N12d Natural Sciences, Part IA Dr M. G. Worster Mathematics (Course B) Lent Term 2005 Examples Sheet 2 Please communicate any errors in this sheet to Dr Worster at M.G.Worster@damtp.cam.ac.uk. Note that there is a different examples sheet for course A Gradient, Divergence and Curl 1. For the function f(x, y, z) = ln(x 2 + y 2 ) + z, find f. Consider a cylinder of radius 5 whose axis is along the z-axis. (i) What is the rate of change of f(x, y, z) in the direction normal to the cylinder at the point (3, 4, 4)? (ii) What is the rate of change of f(x, y, z) in the direction m = i + 2j at the same point, where i, j are unit vectors parallel to the x, y axes respectively. 2. Find the normal to the surface xz + z 2 xy 2 = 5 at the point (1,1,2). Deduce the equation of the tangent plane at this point. 3. Obtain the equation of the plane that is tangent to the surface z = 3x 2 y sin(πx/2) at the point x = y = 1. Take North to be the direction (0, 1, 0) and East to be the direction (1, 0, 0). In which direction will a marble roll if placed on the surface at x = 1, y = 1 2? 4. A vector field F(x) has components (x 3 + 3y + z 2, y 3, x 2 + y 2 + 3z 2 ). Evaluate (i) the divergence of F(x) and (ii) the curl of F(x). 5. Let a and b be fixed vectors. Show that (a.x) = a. The following are vector functions of x: x; a(x.b); a x; x/r 3, where r = x. Evaluate the divergence and the curl of each of these functions. Line integrals 6. Evaluate Γ {P (x, y)dx + Q(x, y)dy} where P = x 2 y, Q = y 2 x and Γ is the closed curve consisting of the semi-circle x 2 +y 2 = a 2 (y > 0), and the segment ( a, a) of the x-axis, described anti-clockwise. Verify that this is equal to { Q x P } dxdy y where D is the plane surface enclosed by Γ. plane.) D (This illustrates Stokes s theorem in the
7. Determine a function f(y) such that [f(y) dx + x cos y dy] = 0 for all closed contours Γ in the (x, y) plane. [Hint: when is the integrand an exact differential?] Γ 8. The work done by a force F acting on a particle which moves along a curve C is defined as the line integral W = F dx. C (i) When F = c v, where c is a constant vector, and v = dx is the velocity of the dt particle, show that the work done is zero. (ii) A particle moves along the helical path given by x = cos t, y = sin t, z = t. Calculate the work W done in the time interval 0 t π by each of the forces F given by (a) yi xj k, (b) xi + yj where i, j, k are unit vectors parallel to the x, y, z axes respectively. 9. Write down a condition obeyed by a conservative vector field V = {P (x, y), Q(x, y)}. Do the following choices for P, Q yield conservative fields? (i) P = x 2 y + y, Q = xy 2 + x ; (ii) P = ye xy + 2x + y, Q = xe xy + x. In the case that V is conservative, find a function f(x, y) such that V = f. Consider the following curves, each joining (0, 0) to (1, 1): (a) C 1 : x = t, y = t (0 t 1) (b) C 2 : x = 0, y = t (0 t 1); x = t, y = 1 (0 t 1). Evaluate the integrals P dx + Qdy and P dx + Qdy for each of the fields (i) and (ii) C 1 C 2 above. 10. Consider the vector field V = (4x 3 z + 2x, z 2 2y, x 4 + 2yz). Evaluate the line integral c V ds along (i) the sequence of straight-line paths joining (0, 0, 0) to (0, 0, 1) to (0, 1, 1) to (1, 1, 1). (ii) the straight line joining (0, 0, 0) to (1, 1, 1), given parametrically by x = y = z = t (0 t 1). Show that V is conservative by finding a function f(x, y, z) such that V = f.
