The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH2021 Vector Calculus and Differential Equations Semester 1, 2020
Lecturer: Zhou Zhang
• Due on Sunday, 24 May 2020 at 11:50PM, Sydney Time, which is the deadline.
• Format: hand-written on physical paper and scanned for submission. It’s fine to take pictures
for pages of solution paper separately and compile them into one file for submission.
• We DON’T accept writing on iPad or tablet, or submission in any other form.
• Submit your assignment through turnitin on Canvas. Double check the status of paper after
submission.
• DON’T include NAME in the title of the submission line for anonymous marking.
• MUST include SID on paper.
1. We have the unit ballB = {(x, y, z) | x2+y2+z2 ≤ 1} in R3 with the unit sphere S = {(x, y, z) | x2+
y2+z2 = 1} as its boundary. S is the union of hemispheres S1 = {(x, y, z) | x2+y2+z2 = 1, z ≥ 0}
and S2 = {(x, y, z) | x2 + y2 + z2 = 1, z ≤ 0}. Consider the vector field V = (x, y, z2).
1) S is a level surface. Explain that n = (x, y, z) is its unit outward normal vector field.
2) Write down the surface integral
∫∫
S1
V · n dS as a double integral by considering S1 as a graph
over xy-plane. You are NOT asked to calculate this double integral.
3) Apply Gauss’ Theorem to calculate
∫∫
S
V · n dS.
Hint: cylindrical coordinates and/or symmetry might be helpful.
2. In the xy-plane R2, we have the unit disk centred at origin, D = {(x, y) | x2 + y2 ≤ 1}, and its
boundary circle C in the counter-clockwise direction. In the xyz-space R3 where the xy-plane is
the plane {z = 0}, we have the unit upper hemisphere S1 = {(x, y, z) | x2 + y2 + z2 ≤ 1, z ≥ 0}
with the same boundary C.
Consider the vector field F (x, y, z) = (y,−x, xy sin z) over R3.
1) Apply Green’s Formula/Theorem to calculate
∮
C
F · dr.
2) Calculate (∇ × F ) · n over S1, where n is the unit normal of S1, pointing out of the unit ball
centred at origin. The answer should be a function with only variables x and y.
3) Apply Stokes’ Theorem to calculate
∫∫
S1
(∇× F ) · n dS.
3. Find the particular solution for the following ODEs with unknown function y(x).
1) y′ + y = 0, y(0) = 1.
2) y′ + y = x, y′(0) = 0.
3) y′′ + 2y′ + 2y = 0, y(0) = y′(pi) = 0.
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