Ø ÔÖÓÚ ÓÖÑÙÐ Ø SCHOOL OF MATHEMATICS AND STATISTICS Mathematics II (Electrical) Autumn Semester 2017 18 2 hours ØØ ÑÔØ ÐÐ Ø ÕÙ Ø ÓÒ º Ì ÐÐÓ Ø ÓÒ Ó Ñ Ö ÓÛÒ Ò Ö Ø º Please leave this exam paper on your desk Do not remove it from the hall Ê ØÖ Ø ÓÒ ÒÙÑ Ö ÖÓÑ Í¹ Ö Ø µ ØÓ ÓÑÔÐ Ø Ý ØÙ ÒØ 1 Turn Over
Blank 2 Continued
½ µ µ Ä Ø f(t) Ø ÙÒØ ÓÒ Ò ÓÖ t 0 Ý f(t) := { t 0 t 1, 1 t > 1. Ë ÓÛ Ø Ø Ø Ä ÔÐ ØÖ Ò ÓÖÑ Ó f(t) Ú Ò Ý L{f(t)} = 1 s 2 e s s 2. µ ÓÒ Ö Ø Ö ÒØ Ð ÕÙ Ø ÓÒ y (t)+4y(t) = f(t) Û Ø Ò Ø Ð ÓÒ Ø ÓÒ y(0) = y (0) = 0 Û Ö f(t) Ø ÙÒØ ÓÒ Ò Ò È ÖØ µº Ë ÓÛ Ø Ø Ø Ä ÔÐ ØÖ Ò ÓÖÑ Ó y(t) Ú Ò Ý Y(s) = 1 s 2 (s 2 +4) e s s 2 (s 2 +4). µ Ø ÖÑ Ò Ø ÙÒØ ÓÒ y(t) ÖÓÑ È ÖØ µ Ø Ø Ø Ø Ö Ò¹ Ø Ð ÕÙ Ø ÓÒ y (t)+4y(t) = f(t) Û Ø Ò Ø Ð ÓÒ Ø ÓÒ y(0) = 0 y (0) = 0º µ ÏÖ Ø ÓÛÒ Ø ÓÖÑÙÐ ÓÖ Ø ÓÒÚÓÐÙØ ÓÒ Ó ØÛÓ ÙÒØ ÓÒ Ò Ò Ò ÜÔÐ Ø ÓÖÑÙÐ ÓÖ g k(t) Û Ò g(t) = H(t)t n Ò k(t) = H(t 1)º ¾ ÓÒ Ö Ø Ô Ö Ó ÙÒØ ÓÒ f(t) Û Ø ÙÒ Ñ ÒØ Ð Ô Ö Ó T = 2π Ò ÓÒ [ π,π) Ý f(t) = π t º µ Ò Ø ÓÙÖ Ö Ö S[f](t) Ó f(t)º ½¾ Ñ Ö µ µ Ë Ø Ø Ö Ô Ó Ø ÓÙÖ Ö Ö ÝÓÙ ÐÙÐ Ø Ò È ÖØ µ ÓÚ Ö Ø ÒØ ÖÚ Ð [ 2π,3π]º Ñ Ö µ µ ÏÖ Ø ÓÛÒ Ø ÜÔÓÒ ÒØ Ð ÓÖÑ Ó Ø ÓÙÖ Ö Ö Ó f(t)º Ñ Ö µ Úµ Ï Ø Ø Ü Ø Ú ÐÙ Ó S[f] ( ) 101π 2 ¾ Ñ Ö µ 3 Turn Over
µ Ä Ø f(x,y) = xsin(xy)º ÐÙÐ Ø Ø Ô ÖØ Ð Ö Ú Ø Ú f x, f y, f xx, f yx, f xy. µ Ò Ò Ð Ý ÐÐ Ø Ö Ø Ð ÔÓ ÒØ Ó Ø ÙÒØ ÓÒ g(x,y) = x4 4 + 2x3 3 +4xy y2. ½¾ Ñ Ö µ µ Ä Ø p(x,y) ÔÓÐÝÒÓÑ Ð Û Ø p(0,y) = y p(x,0) = x Ò p yx = 8x 3 yº Ò p(x,y)º Ñ Ö µ µ Ä Ø R = {(x,y) 1 x 2, 0 y 2}º ÐÙÐ Ø x 3 e y da. R µ Ä Ø P Ø Ö ÓÒ Ó R 3 ÓÚ Ø ÕÙ Ö Ò Ø xy¹ôð Ò Û Ø Ú ÖØ (0,0) (1,0) (0,1) (1,1) Ò ÐÓÛ Ø ÔÐ Ò Ò Ý x+z = 1º Ò x 3 y 2 z dv. P µ ÐÙÐ Ø y=1 x=1 y=0 x= y e x3 dxdy. 4 Continued
µ Ä Ø F : R 3 R 3 Ø Ú ØÓÖ Ð Ò Ý F = ( e xyz, ) 1 xyz, x+y2 +z 3. µ ÐÙÐ Ø Ú º µ ÐÙÐ Ø ÙÖÐ º µ Ä Ø f(x,y) = 6 x 2 y 2 º µ µ ÐÙÐ Ø Ø Ö Ø ÓÒ Ð Ö Ú Ø Ú Ó f(x,y) Ø (x,y) = (1,1) Ò Ø v = (3, 4) Ö Ø ÓÒº ÁÒ Û Ö Ø ÓÒ f(x,y) ÑÓ Ø Ö Ô ÐÝ ÒÖ Ò Ø Ø ÔÓ ÒØ (x,y) = ( 1, 1) Ò Û Ø Ø Ñ Ü ÑÙÑ Ö Ø Ó ÒÖ µ Ä Ø Ö : R 3 R 3 Ø Ú ØÓÖ Ð Ú Ò Ý Ö = (x,y,z) Ð Ø r : R 3 R Ø Ð Ö Ð Ú Ò Ý r(x,y,z) = x 2 +y 2 +z 2 Ò Ð Ø g : R R Ò Ö ØÖ ÖÝ ÙÒØ ÓÒº Ë ÓÛ Ø Ø g(r) = r 1dg dr Ö. End of Question Paper 5
