The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH2021 Vector Calculus and Differential Equations Semester 1, 2021
Lecturer: Zhou Zhang
This assignment is due by Sydney Time, 5 April 2021 at 11:50PM (Monday, Week 6).
Your paper needs to be hand-written on traditional paper and scanned for submission.
DON’T use IPad or Tablet to write your solutions.
Submit your assignment using turnitin on Canvas.
ONLY write SID on your paper and DON’T include your NAME in the title of the submission line
for anonymous marking.
1. (1) In xy-plane, sketch the region R which is enclosed by the following:
ˆ the part of the circle x2 + y2 = 4 with x > 0;
ˆ the line y = 1;
ˆ the part of the circle x2 + y2 = 1 in the second quadrant;
ˆ the line through the points (−1, 0) and (√3,−1).
(2) The boundary of R is a curve C with four pieces corresponding to the four curves in (1). Find
the parametrisation for C by considering it as the following motion:
ˆ starting at the point (0, 1);
ˆ going around the origin in the counter-clockwise direction;
ˆ being unit speed.
Note: The parameter needs to start at 0 and change continuously for the whole curve C. Don’t
worry about the vertices where the motion suddenly changes direction.
2. For the curve C in Problem 1,
(1) calculate the total arc length;
(2) calculate the line integral ∫
C
−→
V · d−→r
for the vector field
−→
V = (y,−x);
(3) calculate the line integral ∫
C
−→
W · d−→r
for the vector field
−→
W = (x, 2y).
(4) decide whether
−→
V and
−→
W are conservative, and explain your answers.
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