Problem 1. (a) State the inverse function theorem for functions of a single variable. (You don't need to prove it.) (b) Let I R be an open interval, and let g : I ! R be a smooth function. Let J be th

Problem 1. (a) State the inverse function theorem for functions of a single variable.

(You don't need to prove it.)

(b) Let I R be an open interval, and let g : I ! R be a smooth function. Let J

be the range of g, i.e. J = g(I). Prove that if g0(t) 6= 0 for all t 2 I, then there

exists a smooth function f : J ! I which is the inverse to g, i.e. g f = 1J , and

f g = 1I . Hint: Use part (a).

(c) Prove that the function f has the property that, for all t 2 J, we have

f0(t) =

1

g0(f(t))

: (1)

Problem 2. Let   : I ! R3 be a regular smooth curve, (not necessarily parametrized

by arc length). Let g : I ! R be the arc length of  . Set J = g(I), and let f

be the inverse of g. For the existence of g, we refer to Problem 1. Recall that the

curve   =   f : J ! R3 is parametrized by arc length, (you don't need to prove

this). By de nition, the curvature of   at s 2 I, is the curvature of   at f 1, i.e.

k (s) = k (f 1(s)). In this problem, we're going to  nd a formula for the curvature of

.

(a) Prove that, for all s 2 I we have

00(f 1(s)) =  00(s) (f0(f 1(s)))2 +  0(s) f00(f 1(s)):

1

(b) Use Eq. 1 to express f0(g(s)) in terms of k 0(s)k, and then use the result to prove

that, for all s 2 I

f00(g(s)) =

00(s)  0(s)

k 0(s)k4 :

HINT: First prove that

d

ds

k 0(s)k =

00(s)  0(s)

k 0(s)k

:

(c) Use parts (a) and (b) to show that

k 00(f 1(s))k2 =

1

k 0(s)k6

k 00(s)k2k 0(s)k2 ( 00(s)  0(s))2

;

and, then take the square root of both sides of this equation, and use an appropriate

formula for the cross product, to prove

k (s) =

k 00(s)  0(s)k

k 0(s)k3 : (2)

Problem 3. Consider the smooth curve   : R ! R2, de ned as

(t) =

e t

p

2 + 1

(cos(t); sin(t)):

(a) Determine its velocity  0(t), and show that   is not parametrized by arc length.

(b) Determine the curvature k (t) of  . HINT: use Equation 2.

(c) Find a reparametrization of   so that it is parametrized by arc length. That is,  nd

a smooth bijective function f : I ! R with smooth inverse f 1 : R ! I, such that

the smooth curve   =   f : I ! R2 is parametrized by arc length. (HINT: Start

by determining the arc length of  .)

(d) Determine the curvature of  , and compare with part (b).

Problem 4. Consider the smooth curve   : (0;1) ! R3, de ned as

(t) = (e t

8 cos(t); e t

8 sin(t); t):

(a) Sketch the trace of  , and in the same  gure, draw the tangent line of   at t = =2.

(b) Compute the curvature k  and the torsion  of  .

(c) Set a(t) =  00(t), and de ne

a?(t) = a(t)

a(t)  0(t)

k 0(t)k

:

2

Recall that the unit normal is then de ned by

n (t) =

a?(t)

ka?(t)k

:

Calculate n (t).

(d) The osculating circle of   at t0 2 (0;1) is the trace of the curve

(s) =

1

k (t0)

cos(s)

0(t0)

k 0(t0)k

+ sin(s)n (t0)

+  (t0) +

1

k (t0)

n (t0):

Set t0 = and draw the corresponding osculating circle Draw the trace of   in the

same picture.