Description
1. (10 pts) Let f : R −→ R be a C1-function, and let C be the plane
curve given by the equation r = f(θ) in polar coordinates. Show that
the arc length of C is given by
� θ
� s(θ) = f(τ )2 + (f�
(τ ))2 dτ. α
2. (5 pts) (3.1.18).
3. (10 pts) (3.1.26).
4. (10 pts) (3.1.30).
5. (5 pts) (3.1.32).
6. (10 pts) (3.2.7).
7. (10 pts) (3.2.12).
8. (10 pts) In this question, we prove the uniqueness statement of
Theorem 2.5 on page 201 of the book. Let �r1 : I −→ R3 and �r2 : I −→
R3 be two smooth regular curves parametrised by arclength. Assume
that κ1(s) = κ2(s) and τ1(s) = τ2(s), for every s ∈ I. Suppose that
there is a point a ∈ I where
�r1(a) = �r2(a), T�1(a) = T�2(a), N� 1(a) = N� 2(a), and B� 1(a) = B� 2(a).
(a) Show that the quantity
�T�1(s) − T�2(s)�2 + �N� 1(s) − N� 2(s)�2 + �B� 1(s) − B� 2(s)�2
,
is a constant function of s. (Hint: Differentiate and use the FrenetSerret formulae.)
(b) Show that �r1(s) = �r2(s), for all s ∈ S.
9. (10 pts) Let �r : R −→ R3 be the helix given by
s s bs ˆ �r(s) = a cos ˆı + a sin jˆ+ k, c c c
where
c2 = a2 + b2 a > 0 and c > 0.
(a) Show that �r is parametrised by arclength.
(b) Find T�(s), N� (s) and B� (s).
(c) Find κ(s) and τ (s).
1
10. (10 pts) Suppose that �r : R −→ R3 is a smooth regular curve
parametrised by arclength. Suppose that the curvature and the torsion
are constant, that is, suppose that there are constants κ and τ such
that κ(s) = κ and τ (s) = τ . Prove that �r is (congruent to) a helix.
11. (10 pts) Let �r : I −→ R3 be a smooth regular curve. Let a ∈ I and
suppose that
2 2 1 1 2 2 dN� � ı+ j− ˆ � ˆ ˆ ˆ T(a) = ˆ ˆ k, B(a) = − ı+ j+ k, (a) = −4ˆı+2jˆ+5k. ˆ 3 3 3 3 3 3 ds
Find
(a) The normal vector N� (a).
(b) The curvature κ(a).
(c) The torsion τ (a).
Just for fun: Show that the implicit function theorem and the inverse
function theorem are equivalent, that is, one can prove either result
assuming the other result.
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18.022 Calculus of Several Variables
