Ma 109b Geometry and Topology
Winter 2010
HOMEWORK 3 SOLUTIONS
1. The dierential of f is the matrix
df(x,y,z) = (0, 0, 2z ),
which is singular at (0, 0, 0), which is in the preimage of 0.
f 1 (0) is the plane z = 0. The map i : (x, y ) (x, y, 0) is a para

PMATH 365 / AMATH 333 (Winter 2009): Assignment #3 (1) Let (s) be a unit-speed parametrized curve in R3 with curvature (s) > 0 for all s (s1 , s2 ). Let X(s) be a unit vector eld dened along the parametrized curve (s). That is, X(s) = aT (s) + bN (s) + cB

Math 143, Winter 2012
Homework 4, Due Thurs, Feb 9
1) Problem 6 on page 151 of the text.
2) Problem 15 on page 152.
3) Problem 2 on page 168.
4) Problem 6 on pages 168-169.
5) Problem 7 parts a, b, and e on pages 169-170.
1

Math 143, Winter 2012
Homework 3, Due Thurs, Feb 2
1) Problem 13 on page 82 of the text.
2) Problem 4 on page 88 of the text. Note that here the tangent plane
at p S contains the point p and is not a linear subspace.
3) Problem 3 on page 99.
4) Problem 7

Math 143, Winter 2012
Homework 2, Due Thurs, Jan 26
1) a) Problem 1 on page 404 of the text.
b) Sketch a regular closed curve in the plane such that there is no point
on the curve at which the unit tangent vector is equal to (0, 1).
c) Show that for any r

PMATH 365 / AMATH 333 (Winter 2009): Assignment #2 Solutions (1) We can think of (t) as being a space curve by taking the third coordinate to be z(t) = 0. We can use the formula | (t) (t)| (t) = | (t)|3 for the curvature of a parametrized space curve. The

Ma 109b Geometry and Topology
Winter 2010
HOMEWORK 5
1. For each of the following parametrisations, nd the values of a, b, c and of u, v
which make the image into a regular surface. Now compute the rst fundamental forms for these regular parametrised surf

Ma 109b Geometry and Topology
Winter 2010
HOMEWORK 4
1. (a) Prove that if L : R3 R3 is a linear map and S R3 is a regular
surface invariant under L (i.e. L(S ) S ) then the restriction of L to S is
a dierentiable map S S and
dLp (w) = L(w)
for all p S and

Ma 109b Geometry and Topology
Winter 2010
HOMEWORK 3
1. Let f (x, y, z ) = z 2 . Show that 0 is not a regular value of f . Prove that f 1 (0)
is a regular surface.
2. (a) Let U = (0, ) (0, 2 ), and let a, b, c = 0. Dene x : U R3 by
x(u, v ) = (a sin(u) co

HOMEWORK 2 - GEOMETRY OF CURVES AND
SURFACES
ANDRE NEVES
Solve problems (2), (4), and (5).
(1) Let S, M be two surfaces in R3 , F : S M a smooth map, and
: U R2 R3 a chart with (0) = p. Recall that the denition
of
DFp : Tp S TF (p) M
was
d(F )
(0) where

MA3D9: Geometry of curves and surfaces
Exercises 3.
(1) Compute the rst fundamental forms of the following parameterised surfaces:
ellipsoid: r(u, v ) = (a cos u cos v, b sin u cos v, c sin v )
elliptic paraboloid: r(u, v ) = (au cos v, bu sin v, u2 ).
hy

MA3D9. Curves and surfaces.
Exercises 2.
(1) Calculate the Frenet frame, curvature and torsion of the curve [t (t, at2 , bt3 )] at the
origin. Verify the Frenet-Serret formula in this case.
(2) (The helix). Consider the curve : R R3 given by
(s) = (a cos

Math 109b - Problem Set 3
Evan Dummit
2.5.6. First, by geometry, the cone with vertex angle 2 a is defined implicitly by the equation x2 + y2 = z2 cot a. Clearly xHu, vL
satisfies this equation for all Hu, vL e 2 so xHu, vL lies on the cone. Also x is cle

Math 109b - Problem Set 2
Evan Dummit
2.2.1. The cylinder S = 9Hx, y, zL x2 + y2 = 1= is a regular surface by Prop. 2.2.2 in Docarmo, because S = f -1 H0L with
f Hx, y, zL = x2 + y2 - 1 - this is true because 0 is a regular value of f as the partial deriv

Ma 109b Geometry and Topology
Winter 2010
HOMEWORK 5 SOLUTIONS
1. Note that in each of these cases we need to nd a, b, c, u, v so that the dierential
is injective. We do this by showing xu xv = 0. Note that it is not necessary
to check that the map is inj

Ma 109b Geometry and Topology
Winter 2010
HOMEWORK 4 SOLUTIONS
1. a) The proof that the restriction of a dierentiable map R3 R3 to a surface
is dierentiable is in example 3 of p.74 of do Carmo.
Given w Tp S , choose a curve : I S over some interval with (

Math 143, Winter 2012
Homework 2 Solutions
1) a) Problem 1 on page 404 of the text.
(a) -3, (b) -4, (c) 1, (d) 0
b) Sketch a regular closed curve in the plane such that there is no point
on the curve at which the unit tangent vector is equal to (0, 1).
c)