DIFFERENTIAL GEOMETRY EXAMPLES 1

DIFFERENTIAL GEOMETRY
EXAMPLES 1
Do try these! Experience shows that those who do the example sheets
regularly do much better in the final exam.
SECTION A: MATH 3113 AND 5113
1. Should be revision! Let R : I → R 2 be the smooth parametrized
curve defined by
γ(t) = ( 2t/(t 2 + 1) , (t 2 − 1)/(t 2 + 1) ) (t ∈ R) .
(i) Show this is a regularly parametrized curve. (ii) Find its total
length. (iii) Find a unit speed reparametrization (please simplify your
answer). (iv) What is its track?
2. (i) Let γ : I → R n be a smooth regularly parametrized curve
( and T(t) = γ ′ /|γ ′ (t)| the unit tangent vector. Show that γ ′′ (t) · T(t) =
d
dt |γ′ (t)| ) . [This says that the tangential component γ ′′ (t) ·T(t) of the
the acceleration vector is the rate of change of the speed.] This directed
magnitude is positive if the tangential part of the acceleration vector
is in the ‘forward’ direction T(t) and negative if it’s in the backward
direction −T(t).]
(ii) Recall the formula
k(t) =
{
1
γ ′′ (t) −
|γ ′ (t)| 2
( ) }
γ ′′ (t) · γ ′ (t)
γ ′ (t) = γ′′ ⊥ (t) (t ∈ I)
|γ ′ (t)| 2 |γ ′ (t)| 2
for the curvature vector of a smooth regularly parametrized curve γ :
I → R n and the formula ˜k(s) = ˜γ ′′ (s) (s ∈ J) for the curvature vector
of its unit speed reparametrization ˜γ : J → R n . Show that these two
formulae agree, i.e., that ˜k(s) = k(τ(s)) ∀s ∈ J where ˜γ = γ ◦ τ.
3. Calculate the signed curvature, total curvature, and thus the
rotation index, of the closed curve γ(t) = ( 5 sin(6πt), 5 cos(6πt) ) (t ∈
[0, 1]). Explain your result geometrically.
4. Consider the curve given by
γ(t) = ( 20 cos(t) − 6 cos(10t) , 20 sin(t) − 6 sin(10t) ) .
This is a curve called an epitrochoid. (i) Show that this curve is periodic
and give its period. It thus defines a smooth closed curve. (ii) Show
that it is regular. (iii) Show that the signed curvature is always positive.
(iv) Sketch the curve and determine its rotation index geometrically.
[You may use Maple to get an accurate picture of the curve but try to
work out what it looks like first by hand.] (v) The curve is given by the
trace of a point on the flange of a wheel which rolls without slipping
around a larger circle. Can you sketch this and work out the radii of
1

2
the circle and wheel and the distance of the point from the centre of
the wheel which give the above formula?
5. (i) The winding number. Let γ : [a, b] → R be a smooth closed
plane curve and let p be a point in the plane which is not on the
track of γ. Then the winding number of γ with respect to p is the
number of times the point γ(t) goes round p. More precisely, we can
write γ(t) − p = r(t) (cosϕ(t), sin ϕ(t)) (t ∈ [a, b]) for some smooth
functions r : [a, b] → (0, ∞) and ϕ : [a, b] → R; then the winding
number is (1/(2π))(ϕ(b)−ϕ(a)). The winding number depends only on
which component of R 2 −γ[a, b] the point p lies in (proof not required).
Write down the winding number for each such component for (A) the
epitrochoid above with (a) p = (0, 0), (b) p = (20, 0), (c) p = (40, 0);
(B) the curves shown.
(ii) Write down the rotation indices for the four curves shown (these
do not depend on a point p).
Figure 1. Curves for Q. 5.

DIFFERENTIAL GEOMETRY
EXAMPLES 1: HINTS
3
PLEASE TRY THEM ON TWO DIFFERENT
OCCASIONS BEFORE READING THE HINTS
1. (i) The total length is the integral over the whole parameter interval,
i.e., from −∞ to +∞, of the speed. (iv) This should be obvious if you’ve
simplified the answer to (iii).
2. (i) Calculate d dt 〈γ′ (t),γ ′ (t)〉 = d dt |γ′ (t)| 2 in two different ways. (ii) We
have a composition of functions (function of a function): ˜γ = γ ◦ τ, i.e.
˜γ(s) = γ(t) where t = τ(s). Write down the chain rule expression for d˜γ/ds
and think about what dt/ds is equal to; then differentiate again to get an
expression for ˜γ ′′ (s) = d 2˜γ/ds 2 .
3. Use the formula for the signed curvature.
4. (ii) Calculate |γ ′ (t)| 2 . Use fact that the sin and cos of an angle vary
between −1 and +1 to show that this is never zero. (iii) Again estimate
and don’t work out unnecessary quantities. (iv) Don’t try to work out the
signed curvature and integrate it — this is hard; just think geometrically
about how many times the tangent to the curve rotates during one circuit.
Is it 9 or 10? (v) Does the wheel rotate round the large circle 10 times or 9
times? Why? The three numbers asked for are whole numbers.
5. (i) Just think geometrically about how many times the ‘radius’ γ(t)−p
rotates.
(ii) This time, think about how many times the unit positive tangent
rotates.

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DIFFERENTIAL GEOMETRY
EXAMPLES 1
SECTION B: MATH 5113 ONLY
1. Find a smooth parametrized curve with track given by
{(0, y) : y ≥ 0} ∪ {(x, 0) : x ≥ 0} .
Is your curve regular? Might there exist a regular curve with this track?
2. (i) Prove the AM-GM inequality:
√
ab ≤
1(a + b) for all non-negative numbers a and b.
2
Prove that equality occurs if and only if a = b.
*(ii) What is the generalization to n numbers. Can you prove it?
3. (i) Prove that, for any real numbers a 1 , a 2 , b 1 , b 2 ,
( 2∑
) 2 2∑ 2∑
a i b i ≤ a 2 i b 2 i.
i=1 i=1 i=1
When does equality occur?
*(ii) What is the generalization to 2n numbers. Can you prove it?

DIFFERENTIAL GEOMETRY
EXAMPLES 1: HINTS
5
PLEASE TRY THEM ON TWO DIFFERENT
OCCASIONS BEFORE READING THE HINTS
1. There is a smooth function f : R → R which is identically zero on
{x ∈ R : x ≤ 0},and positive on {x ∈ R : x > 0}. Sketch the graph of
such a function. What are the values of f(0), f ′ (0), f ′′ (0),...? Can you
give a formula for such an f? If not, search some analysis books. Use f to
manufacture the desired curve.
2. (i) Square up ...
3. (i) Write RHS − LHS as a perfect square.