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MATH2061: Linear Mathematics and Vector Calculus

创建日期：2024-06-17 16:38:42

所属分类：数学mathematics

标签： MATH2061

Linear Mathematics and Vector Calculus

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MATH2061: Linear Mathematics and Vector Calculus

The assignment will be due at 11:59pm on Friday 10 May 2024, and is worth 5%
of your final raw mark. The assignment must be scanned/imaged and submitted in
PDF format online via Canvas with Turnitin. Please
ensure your submitted pdf is legible and keep your original handwritten version.
1. Consider the curve C in the xy-plane given by
(x(t), y(t)) = (a(1− cos t) cos t, a(1− cos t) sin t) , 0 ≤ t ≤ 2π,
where a is a fixed positive number. Write r(t) = x(t)i+ y(t)j.
1. Sketch the curve C, and calculate dr(t)
dt
and
∣∣∣dr(t)dt ∣∣∣.
2. Calculate the arc length of C.
3. Calculate the line integral
´
C
F · dr for F = −yi+ xj.
[Warning: You are NOT allowed to pick a value for a in your solution.]
2. Let a and c be fixed positive numbers. Consider the two surfaces
S˜1 : z =
√
c2 −
( c
a
)2
(x2 + y2) and
S˜2 : z =
√
x2 + y2
in R3. Let V be the solid body bounded by the two surfaces, i.e., the finite region above
S˜2 and below S˜1 in R3. The boundary surface S of V is the union of S1 = S˜1 ∩ S and
S2 = S˜2 ∩ S (where S˜i ∩ S is the part of S belonging to S˜i for i = 1, 2).
1. Calculate the volume of V .
2. Calculate the outward pointing unit normal vectors for S1 and for S2.
3. Calculate the outward flux cross S of the vector field F = −y
a
i+
x
a
j+
z2
c2
k.
[Warning: You are NOT allowed to pick values for a and c in your solution.]
3. Let C be a simple closed curve in the xy-plane such that the point (1,−1) is not in C,
and let R be the region bounded by C. Denote by n the outward pointing unit normal
vector of C. Consider the function ϕ = 1
2
ln ((x− 1)2 + (y + 1)2) on R2\{(1,−1)}, and
let F = ∇ϕ.
1. Calculate F and ∇ · F on R2\{(1,−1)}.
2. Assume that (1,−1) is not in R. Calculate ¸
C
F · n ds.
3. Assume that (1,−1) is in the interior of R. Calculate ¸
C
F · n ds.
[Notation: lnw is the natural logarithmic function for w > 0.]

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