Resources (Updated 2/3)
Stewart Textbook Website (Calculus III)
Stewart Textbook Website (Calculus I & II)
Paul's Online Math Notes - Calculus I
Paul's Online Math Notes - Calculus II
Paul's Online Math Notes - Calculus III
Paul's Online Math Notes - Differential Equations
Announcements (Updated 6/12 9:50 AM)
The Final Exam will consist of 4 Take-Home Problems (Part 1) and 10 In-Class Problems (Part 2). Part 1 of the Final Exam can be downloaded here and is due Friday 6/13 at 12:40 PM.
The Final Exam will be Friday 6/13 from 12:40 PM to 2:25 PM (Periods 5 & 6) in Room 620.
Blank Copies of Exams & Quizzes
There will be an optional Review Session on Tuesday 6/10 at 9:00 AM in Room 735.
Solutions (corrected 6/4) and Statistics for Exam 3 are posted below.
There will not be a Quiz on Friday 6/6.
Tentative Exam Dates
Exam 3: Thursday 5/29 (Note Change!)
Final Exam: Friday 6/13
Solutions and Statistics for Exam 2 are posted below.
Exam 2 will consist of 7 Problems x 15 Points each = 105 Points (5 Points Extra Credit). The first problem is a take-home problem and can be found here, and is due at the beginning of class tomorrow. There will be 6 problems on the in-class portion of the exam.
Exam 2 will be on be on Friday April 25, and will cover HW 11-17 (and Quizzes 5-8).
The Quiz for Friday 4/11 is a Take Home Quiz. It is due at the beginning of class tomorrow Friday 4/11.
The Quiz on Friday will consist of one Linear Homogeneous ODE and one Complex Variables Problem (HW 12 and HW 13). The Method of Undetermined Coefficients will not be on this Quiz. You do need to know how to go back and forth between exponential functions with complex argument and the circular and hyperbolic sine and cosine functions.
Solutions and Statistics for Exam 1 are posted below.
Exam 1 will be on be on Friday March 14, and will cover all the course material through Wednesday March 12. Thursday March 13 will be a Review Day.
There will be a Quiz on Friday 2/14 (or if schools are closed that day, the first day that schools are open). The Quiz will consist of a single ODE, which you will be asked to solve using two different methods which I will specify (Separable Method, Homogeneous Method, or Linear Method).
The Quiz on Friday 2/7 will consist of one problem from HW 2A and one problem similar to a problem worked in class.
Welcome to Advanced Applied Math.
Homework (Updated 6/6)
(Note: You are responsible for all Homework posted and/or updated before 5:00 PM the school day before the due date. Due dates are subject to change, so please check them carefully.)
"ODE" denotes the Ordinary Differential Equations textbook
"FCA" denotes the Fundamentals of Complex Analysis textbook
Notes:
(1) Some of these ODE's are in differential form (i.e., the ODE has already been "multiplied" through by dx). To convert back to the more familiar Leibnitz form, just "divide" the ODE through by dx. Also consider regarding x as the dependent variable and "dividing" through by dy, which may result in an ODE which is easier to solve.
(2) Whenever possible, solve for an explicit solution (isolate the dependent variable), even though the answers in the textbook are often left in implicit form. You will generally be required to find explicit solutions on Exams and Quizzes.
(3) In the Textbook (as well as in Wolfram Alpha, and many other places), "log" denotes the natural logarithm that we normally denote by "ln".
HW 33 (Complex Integration Part 2) - The Last Homework!
FCA Page 170 Problems 3, 6, 7, 13; Page 178 Problem 1; Page 201 Problem 10
Due Mon 6/9
HW 32 (Complex Integration Part 1)
FCA Page 159 Problems 1, 7, 8
Due Fri 6/6
HW 31 (Polynomial Differential Operators)
ODE Page 266 Problems 6, 13, 29
Due Wed 5/4
HW 30 (Complex Powers and Inverse Trig Functions Part 2)
FCA Page 136 Problems 10, 11, 12, 15
Due Mon 6/2
HW 29 (Complex Powers and Inverse Trig Functions Part 1)
FCA Page 136 Problems 1, 3, 4, 5, 7
Also, familiarize yourself with the log function on Wolfram Alpha for complex arguments, which agrees with the principal value function Log z we have been discussing. Try to reproduce the results of the second Quiz problem (with branch cut at pi/6) that we went over in class on Friday. You may need to use Wolfram Alpha on Exam 3 and/or the Final Exam.
