Mathematical Methods I Dr M B Wingate Natural Sciences Tripos, Part IB Michaelmas Term 2012 Example Sheet 0 Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. This is a revision sheet. If you did NST Mathematics A or B last year you should be able to do the questions already (let me know if I have made an incorrect assumption here, especially if you did Course A). If you did the Mathematical Tripos last year you will have to read up on Fourier Series for question 4. Some of the material will be touched on in the first couple of lectures, so you might prefer to wait until then. 1. Let f (x) be a function of one variable. Working to O((δx)3 ), write down the Taylor expansion of f (x + δx). Let g(x, y, z) be a function of three variables. Working to O(δx, δy, δz), write down the Taylor expansion of g(x + δx, y + δy, z + δz). Express the latter using vector notation. 2. (a) Let h be a function of one variable. Working from first principles, differentiate with respect to x the function Z x h(y) dy , I(x) = a where a is a constant. (b) Let f (x, y) be a function of two variables. Working from first principles, differentiate with respect to x the function Z x f (x, y) dy . J(x) = a 3. Find functions y(x) which satisfy x2 d2 y dy + 3x − 15y = 0 . 2 dx dx 4. (a) Let f (y) be an odd periodic function of y with period 2, i.e. f (−y) = −f (y) and f (y + 2) = f (y). Given that f (y) = 12 y(y − 1) for 0 6 y 6 1, sketch the function f (y) for −2 6 y 6 2, and find the Fourier (sine) series for f . (b) Let g(x) be an even periodic function of x with period 4L. Given that g(x) = (x − L)/L for 0 6 x 6 2L, sketch g(x) for −4L 6 x 6 4L and find the Fourier series for g. (c) Let h(x) be a periodic function of x with period 2 ln 2, where ½ 1 for − ln 2 < x 6 0 h(x) = x e − 1 for 0 < x 6 ln 2 . Find the Fourier series for h. 1 5. Show the following: (a) For vector components ai (i = 1, 2, 3), 3 X aj δij = ai ; j=1 where δij is the Kronecker delta, Copyright © 2012 University of Cambridge. Not to be quoted or reproduced without permission. (b) For independent variables qi (i = 1, 2, 3), ∂qi = δij . ∂qj 6. Show that, if ei · ej = δij (i, j = 1, 2, 3) and e1 × e2 = e3 , then e2 × e3 = e1 and e3 × e1 = e2 . 7. Using a coordinate system of your choice, write down an example function possessing each of the following kinds of symmetry: (a) A function of 2 variables which has circular symmetry (b) A function of 3 variables which is spherically symmetric (c) A function of 3 variables which has axial symmetry 8. Evaluate ∇ · r and ∇ × r, where r is the position vector. 9. Show that, if v = ω × r, where ω is a constant vector and r is the position vector, then ∇ × v = 2ω. (Feel free to use suffix notation.) Comment. 10. Let ψ(r) and χ(r) be scalar fields, and u(r) and v(r) vector fields. Show the following (feel free to use suffix notation): (a) ∇ × ψv = ∇ψ × v + ψ∇ × v , (b) ∇·u×v = v·∇×u−u·∇×v, and hence that ∇ · (∇ψ × ∇χ) = 0 . This example sheet is available at http://www.damtp.cam.ac.uk/user/examples/ Hints and answers will be posted to CamTools. Supervisors may request these early. Comments/corrections to M.Wingate@damtp.cam.ac.uk. 2

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