20
Conformal Mappings and Applications
Exercises 20.1
1. For w =
1 -y 1 1 x 1 1 and v = 2 . If y = x, u = ,u= 2 ,v=- , and so v = -u. The image is the 2 2 z x +y x +y 2 x 2 x
line v = -u (with the origin (0, 0) excluded.) 2. If y = 1, u = x 1 -1

1
Introduction to Differential Equations
Exercises 1.1
1. Second-order; linear. 2. Third-order; nonlinear because of (dy/dx)4 . 3. The differential equation is first-order. Writing it in the form x(dy/dx) + y 2 = 1, we see that it is nonlinear in y

Concordia University
Course Number: ENGR 233 Date: Wednesday, April 2, 2008 Total Marks: 100 MOCK FINAL EXAMINATION Special Instructions: Use of calculators and outside materials is NOT permitted. __ Question 1. Time: 10:15-11:30 hrs. Pages: 2
Find

Name: _ Date: _ Use the following to answer question 1:
1. In figure 3.8, what would be the result if a price ceiling is set which is $2 different from the equilibrium price and demand increased by 30? A) The price would be above equilibrium and a s

Stokes Theorem: Background
Similar to Green's Theorem: a line integral equals a surface integral.
[Pdx + Qdy ] = R [Qx Py ]dxdy C
The line integral is still the work around a curve:
R C
r r F dr ,
C
r F = Pi + Q j
The surface integral in Gre

r r cos = a b /(| a | b |)
compb a =| a | cos = a b
r projb a = a b b
( )
Area of a parallelogram = a b
Volume of a parallelepiped =a (bc)
Equation of a line : r r r r r r r = r2 + t (r2 r1 ) = r2 + ta
Equation of a plane : a x + b

Double Integrals and Greens Theorem
This unit is based on Sections 9.10 through 9.12 , Chapter 9. All assigned readings and exercises are from the textbook Objectives: Make certain that you can define, and use in context, the terms, concepts and form

Triple Integrals
This unit is based on Sections 9.15 Chapter 9. All assigned readings and exercises are from the textbook Objectives: Make certain that you can define, and use in context, the terms, concepts and formulas listed below: Evaluate tripl

Change of Variables in Double integrals
Double integrals in terms of two variables x and y may be written in the form
R
f ( x, y )dA = f ( x, y )dxdy
R
It is sometimes convenient or necessary to change the variables x, y to u, v:
x = g (u , v)

THE DIVERGENCE THEOREM
It is another generalization of Greens theorem to 3D. Let D be a simple solid region (For example, regions bounded by ellipsoids or rectangular boxes are simple solid regions). Let S be the boundary surface of D, given with pos

Concordia University
EMAT 233 - Final Exam
Instructors: Dafni, Dryanov, Enolskii, Keviczky, Kisilevsky, Korotkin, Shnirelman Course Examiner: M. Bertola Date: May 2006. Time allowed: 3 hours. [10] Problem 1. Compute the curvature (t) of the curve C d

Concordia University
Faculty of Engineering and Computer Science Department of Mechanical Engineering Final Examination
Course: Number: Date: Time and Place: Number of Page: Instructors: Material allowed: Calculators Allowed: Special instructions:

cos = a b /(| a | b |)
compb a =| a | cos = a b
projb a = a b b
Equation of a line : r = r2 + t (r2 r1 ) = r2 + ta
Area of a parallelogram = a b
Equation of a plane : a x + b y + cz + d = 0 also : [ (r2 r1 ) (r3 r1

Concordia University
Department of Electrical and Computer Engineering ENGR 233: Applied Advanced Calculus
Fall 2007 Mid-Term Exam Solution Time: 75 Minutes
Question 1: Express the vector x in terms of the vectors a and b shown in the following figu

Surface Integrals and Stokes Theorem
This unit is based on Sections 9.13 and 9.14 , Chapter 9. All assigned readings and exercises are from the textbook Objectives: Make certain that you can define, and use in context, the terms, concepts and formula

19
Series and Residues
Exercises 19.1
1. 5i, 5, 5i, 5, 5i 3. 0, 2, 0, 2, 0 5. Converges. To see this write the general term as 6. Converges. To see this write the general term as 7. Converges. To see this write the general term as 8. Diverges. To s

3
Higher-Order Differential Equations
Exercises 3.1
1. From y = c1 ex + c2 e-x we find y = c1 ex - c2 e-x . Then y(0) = c1 + c2 = 0, y (0) = c1 - c2 = 1 so that c1 = 1/2 and c2 = -1/2. The solution is y = 1 ex - 1 e-x . 2 2 2. From y = c1 e4x + c2

5
Series Solutions of Linear Equations
Exercises 5.1
1. lim
n
an+1 2n+1 xn+1 /(n + 1) 2n = lim = lim |x| = 2|x| n n n + 1 an 2n xn /n (-1)n converges by n n=1
The series is absolutely convergent for 2|x| < 1 or |x| < 1/2. At x = -1/2, the series

6
Numerical Solutions of Ordinary Differential Equations
Exercises 6.1
All tables in this chapter were constructed in a spreadsheet program which does not support subscripts. Consequently, xn and yn will be indicated as x(n) and y(n), respectively.

8
Matrices
Exercises 8.1
1. 2 4 6. 8 1
2. 3 2 7. Not equal
3. 3 3 8. Not equal
4. 1 3 9. Not equal
5. 3 4 10. Not equal
11. Solving x = y - 2, y = 3x - 2 we obtain x = 2, y = 4. 12. Solving x2 = 9, y = 4x we obtain x = 3, y = 12 and x =

9
1.
Vector Calculus
Exercises 9.1
2. 3.
4.
5.
6.
7.
8.
9.
Note: the scale is distorted in this graph. For t = 0, the graph starts at (1, 0, 1). The upper loop shown intersects the xz-plane at about (286751, 0, 286751). 10.
375
Exercises 9.

10
Systems of Linear Differential Equations
Exercises 10.1
1. Let X =
x . Then y X = x . Then y X = 4 5 -7 0 X. 3 4 -5 8 X.
2. Let X =
x 3. Let X = y . Then z
-3 X = 6 10 x 4. Let X = y . Then z
4 -1 4
-9 0 X. 3
1 X = 1 -1

11
Systems of Nonlinear Differential Equations
Exercises 11.1
1. The corresponding plane autonomous system is x = y, y = -9 sin x.
If (x, y) is a critical point, y = 0 and -9 sin x = 0. Therefore x = n and so the critical points are (n, 0) for n =

13
Boundary-Value Problems in Rectangular Coordinates
Exercises 13.1
1. If u = XY then ux = X Y, uy = XY , X Y = XY , and X Y = = 2 . X Y Then X 2 X = 0 so that X = A1 e x ,
2
and Y 2 Y = 0
Y = A2 e y ,
2
and u = XY = c1 ec2 (x+y) . 2. If u =