Errata for Differential Geometry and its Applications

It seems that no matter how careful I thought I was with the book, there are still misprints, misplacements and just plain miss-ings which this page is devoted to exposing. This page will be updated periodically with new corrections. I would really appreciate it if anyone who finds a typo or mistake would email me at oprea@math.csuohio.edu . Thanks!

Chapter 1

Joel Langer and Ted Shifrin don't like Proposition 1.1. While it's true that lines may have many parametrizations, I only define a line as p + tv in Chapter 1. I do this purposely to obtain Prop. 1.1 as motivation for understanding geodesics later. Of course, geodesics are constant speed, so the only parametrization of the line which is appropriate in this context is the one given in Chapter 1. What are other people's feelings about this?

On p.7, the definition of u should be (q-p)/|q-p|. The parentheses were left out.

In Example 1.2, the standard picture of the circle with a triangle defining sin(theta) and cos(theta) is missing. The point P referred to is just the point on the unit circle given by the intersection of a radial line from the origin at angle theta (with respect to the x-axis).

In Exercise 1.8, it should say `Let t denote the angle between ...'. The word angle was omitted, but it is depicted.

Exercises 3.19 and 3.20 should both say that readers can use the Maple procedures in section 6 to calculate curvature and torsion of non-unit speed curves or use the formulas of the next section (section 4). These exercises were put here because they are examples of the spherical curves described in Exercises 3.17 and 3.18, but they are not unit speed, so, at this point, they must be handled by Maple.

On p. 31, in the middle of the page, the power of nu should be 2 in the middle line until the last line when it becomes 3.

Exercise 5.11 on Euler's spiral only asks for the formulas describing the spiral and should refer to the Maple procedure `recreate' of section 6 for graphing.

On p. 41, the tor procedure (to calculate torsion) should have `num' listed as a local variable (after alphappp say). The procedure still works without this since Maple implicitly makes variables local, but Maple will give a warning message.
Chapter 2

On p. 61, remove the (5) before Example 1.9.

On p. 63, the word `orthogonal' should be removed from Exercise 1.14.

On p. 64, the hint for Exercise 2.1.15 should be gamma' dot delta' just as it is earlier, not gamma' dot delta.

On p. 66, at the bottom, the x_u should be bold.

On p. 67, in Example 2.1, +z should be replaced by -z in the final tangent plane equation and then should give 2x-z=1.

On p. 68, the middle coordinate of the Moebius strip should have a minus sign to get exactly the same surface as before.

On p. 71, Exercise 2.4 asks to show that (v+w)[f] = v[f] + w[f]. The gamma(t) hinted at there doesn't quite work because it doesn't have gamma(0)=p! Instead, take
gamma(t) = x( (u(2t)+bar u(2t))/2, (v(2t)+bar v(2t))/2 ).
Then gamma(0)=p and gamma'(0) = v + w. Thanks to Dave Arnold for spotting this.

On p. 86, the plot3d command should be `plot3d(helixtandev...)' instead of `plot(tandev...)'. The name of the tangent developable was changed to make it more descriptive --- without changing the relevant plot! Also, see below for a comment on Exercise 3.2.17.
Chapter 3

On p. 98, Exercise 2.17 talks about tangent developables. The picture is taken from the plot3d on p. 86. Ted Shifrin has pointed out that the interesting feature of these developables is that they have two `sheets' split by the line of striction (see exercise 2.1.15). The picture doesn't show this, but to see it, just let v=-6..6 instead of v=0..6.

On p. 122, in order to have Gaussplot actually plot a graph, you must specify a number for R in Rsphere.

On p. 123, Maple changed its `pointplot' procedure to `pointplot3d' in going from version 3 to 4, so make the change in the procedure `plotconstcurv'.
Chapter 4

On p. 128, Dave Arnold points out that it would be better to motivate the area integral by taking a parallelogram with sides x_u du and x_v dv.

On p. 134, the parametrization of Henneberg's surface is incorrect. The x^2 component should have a plus sign (+) in between terms. The parametrization is given correctly on p. 229 and (in slightly modified form) as a solution to Bjorling's problem on p. 241-242.
Chapter 5

In Example 2.1, the second geodesic equation should have an `= 0'.

