Monash University
School of Mathematical Sciences ENG2091
Assignment • This assignment is due by 12 noon on Wed, 11 October 2006. Deposit your assignment in the Assignment Box labelled with the name of your tutor on the 3rd floor of Building 28 opposite the lifts. • A completed and signed Assessment Cover Sheet, available from the unit MUSO site, must be attached to your assignment. • Late assignments must be handed in directly to your lecturer (otherwise it will not be marked) and will be penalised at 5% for each calendar day.
1. Evaluate the flux integral
RR
F = [3x2 , y 2 , 0],
FndS. S : r = [u, v, 2u + 3v], 0 ≤ u ≤ 2, −1 ≤ v ≤ 1
2. Find the Fourier Series for f (x) = x2 /2, −π ≤ x ≤ π. Show whether f (x) is an odd or even function. 3. By choosing an appropriate value for x in exercise 2, find the sum of the series ∞ X
1 . 2 n=1 n 4. Use the Fourier Series in exercise 2 to evaluate 1 − 14 + 19 −
1 16
+ ...
5. Compute the circulation of the vector a = yi + x2 j − zk around the contour L : {x2 + y 2 = 4, z = 3}. Plot the contour and show its orientation. (a) directly (b) via the Stokes Theorem. 6. Showing the details of your work, find the Fourier series of the function f (x), which is assumed to have the period 2π, and plot accurate graphs of the first three partial sums, where f (x) = x, (−π < x < π). 7. Plot the surface S : {y = x2 , −2 ≤ x ≤ 0, −3 ≤ z ≤ 3}