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Math 2XX3 - Assignment 4
Due: April 8, 11:59 pm (EST) Please submit scans / photographs of your full solutions (if handwritten) or pdfs (if typed) to the Crowdmark dropbox by the above date. You are welcome to discuss these problems with your classmates, but your submission must be your own work and written in your own words. Also please include an acknowledgement naming any classmate who helped you and citing any outside sources you consulted. You are not required to use exactly the same notations that are used in the lectures and course notes, but if you wish to modify or introduce your own notation you must first clearly define it. 1. Draw pictures to illustrate the case n = 1 and m = 2 of Exercise 8 on page 118 of the Lecture Notes, and explain how your pictures relate to the four hints given there. 2. Let L(u) = ∫ 1 0 F (u ′(s), u(s), s) ds for u ∈ C2([0, 1]), where F (v, w, s) = √ 1+v2 w . Find the Euler-Lagrange equation, and solve it to find a general solution for the local extremals of L(u). Hint: note that F is independent of the variable x. 3. Justify in detail each line of the tricky application of the chain rule in display (2.2) on page 126 of the Lecture Notes. 4. Let f(u) = √ 2u4 1+u4 in Subsection 3.2 on page 130 of the Lecture Notes. Use an online graphing program (such as Desmos) to help sketch a variety of geodesics with differing parameters λ as in the picture on page 132 of the Lecture Notes. Extend your sketches to show the geodesics returning to the y-axis as well (when applicable). Make sure your examples demonstrate each of behaviours (2), (3), and (4) in Summary 1 on page 131 of the Lecture Notes. You may submit either hand-drawn sketches or printouts from your graphing calculator for this problem. 5. Let Yn(x) = √ 2 cosnx and Y0(x) = 1, so that {Yn}∞n=1 is an orthonormal basis for L2([0, pi]) with respect to inner product 〈P,Q〉 = 1pi ∫ pi 0 P (x)Q(x) dx. Then for φ ∈ L2([0, pi]), the cosine series of φ is given by φ(x) = ∞∑ n=0 anYn(x) = a0 + ∞∑ n=0 an √ 2 cosnx, 0 ≤ x ≤ pi, where equality in the display above holds in the sense of mean square convergence (you may simply assume this as it is analogous to the situation for sine series). Calculate the Fourier cosine series of x2 on the interval [0, pi], and then use Parseval’s identity to prove that ∞∑ n=1 1 n4 = pi4 90 .