On p. 160, the formula for v is incorrect. There should only be a cr in the numerator of the integrand and a factor R+rcos u times the square root in the denominator.

On p. 164, in the discussion of the angular momentum interpretation of the Claiaut relation, only the z-component of the angular momentum is conserved. This is a bit too complicated to explain at this point because it involves a `cyclic' variable in the Lagrangian. So, I wrongly tried to make the whole angular momentum conserved. Thanks to Ted Shifrin for pointing this out.

Ted Shifrin also pointed out that the basis T, T x U, U is a left handed system, so the sign of geodesic curvature is affected. (See Exercise 6.3.3 for an indication of this.) Note that the Gauss-Bonnet material using w_21 is still OK.

On p. 177, in Exercise 5.5.1, there should be a minus sign in front of the K listed.
Chapter 6

On p. 206, Euler's formula is proved and related to the Euler characteristic to better understand the Gauss-Bonnet theorem. The Euler characteristic requires surfaces to be cut up into pieces which are like 2-disks, so the Euler characteristic of the plane (as for any contractible space) is 1. The reason that Euler's (plane) formula is V-E+F=2 is that the face at infinity is not disk-like. When the point at infinity is added, the plane becomes a sphere and Euler's formula becomes the Euler characteristic of the sphere --- 2! I hope this makes the discussion a little clearer. There always seems to confusion between the formula and the characteristic.
Chapter 7
Chapter 8

On p. 256 the Euler-Lagrange equation is derived in the usual way involving integration by parts. Of course, we should have dv = dn/dt dt and v = n (where, in the book, dots above the n's denote the t derivative). Unfortunately, the v = n came out as v = dn/dt (i.e. n dot in the book).

In the solution of the Brachistochrone Example 2.1, the usual substitution cos^2(theta) = 1/2 + cos(2 theta)/2 on p. 263 has a cos(theta) instead of cos(2 theta).

On p. 266, near the bottom, the sentence should read `Denote the values which J has along the curves ...'.

The answer for Exercise 3.3 should be c*sin(t).

The first sentence of section 6 may be `sightly', but the word there should be `slightly'!
Chapter 9
Maple
The book was written using Maple V version 3, so version 4 users should know that almost everything works exactly as in the text. There are a few differences however.

In the book, a great deal is made of the differences between Maple on the PC and Maple on the Mac or Power Mac. Thankfully, version 4 has brought the Windows and Mac interfaces very close together. Now, it is virtually impossible to get confused going from one interface to the other. In particular, Shift-Return always takes you to a new line (of a procedure say) without trying to execute.

Here is a tip due to Robert Israel told to me by Dave Arnold. For procedures such as those to find curvature and torsion of curves and Gauss and Mean curvatures of surfaces, instead of ending the procedures with `simplify(...)', use `simplify(...,symbolic)' and you will get answers without the annoying `csgn' and possible square roots.

On p. 41, add `num' to the local variable list in `tor'. If you forget, this won't affect your procedure, but it will give a warning message.

On p. 46, the definition of astr3 uses evalm and a * to evaluate a matrix on a vector. This doesn't seem to work in version 4. Replace the * by &* and everything is OK. (Thanks to David Arnold for spotting this.)

On p. 50, for `vi', use limits for the plot t=-2*Pi..2*Pi instead of -Pi..Pi.

For some reason, Maple V version 4 uses a different syntax for the phaseportrait command used in Chapter 8 when discussing the Pontryagin Maximum Principle. Version 3 used some shorthand for the differential equations involved while version 4 requires the equations put in as they would be in dsolve. The initial conditions also must be put in slightly differently. See the Maple Help ?phaseportrait (or ?DEtools[DEplot] which is the same).

On p. 122, in order to have Gaussplot actually plot a graph, you must specify a number for R in Rsphere.

On p. 123, Maple changed its `pointplot' procedure to `pointplot3d' in going from version 3 to 4, so make the change in the procedure `plotconstcurv'.

At the bottom of p. 300, the dsolve command won't work with EL1(f)=0. This is because EL1 returns an `= 0'. So, just change the dsolve to `dsolve({EL1(f),x(0)=1 ...)'.

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