School of Mathematics and Statistics
MAST20009 Vector Calculus, Semester 1 2021
Assignment 4 and Cover Sheet
Student Name Student Number
Tutor’s Name Tutorial Day/Time
Submit your assignment via Gradescope accessed via the MAST20009 Canvas page
before 12pm on Tuesday 18 May.
• This assignment is worth 5% of your final MAST20009 mark.
• Assignments must be neatly handwritten in blue or black pen on A4 paper or may be typeset using
LATEX. Diagrams can be drawn in pencil.
• You must complete the plagiarism declaration on the LMS before submitting your assignment.
• Full working must be shown in your solutions.
• Marks will be deducted for incomplete working, insufficient justification of steps, incorrect
mathematical notation and for messy presentation of solutions.
1. Let S be the surface given by
x2 + 2y4 + 3z6 = 1 with z ≥ 0.
Let Γ be the curve
γ(t) =
(
cos(pit),
√
t+ 1,
2t
t2 + 1
)
, for 0 ≤ t ≤ 1
and let F be the vector field
F(x, y, z) = (a− 4z2x)i+ 2byj+ cx2zk
where a, b and c are real numbers.
(a) Compute ∫∫
S
(∇× F) · dS.
(b) Find all values of a, b and c such that the vector field F is conservative.
(c) If a, b and c are chosen such that F is conservative, compute the work done by F to move a
particle along Γ in the direction of increasing t.
2. Let T be the triangle with vertices (−1, 0), (1, 1) and (0, 2) traversed in the anticlockwise direction.
Let nˆ be the outward unit normal to T in the x-y plane.
(a) Sketch T , indicating nˆ on each part of T .
(b) Evaluate ∫
T
F · nˆ ds
where
F(x, y) = (sinh3(2y) + 2x2 + 3x+ 2ye−3x, 4e2x sinhx− 3y + 3y2e−3x).
3. Let R be the region of the plane x− 2y+ 2z = 1 cut out by the “cylinder” whose walls are given by
the equation 4x2 = (1− y)2(1− y2). Find the area of R.
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