Laplace transform: FORMULA SHEET The Laplace transform of a function f(t) is given by: L{f(t)}(s) := 0 e st f(t)dt. Properties of the Laplace transform: L{f(t)} = F(s) in the following table. L{af(t) + bg(t)} = al{f(t)} + bl{g(t)} L{f (t)} = sf(s) f(0) L{f (t)} = s 2 F(s) sf(0) f (0) L{e kt f(t)} = F(k +s) linearity differentiation w.r.t. t second differentiation w.r.t. t frequency shift L{f(t a)h(t a)} = e as F(s) (for a > 0) time shift L{f(at)} = 1 a F(s ) (for a > 0) scaling a L{f g(t)} = L{f(t)}L{g(t)} (for f(t), g(t) causal) Table of standard Laplace transforms: convolution f(t) L{f(t)}(s) Region of validity Fourier transform: t n (for n 0) n! s n+1 Re(s) > 0 sin(kt) cos(kt) k Re(s) > 0 s 2 +k 2 s Re(s) > 0 s 2 +k 2 H(t T) (for T 0) e st s Re(s) > 0 δ(t T) (for T 0) e st s C The Fourier transform and inverse Fourier transforms are given by: F{f(t)}(ω) = F(ω) := f(t)e jωt dt, f(t) = F 1 {F(ω)} = 1 2π F(ω)e jωt dω.
Properties of the Fourier transform: F{f(t)} = F(ω) in the following table: F{e jθt f(t)} = F(ω θ) F{f(t T)} = e jωt F(ω) F{f (n) (t)} = (jω) n F(ω) F{F(t)} = 2πf( ω) F{f(at)} = 1 a F(ω a ) F{f g(t)} = F{f(t)}F{g(t)} frequency shift time shift differentiation symmetry scaling convolution Table of standard Fourier transforms: f(t) F{f(t)}(ω) e a t (for a > 0) 2a a 2 +ω 2 rect T (t) sinc( Tω 2 ) 1 2πδ(ω) Fourier series: The Fourier series of a periodic function f(t) with fundamental period T is given by where S[f] = a 0 2 + ( ) a n cos(ω n t)+b n sin(ω n t) n=1 ω n = 2πn T, a n = 2 T T/2 T/2f(t)cos(ω n t)dt, b n = 2 T T/2 T/2 f(t)sin(ω n t)dt. Coordinate systems: Cylindrical polar coordinates (x,y,z) = (rcos(θ),rsin(θ),z) ( ) (r,θ,z) = x2 +y 2,arctan( y ),z x dv = rdrdθdz. Spherical polar coordinates (x,y,z) = (ρsin(φ)cos(θ),ρsin(φ)sin(θ),ρcos(φ)) ( ) (ρ,θ,φ) = x2 +y 2 +z 2,arctan( y x ),arccos(z) ρ dv = ρ 2 sin(φ)dρdφdθ.