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MATH2021: Vector Calculus and Differential Equations

创建日期：2024-04-29 18:12:27

所属分类：数学mathematics

标签： MATH2021

Vector Calculus and Differential Equations

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MATH2021: Vector Calculus and Differential Equations
This individual assignment is due by 11:59pm Thursday 31 March 2022, via Canvas. Late assign-
ments will receive a penalty of 5% per day until the closing date.
A single PDF copy of your answers must be uploaded in Canvas. Please make sure you review your
submission carefully. What you see is exactly how the marker will see your assignment.
To ensure compliance with our anonymous marking obligations, please do not under any circumstances
include your name in any area of your assignment; only your SID should be present.
The School of Mathematics and Statistics encourages some collaboration between students when work-
ing on problems, but students must write up and submit their own version of the solutions.
This assignment is worth 5% of your final assessment for this course. Your answers should be well written,
neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any resources used and show all
working. Present your arguments clearly using words or explanations and diagrams where relevant. After all,
mathematics is about communicating your ideas. This is a worthwhile skill which takes time and effort to mas-
ter. The marker will give you feedback and allocate an overall letter grade and mark to your assignment using
the following criteria:
Mark Grade Criterion
8 A Outstanding and scholarly work, answering all parts correctly, with clear accurate expla-
nations and all relevant diagrams and working. There are at most only minor or trivial
errors or omissions.
7 B Very good work, making excellent progress, but with one or two substantial errors, mis-
understandings or omissions throughout the assignment.
6 C Good work, making good progress, but making more than two distinct substantial errors,
misunderstandings or omissions throughout the assignment.
5 D A reasonable attempt, but making more than three distinct substantial errors, misunder-
standings or omissions throughout the assignment.
3 E Some attempt, with limited progress made.
1 F Extremely limited attempt.
0 Z No credit awarded.

1. Let C be the part of the graph of y =
√
1 + x2 between x = 0 and x = 1.
(a) Write down a parametrisation of C, and find the velocity vector of your parametrisation.
(b) Find the line integral ∫
C
xy ds.
2. Let F (x, y, z) = (1 + 2xyz3, z + x2z3, y + 3x2yz2).
(a) Show that curl(F) = 0.
(b) Find a function f : R3 → R such that F = ∇ f .
(c) Hence, or otherwise, find the value of the integral∫
C
F · d s,
whereC is the part of the helix with parametrisation γ(t) = (cos t,sin t, t) with t ∈ [0, pi].
3. (a) Write the iterated integral ∫ 1
0
∫ x
x2
f (x, y) dydx
as an iterated integral in the reverse order (that is, with inner integral with respect to x
and outer integral with respect to y). Include in your solution a sketch of the domain of
integration of the iterated integral.
(b) Let P be the parallelogram in R2 with vertices (0,0), (1,1), (4,−2), and (5,−1). Calcu-
late the double integral "
P
(x − y) dA.
4. Calculate the area enclosed by the curve
γ(t) = (4 cos t − cos(4t),4 sin t − sin(4t)), t ∈ [0,2pi].
A sketch of the curve is shown below. Hint: Green’s Theorem may help.
MATH1903 Integral Calculus 33
Example 3.19. An epicycloid is a curve traced out by a fixed point on the rim of a circle as
it rolls around on the outside of a larger circle. There is a good ill stration on the webpage
http://en.wikipedia.org/wiki/Epicycloid. If the radius of the rolling circle is 1 and the
radius of the larger circle is k then th corresponding epicycloid has parametric equation
x(t) = (k + 1) cos t− cos((k + 1)t)
y(t) = (k + 1) sin t− sin((k + 1)t) t ∈ [0, 2pi]. (3.1)
Some examples are shown in Figure 3.9.
−3 −2 −1 0 1 2
−3
−2
−1
0
1
2
3
−3 −2 −1 0 1 2 3
−4
−3
−2
−1
0
1
2
3
4
−5 0 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
−6 −4 −2 0 2 4 6
−6
−4
−2
0
2
4
6
Figure 3.9: Epicycloids for k = 1 (cardioid), k = 2, 3 and k = 4
Compute the length of the epicycloid given by (3.1).
Solution. A similar calculation to the hypocycloid computation gives
x′(t)2 + y′(t)2 = 4(k + 1)2 sin2(kt/2).
Therefore
L = 2(k + 1)
∫ 2pi
0
| sin(kt/2)| dt = 2k(k + 1)
∫ 2pi/k
0
sin(kt/2) dt = 8(k + 1).
Remark 3.20. The epicycloid with one cusp (corresponding to k = 1) is called the cardioid .
Its parametric equation can also be written as
(x(t), y(t)) = (2(1 − cos t) cos t, 2(1 − cos t) sin t), t ∈ [0, 2pi].
Remark 3.21. In the Ptolemaic (Earth-centred) system of the solar-system the planets orbits
are epitrochoids. These are generalised versions of epicycloids, where the point is not necessarily
on the rim of the rolling circle.
Example 3.22. The Euler spiral has parametric equation in terms of the Fresnel integrals:
x(t) =
∫ t
0
cos(s2) ds, y(t) =
∫ t
0
sin(s2) ds, t ∈ R.
Write r(t) = (x(t), y(t)), and let τ (t) = r′(t)/‖r′(t)‖ be the unit tangent vector. The curvature
of C is, by definition, ‖τ ′(t)‖. Calculate the length of the Euler spiral between t = 0 and t = T .
Calculate the curvature of the Euler spiral at t = T . Hence show that the curvature of the Euler
spiral varies linearly with arc length.

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