ERRATA IN DO CARMO, DIFFERENTIAL （在做u2022勘误表,微分）.pdfVIP

ERRATA IN DO CARMO, DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES BJORN POONEN This is a list of errata in do Carmo, Differential Geometry of Curves and Surfaces, Prentice- Hall, 1976 (25th printing). The errata were discovered by Bjorn Poonen and some students in his Math 140 class, Spring 2004: Dmitriy Ivanov, Michael Manapat, Gabriel Pretel, Lauren Tompkins, and Po Yee Wong. Some errata were discovered later by Brent Doyle. • p. 5, line 9, “using properties 3 and 4”: Actually, property 2 also is being used, since property 4 gives linearity in the second variable only. • p. 5, bottom: The definition of the tangent line is confusing. It is not the line passing through the points α(t) and α (t) but rather the line through α(t) in the direction of α (t), namely { α(t) + λα (t) : λ ∈ R }. • p. 7, Exercise 3, “On 0B mark off the segment 0p = CB .” It would be better to say “On 0B mark a point p such that 0p = CB .” • p. 8, Figure 1-8: The labelling is wrong: the points p and C should lie on the same half-line r through 0 as B . • p. 8, Figure 1-8: It might be good to add a point labelled Y on the y-axis. • p. 8, Figure 1-9: The angle labelled t is in the wrong place. (See problem 4 on p. 7.) • p. 13, line 4: Change “It follows from property 4 that the vector product u ∧ v = 0 is normal to a plane generated by u and v .” to “It follows from property 4 that if u and v are linearly independent, then u ∧ v is normal to the plane spanned by u and v .” or better yet, “It follows from properties 3 and 4 that if u and v are linearly independent, then u ∧ v is a nonzero vector normal to the plane spanned by u and v .”