No: CITY UNIVERSITY LONDON BEng (Hons) in Electrical and Electronic Engineering PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2003 Date: August 2004 Time: 3 hours Attempt Five out of EIGHT questions
Question 1 (a) A scalar vector field is given in rectangular coordinates as f(x, y) = 1 x2 + y 2 ( Calculate the gradient of this field at the point (x, y) = 1 1 2, 2 ). Sketch this along with the level contours of the field f(x, y) = 2, f(x, y) = 1 and f(x, y) = 1. (In your sketch indicate 2 clearly the direction of f(x, y) relative to the three level contours). [7 marks] (b) Consider the function f(x, y) = ax 2 + 2bxy + y 2 in which a and b are real parameters. Show that if a b 2 then f(x, y) has a unique stationary point. Classify this point as a local minimum, local maximum or a saddle point. (Your classification should depend on a specific relation between a and b which you must clearly indicate). What is the nature of the stationary points of f(x, y) if a = b 2? [7 marks] (c) Obtain the Taylor series expansion of the function f(x, y) = e x sin y around the point (x, y) = (0, 0) up to - and including - quadratic terms. Use your result to estimate the value of f( 0.1, 0.1) from the value of f(x, y) and its partial derivatives at (x, y) = (0, 0), and compare your estimate with the exact value. Question 2 The equation of an ellipse is given in Cartesian coordinates as x2 + y2 = 1, where a and b are a 2 b 2 the lengths of the ellipse s semi-axes. Show that the equation of the ellipse can be written in polar form as: r = γ 1 + δ cos(2θ) where γ = 2ab a2 + b 2 and δ = b2 a 2 b 2 + a 2 Using an appropriate double integral show that the area S enclosed by the ellipse is given by: S = 2γ 2 π/2 0 dθ 1 + δ cos(2θ) 1 δ Evaluate this integral by making the substitution z = tan θ and hence (or otherwise) 1+δ show that S = πab. In your derivation you may find the following three identities useful: cos(2θ) = 2 cos 2 θ 1 = cos 2 θ sin 2 d θ; dθ (tan θ) = dz sec2 (θ); 1 + z = arctan(z) 2 [10 marks] 2 of 7
Question 3 (a) Give a coordinate free definition of the Curl of a vector field F. Using your definition and a limiting argument derive an expression of one component of F in Cartesian co-ordinates (e.g. ( F) x ). Write down the complete expression of F in Cartesian coordinates in terms of a 3 3 determinant. (b) Explain what is a conservative vector field. Show that the Curl of a conservative field is zero. (c) By constructing an appropriate potential function, show that the vector field F(x, y, z) = (2xy + yz 2 )i + (x 2 + xz 2 )j + 2xyzk is conservative. Check your result by: (i) Calculating F; (ii) Calculating directly the line integral I = F dl along the straight line segment C on the (x, y)-plane connecting points C (1, 0) and (0, 1), and (iii) Evaluating the line integral I indirectly using the potential function you have constructed. Question 4 (a) Let f(z) be a complex-valued function of the the complex variable z = x+jy. Explain what is meant by the statement: f(z) is analytic in a region D of the complex plane. If f(z) is written in terms of its real and imaginary part as f(z) = u(x, y) + jv(x, y), state two equations involving u(x, y) and v(x, y) which can be used to test the analyticity of f(z) (Cauchy-Riemann equations). (b) By using the Cauchy-Riemann equations (or otherwise) determine whether the functions f(z) = e z and g(z) = z are analytic in the complex plane. Here z denotes the complex conjugate of z, i.e. z = x jy if z = x + jy. (c) It may be shown that the covariance sequence {R(0), R(1), R(2),...} of a random process can be obtained from the spectral density function Φ(z) of the process via the contour integral: R(k) = 1 Φ(z)z k 1 dz k = 0, 1, 2,... 2πj where the contour integral is calculated along the unit circle z = 1 in the complex plane (in the anti-clockwise direction). The spectral density function of a first-order auto-regressive process is given as: Φ(z) = 1 (1 az 1 )(1 az) where 1 < a < 1. By applying Cauchy s residue theorem find the corresponding covariance sequence {R(k)} (k = 0, 1, 2,...) in closed-form. [10 marks] 3 of 7
