2M2. 2011. Coursework.
                               Fourier Series and Vector Calculus
         Please hand in to the coursework box in the Renold building by 4pm Friday 25th March. Make
      sure you name and student number are written clearly on your answer and all pages are securly
      fastened together.
         1. A periodic function of period 1 is specified by its values on the interval 0 < x < 1: f(x) = 3
           for 0 < x < 1/2 and f(x) = 0 for 1/2 < x < 1. Expand f(x) as a Fourier series. You should
           give explicitly at least the first three non-zero terms, as well as a formula for the general term.
           To what limit does the series converge at the points x = 0, x = 1/3 and x = 1/2? (Justify
           your answer.)
         2. Find the complex Fourier series of the function in Question 1 giving a general formula for ck.
         3. Given F = x2i+yzj,G = (x+3y)i+y2j+xzk,φ = xycos(z2 +x) calculate
            (a) ∇×G
            (b) ∇·F
            (c) ∇φ
         4. Given V = cosxi+j
            (a) Find curlV (you can express the answer as a multiple of k)
            (b) Evaluate                         Z
                                                  C V·dx
                where C is the curve consisting of the line from (0,0) to (0,a) followed by the line from
                (0,a) to (a,b).
            (c) Let ψ(x,y) be the integral you found in (b) with x = a,y = b, find ∇ψ.
            (d) Would the result of (c) be different if C were replaced by a different curve from (0,0) to
                (a,b) (justify your answer)?
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