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    Devry
    Jul-1996 - Jul-2000

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Category > Math Posted 23 Apr 2017 My Price 8.00

USYD math2069/2067 vector calculus assignment

USYD math2069/2067 vector calculus assignment

 

This assignment is due by Friday 20 May 2016 at 4:00pm
and is worth 5% of your assessment for Vector Calculus.
Submit assignments using turnitin.
Check that EVERY page of your assignment is legible and the correct way up
before hitting the CONFIRM button
>
>C
1
C2
R
O
Let R be the region shown above bounded by the curve C = C1 [ C2.
C1 is a semicircle with centre at the origin O and radius
9
5
.
C2 is part of an ellipse with centre at (4; 0), horizontal semi-axis a = 5 and vertical
semi-axis b = 3.
1. (a) Parametrise C1 and C2. Hint: Use t : ????t0 ! t0 as limits when parametrising
C2 and explain why cos(t0) = ????
4
5
and sin(t0) =
3
5
.
(b) Calculate I
C
v dr
where v =
1
2
(????yi + xj).
(c) Use Green's theorem and your answer from 1(b) to determine the area of R
and then verify that it is less than ab.
2. (a) Give the cartesian equation for the ellipse used to dene C2.
(b) Show that 9 + 4r cos = 5r is the equation of that ellipse when written in
polar coordinates (r; ). Hint: Square both sides rst.
(c) Calculate ZZ
R
1
r3dA
using polar coordinates. Hint: Integrate with respect to r rst and then .
Explain why the limits on the outer integral should be = 

2
.
3. If T(r) = T0=r3 is the temperature prole in the region R, then use the previous
results to calculate the average temperature in R when T0 = 1000. Verify that the
average temperature is between the minimum and maximum temperatures in R.

 

 
The University of Sydney
School of Mathematics and Statistics Vector Calculus Assignment
MATH2061/2067: Vector Calculus Semester 1, 2016 This assignment is due by Friday 20 May 2016 at 4:00pm
and is worth 5% of your assessment for Vector Calculus.
Submit assignments using turnitin.
Check that EVERY page of your assignment is legible and the correct way up
before hitting the CONFIRM button
<
R
> O C1
C2 > Let R be the region shown above bounded by the curve C = C1 ∪ C2 .
9
C1 is a semicircle with centre at the origin O and radius .
5
C2 is part of an ellipse with centre at (4, 0), horizontal semi-axis a = 5 and vertical
semi-axis b = 3.
1. (a) Parametrise C1 and C2 . Hint: Use t : −t0 → t0 as limits when parametrising
3
4
C2 and explain why cos(t0 ) = − and sin(t0 ) = .
5
5
(b) Calculate
I
v · dr
C 1
where v = (−yi + xj).
2
(c) Use Green’s theorem and your answer from 1(b) to determine the area of R
and then verify that it is less than πab.
2. (a) Give the cartesian equation for the ellipse used to define C2 .
(b) Show that 9 + 4r cos θ = 5r is the equation of that ellipse when written in
polar coordinates (r, θ). Hint: Square both sides first.
(c) Calculate
ZZ 1
dA
3
R r
using polar coordinates. Hint: Integrate with respect to r first and then θ.
Ï€
Explain why the limits on the outer integral should be θ = ± .
2
3. If T (r) = T0 /r3 is the temperature profile in the region R, then use the previous
results to calculate the average temperature in R when T0 = 1000. Verify that the
average temperature is between the minimum and maximum temperatures in R.
c 2016 The University of Sydney
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ComputerScienceExpert
(11)
Status NEW Posted 23 Apr 2017 12:04 PM My Price 8.00

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