,z), which vanishes at the origin, by (1,0,y) and (0,1,x).
  458/-12      The answer to Ex.5 has six subscripts: i,i,i,i,i,j. Change these to i,j,i,j,i,j.
  459/-6        This line should begin "dual of curl V is d".
  463/9          Each U should be changed to U'  (adding a prime).
  464             In line 1, replace "(a)" by "(b)". In line 2, replace "Ex. 6.6" by "Ex. 7.6".
  465/middle  In the answer to Ex.1, the second omega should have subscripts "13".
  466/-5         Change "(Ex. 7.8)" to "(Ex. 7.6, on p. 292)".
   469/5         Replace  13(Y)E 3 by 12(Y) E3, and subscript "y" by "Y".
   474/-5        Change Lemma 7.3 to Lemma 7.4.
Index
   481/24 (right col)   Should read:  ...Surface of revolution,130, 241-250,  not 234-242.
   481/26 (left col)      Should read:  ...Smooth disk, 185, (Ex.6),  not Ex.5.
   482/-1 (right col)   Add to "Winding number" references: 185 (Ex.6), 372(Ex.9)
Figures:  On p. 35, replace Fig 1.14--a duplicate of Fig.1.13--by this figure.

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