Surface integrals 11. Let E = ( ye 2t, xe 2t, 0) and B = (0, 0, e 2t ). Evaluate S B ds, C E dx, where S is the surface of the circular disc x 2 + y 2 < 1, z = 0 and C is the curve bounding S. Show that dx = CE d B ds. dt S 12. A cube is defined by 0 x 1, 0 y 1, 0 z 1. Evaluate the surface integral F nds over the surface of the cube where F = (x 2 +ay 2, 3xy, 6z) and a is a constant. [n is a unit vector in the direction of the outward normal from the volume across a surface element ds; e.g. on the x = 0 face, ds = dydz and n = ( 1, 0, 0).] Evaluate also fdxdydz over the volume of the same cube, where f = bx + 6 and b is a constant. For what values of a and b do these integrals have the same value? (How is this result related to Gauss s law in electricity?) 13. Let u be the vector field u = Qx/4πɛ 0 r 3 in three dimensions, where x is the position vector and r = x (Q, ɛ 0 constants). Show that S u nds = Q/ɛ 0, where S is a sphere of radius a centred on the origin, and n is a unit normal vector pointing radially outward from the sphere. [This is Gauss s law for a point charge.] 14. (i) Find the values of a and b in Question 12 by using the divergence theorem and imposing the requirement that curl F = 0. (ii) For E and B as in Question 11 show show that curl E = B t and apply Stokes s theorem to deduce the equality of the integrals.
15. State the divergence theorem. Let F(r) = (x 3 + 3y + z 2, y 3, x 2 + y 2 + 3z 2 ) and let S be the (open) surface 1 z = x 2 + y 2, 0 z 1. Evaluate F nds, where n is the unit normal on S having a positive component in the S z-direction. [Hint: construct a closed surface including S and use the divergence theorem.] 16. State Stokes s theorem, and verify it for the hemispherical surface r = 1, z 0, and the vector field A(x) = (y, x, z). Fourier series 17. Show that the functions 1, x, 1 2 (3x2 1), 1 2 (5x3 3x) are orthogonal on the interval [ 1, 1]. 18. Show that a sin nx sin mx dx = 0 0 if m and n are integers with m n and a = kπ, where k is an integer (i.e. a is an integer multiple of half wavelengths of the fundamental mode, sin x). 19. Prove that for n m (where n and m are integers) T T Find the value of the integral when n = m. sin(mπθ/t ) sin(nπθ/t )dθ = 0. 20. Write down the Fourier series on ( π, π) (with period 2π) for (i) sin 2θ and (ii) cos 2 θ. Obtain the Fourier series for sin 3 θ. 21. Find the Fourier series for the function which equals x when l x l and is periodic with period 2l. If this series is integrated, what will the resulting series sum to? If the series is differentiated, what will the resulting series sum to (if anything)? Give sketches. * Is the rate of convergence of each series roughly what you might expect from considering the smoothness properties of the function? 22. Find the Fourier series with period 2π which converges to e x for π < x < π. To what does it converge when x = π and x = π? * Show that any function f(x) can be written (in terms of f(x) and f( x)) as the sum of an odd function and an even function. Deduce from the first part of the question the Fourier series for cosh x and sinh x with the same range and periodicity.
23. Let f(x) = b n sin nx and let g(x) = B m sin mx. Show that m=1 π π f(x)g(x)dx = π b n B n. Without detailed calculation, give the corresponding result when f(x) = a 0 2 + (a n cos nx + b n sin nx) = g(x). 24. Let f(x) = a 0 2 + (a n cos nx + b n sin nx). What can be said about the coefficients a n and b n if: (i) f(x) = f( x) (ii) f(x) = f( x) (iii) f(x) = f(π x) (iv) f(x) = f(π x) (v) f(x) = f(π/2 + x) (vi) f(x) = f(π/2 x) (vii) f(x) = f(2x) (viii) f(x) = f( x) = f(π/2 x). 25. Assume that x 2 = n= c n e inx for π x π, for some complex coefficients c n. By multiplying both sides by e imx and integrating, find the coefficients c n. Write down the (real) Fourier series for x 2 on ( π, π) with period 2π. 26. Let f(x) = x for 0 x < π. Sketch: (i) the odd function; and (ii) the even function that are periodic of period 2π and equal to f(x) for 0 x < π. Show that the function f(x) can be represented for 0 x < π either as or as f(x) = f(x) = 2 π 2( 1) r+1 sin rx r r=1 [ π 2 4 k=0 ] 2 cos(2k + 1)x (2k + 1) 2.