文档介绍：Math 401 - Solutions to Final Exam
1. Suppose Φ(x, y, z) = (λx, λy, λz) for some fixed λ> 0, C is a regular curve with
nonvanishing curvature, S is a regular surface, and C ⊂ S.
(a) Find the curvature and torsion of Φ(C) at Φ(p) in terms of the curvature
and torsion of C at p.
1 1
κ
Φ(p) = κ(p), τ
Φ(p) = τ(p) .
Φ(C) λ C Φ(C) λ C
(b) Find the mean curvature and Gauss curvature of Φ(S) at Φ(p) in terms of
the mean curvature and Gauss curvature of S at p.
1 1
H
Φ(p) = H (p), K
Φ(p) = K (p) .
Φ(S) λ S Φ(S) λ2 S
(c) Find the geodesic curvature of Φ(C) in Φ(S) at Φ(p) in terms of the geodesic
curvature of C in S at p.
1
κ
Φ(p) = κ(p) .
gΦ(C) λ gC
(d) Find the length of Φ(C) and area of Φ(S) in terms of the length of C and
the area of S.
length Φ(C) = λlength C, area Φ(S) = λ2area S .
2. Suppose α and β are smooth curves defined on R.
(a) Show that, if α0(0) and β0(0) are linearly independent, then, for some
sufficiently small positive r, ~x(u, v) = α(u) + β(v) defines a regular parametrized
surface for −r < u < r, −r < v < r. Here ~x is a smooth function from R2 to R3.
Since
0 0
~xu(0, 0) ∧~xv(0, 0) = α(0) ∧β(0) 6= 0 ,
the inverse function theorem implies that on a sufficiently small neighborhood of
(0, 0), ~x is one-one with a smooth inverse and so defines a regular parametrized
surface.
(b) Find a condition on α(0), α0(0), β(0),