Question 5 (a) Using Laplace transforms (or otherwise) solve the following differential equation: subject to the initial conditions: d 2 y(t) dt 2 + 2 dy(t) dt + y(t) = e t y(0) = 1 and dy(0) dt = 1 Check your solution by substituting into the differential equation and by verifying that the initial conditions are satisfied. A table of Laplace transforms is provided at the end of the paper. [10 marks] (b) Show from first principles (i.e. without reference to the table of Laplace transforms) that: L(cos ωt) = s s 2 + ω 2 Indicate clearly the region of convergence of the transform, i.e. the range of values of s for which your result is valid. Hint: Write cos ωt = 1 2 (ejωt + e jωt ) and apply the definition of the transform. (c) Consider the function f(t) = t sin ωt (t 0), f(t) = 0 (t < 0). Show by direct differentiation that f (t) = 2ω cos ωt ω 2 f(t) By using the properties of Laplace transforms of derivatives, show that L(t sin ωt) = 2ωs (s 2 + ω 2 ) 2 In your derivation you will need to use the Laplace transform of the function cos ωt you obtained in part (b). Question 6 (a) On the linear space R[0, 2π] (real valued-functions defined on the interval [0, 2π]) define the inner product of two functions f and g as: f, g = 2π 0 f(x)g(x)dx A set S of functions in R[0, 2π] is said to be orthonormal if: (i) f, g = 0 for any two functions f and g in S such that f g, and (ii) f, f = 1 for every function in f in S. Prove that the set of functions: S = { 1 2π, cos x π, } sin x π is orthonormal. 4 of 7
(b) Consider the periodic function f(t) with period T = 2π, defined as f(t) = t 2 in the interval π < t π. The Fourier series expansion of f(t) is of the form: f(t) = a 0 2 + a n cos nt + where the a n s and b n s are unspecified coefficients. b n sin nt Show that f(t) is an even function and, as a result, b n = 0 for all n > 0 in the Fourier series expansion of f(t). Calculate the a n s (n 0) in closed form and hence show that: f(t) = π2 3 + 4( 1) n cos(nt) n 2 [2 marks] Hint: You need to integrate by parts (twice). Show that: ( 1) n+1 Hint: Set t = 0 in your Fourier series expansion. n 2 = 1 1 2 1 2 2 + 1 3 2 +... = π2 12 [2 marks] Question 7 (a) Define the following terms of linear-algebra: A subspace of a vector space V. Direct sum of two subspaces. Linear independence of a list of vectors. Linear span of a list of vectors. Basis of a vector space V. Range and Kernel of a linear transformation. Give simple examples to illustrate your definitions. (b) Show that: (i) The intersection of two subspaces of a vector space V is a subspace of V. (ii) A list of vectors containing two identical vectors is linearly dependent. (c) Let S 2 2 denote the set of 2 2 symmetric matrices with real entries. (A matrix A is called symmetric if A = A T where A T is the transpose of A). For a fixed 2 2 symmetric matrix A define the transformation Π A : S 2 2 S 2 2 which maps 2 2 symmetric matrices 5 of 7
X to 2 2 symmetric matrices Y according to the rule Y = AXA T. Show that Π A is a linear transformation. If ( A = ) 1 1 1 1 find the Range and Kernel of Π A, and hence verify the rank-nullity theorem. Question 8 (a) We wish to perform the following three elementary operations on the rows of a 3 3 matrix A: Multiply the first row by 2. Interchange the first and third rows. Add twice the second row to the third row. Write down the three elementary matrices by which A must be pre-multiplied to perform each operation. If the three operations must be performed in sequence, find the overall transformation matrix and its inverse. (b) The matrix inversion lemma states that for four matrices A, B, C and D of compatible dimensions the following identity holds, provided the indicated inverses exist: (A BD 1 C) 1 = A 1 + A 1 B(D CA 1 B) 1 CA 1 By multiplying the matrix in the right-hand-side of the above equation by A BD 1 C (either from the left or the right) show that this identity is valid. What are the computational advantages of using this identity when A = I n, B is a column vector and C is a row vector (and hence D is a scalar)? (c) Give sufficient and necessary conditions for the linear system of equations Ax = b, where A R m n and x is the vector of unknowns, to have: (i) At least one solution, and (ii) Exactly one solution. (d) Find all solutions of the system of equations: 1 2 1 x 1 1 3 2 y = 4 0 1 1 z 3 Explain clearly why your result is consistent with the conditions you gave in part (c) above. 6 of 7
Table of Laplace Transforms f(t) F (s) f(t) F (s) s δ(t) 1 cos ωt s 2 +ω 2 ω 1 1/s sin ωt s 2 +ω 2 t 1/s 2 s cosh at s 2 a 2 t 2 2/s 3 a sinh at s 2 a 2 t n n! e at s a cos ωt s n+1 (s a) 2 +ω 2 e at 1 e at ω sin ωt s a (s a) 2 +ω 2 te at 1 t n 1 e at (n 1)! (s+a) 2 (s+a) n External Examiners: Prof. P.M. Taylor, Prof. M. Cripps Internal Examiner: Dr G. Halikias 7 of 7