ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday,...
Transcript of ME 565, , Due on Friday, January 18, 2019 Homework #1 · 2019-01-25 · ME 565, , Due on Friday,...
ME 565, , Due on Friday, January 18, 2019 Homework #1
Exercise 1-1 Express the complex numbers in the form a+ bi and Reiθ:(a) 1
4−3i
1
4− 3i=
1
4− 3i
4 + 3i
4 + 3i=
4 + 3i
16− 12i+ 12 + 9=
4 + 3i
25=
4
25+
3
25i
a =4
25, b =
3
25
R =√a2 + b2 = 0.2
θ = arctan(325425
) = arctan(3
4) ≈ 0.64
1
4− 3i=
4
25+
3
25i = 0.2ei0.64
(b)(√
32− 1
2i)4
(√3
2− 1
2i
)4
= z4 =(Reiθ
)4
R =
√(
√3
2)2 + (
1
2)2 = 1
θ = arctan(
√3
2
−12
) = −π3
z4 = e−i4π3 = cos(−4π
3) + i sin(
4π
3)(√
3
2− 1
2i
)4
= −0.5− i√
3
2= e−i
4π3
(c)in
i2, i3, i4, i5, ... = zn
z = i = Reiθ
R = 1
θ =π
2(eiπ2
)n= ei
πn2
in = cos(πn
2) + i sin(
πn
2) = ei
πn2
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ME 565, , Due on Friday, January 18, 2019 Homework #1
Exercise 1-2 Find all solutions of:(a)ez = i
ez = ea+ib
= eaeib
i = Reiθ
= eiπ2
eaeib = eiπ2
eib = eiπ2
z = i(π
2+ 2πn
), with n ∈ Z
(b)ez = −1
−1 = Reiθ
= eiπ
ez = eiπ
eaeib = eiπ
z = i(π + 2πn
), with n ∈ Z
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ME 565, , Due on Friday, January 18, 2019 Homework #1
Exercise 1-3 Find all solutions of:(a)z4 = 1
z4 =(Reiθ
)4
= R4ei4θ
1 = Reiθ
= ei(2πn)
R4ei4θ = ei(2πn), with n ∈ ZR = 14θ = 2πn
θ =πn
2z = ei
πn2 , with n ∈ Z
(b)z2 = 4i
z2 =(Reiθ
)2
= R2ei2θ
4i = Reiθ
= 4ei(π2
+2πn), with n ∈ ZR2ei2θ = 4ei(
π2
+2πn)
R2 = 4R = 2
2θ =π
2+ 2πn
θ =π
4+ πn
z = 2ei(π4
+πn), with n ∈ Z(c)z2 = 1− i
z2 =(Reiθ
)2
= R2ei2θ
1− i = Reiθ
=√
2ei(−π4
+2πn), with n ∈ ZR2ei2θ =
√2ei(−
π4
+2πn)
R2 =√
2
R = 214
2θ = −π4
+ 2πn
θ = −π8
+ πn
z = 214 e(−
π8
+πn), with n ∈ Z
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DATE : 17-Jan-2019 12:39
Table of ContentsRevision : 1.00 .................................................................................................................... 1MATLAB Ver: 9.2.0.556344 (R2017a) ................................................................................... 1FILENAME : hw1_pr6.m ..................................................................................................... 1HW 1, Problem 6, ME 565 Winter 2019 ................................................................................. 1
Revision : 1.00
MATLAB Ver: 9.2.0.556344 (R2017a)
FILENAME : hw1_pr6.m
HW 1, Problem 6, ME 565 Winter 2019clear all; close all;
x = 0:0.05:2;y = 0:0.05:1;[X,Y] = meshgrid(x,y);
V1 = pi*sin(pi*X).*cos(pi*Y);V2 = -pi*cos(pi*X).*sin(pi*Y);
init_posx = linspace(0,2,10);init_pos = [init_posx',0.5*ones(10,1)];part_color = jet(10);tspan = [0,10];
func = @(t,z) [pi*sin(pi*z(1)).*cos(pi*z(2)); -pi*cos(pi*z(1)).*sin(pi*z(2))];
figure(1)hold onquiver(X,Y,V1,V2)for jj=1:10 [t,z] = ode45(func,tspan,init_pos(jj,:)); plot(z(:,1),z(:,2),'color',part_color(jj,:),'linewidth',1.5); scatter(z(1,1),z(1,2),'k');endylim([0 1])xlim([0 2])xlabel('x axis')ylabel('y axis')
1
DATE : 17-Jan-2019 12:39
Published with MATLAB® R2017a
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