Due Wed 5/28
HW 28 (Complex Logarithms Part 2)
FCA Page 123 Problems 9, 10, and these Problems
Due Thu 5/22
HW 27 (Complex Logarithms Part 1)
FCA Page 123 Problems 1, 3, 4, 5, 6
Due Wed 5/21
HW 26 (Exp, Trig, & Hyp Functions)
FCA Page 115 Problems 1, 5, 11, 12, 13, 15, 17, 20
Due Mon 5/19
The Quiz on Friday 5/16 will cover HW's 23, 24, & 25. Please bring any issues you have with these HW's to class on Thursday 5/15.
HW 25 (Rational Functions)
FCA Page 108 Problems 11, 13, 14
Due Thu 5/14
HW 24 (Polynomials)
FCA Page 108 Problems 1, 2, 3, 4, 5
Due Wed 5/14
HW 23 (Harmonic Functions)
FCA Page 84 Problems 1, 2, 3, 5, 6, 14
Due Mon 5/12
HW 22 (Cauchy-Riemann Equations)
FCA Page 77 Problems 1, 3, 5, 9, 10
Due Thu 5/8
HW 21 (Analyticity)
FCA Page 70 Problems 4, 7, 9, 11, 13
Due Wed 5/7
HW 20 (Limits and Continuity)
FCA Page 63 Problems 7, 11, 12, 13, 17, 21, 25
Due Fri 5/2
HW 19 (Functions of a Complex Variable)
FCA Page 56 Problems 1-5, 7b, 8b, 9b, 10d, 11d
Due Wed 4/30
HW 18 (Planar Sets)
FCA Page 42 Problems 2-8 (We did not cover the definitions of "bounded" and "region" in class - they are on Page 42); Problems 11, 12, 15, 16, 17, 21. Also, are there any subsets of the complex numbers which are both open and closed? Prove your answer is correct.
Due Mon 4/28
HW 17 (Series Solutions Near an Ordinary Point)
ODE Page 546 Problems 3, 5, 7, 9, 11
Due Wed 4/9
HW 16 (Method of Variation of Parameters)
ODE Page 240 Problems 2, 3, 4, 5, 12, 15, 18, 20
Due Mon 4/7
HW 15 (Complex Exponentials are Your Friends)
Do these Problems.
Due Wed 4/2
HW 14 (Method of Undetermined Coefficients)
ODE Page 231 Problems 3, 6, 7, 9, 11, 13, 15, 17, 19, 23, 30, 32
Due Mon 3/31
HW 13 (Linear Homogeneous ODE's with Constant Coefficients - No Restrictions on Roots of Characteristic Equation)
ODE Page 220 Problems 24, 25, 26, 27, 28, 29, 30, 33, 34, 35
Due Wed 3/26
HW Quiz 5 Review
Here are the Solutions to Quiz 5. Please review them. If anything is not crystal clear, ask on Monday.
If you are skeptical of the answer to Problem 1, cut and paste "Arg((3 e^(-2+ i 4 pi / 9))^(-2) / (1 - i^3))" into Wolfram Alpha.
Due Mon 3/24
HW 12 (Powers and Roots)
FCA Page 37 Problems 4, 5, 7, 9, 10, 11, 15, 16
Due Mon 3/24
HW 11 (Complex Exponentials)
FCA Page 31 Problems 1, 2, 3, 4, 10, 12, 16, 17, 18
Also, let z = x + iy and calculate Re[cos(z)], Im[cos(z)], Re[sin(z)], Im[sin(z)], Re[cosh(z)], Im[cosh(z)], Re[sinh(z)], & Im[sinh(z)]. Express your answers in terms of the functions cos, sin, cosh, and sinh with real arguments.
Due Thu 3/20
HW Exam 1 Solutions Review
Review the Solutions for Exam 1 posted below, until you are comfortable with all the problems.
Due Mon 3/17
Please have HW 9A, 9B, & 10 ready to hand in and/or present at the board on Wed 3/12.
Please have HW MFO, 7, & 8 ready to hand in and/or present at the board on Thu 3/13.
HW 10 (arg! Arg!)
FCA Page 22 Problems 5, 6, 7, 11, 12, 13
Note: For Problem 5, if you choose to do the absolute value last, you will do a lot of unnecessary work.
Due Wed 3/12
HW 9B (Fun with Finite Fields)
Consider the set {0, 1, 2, 3, 4} with addition and multiplication defined "modulo 5", which means whenever you obtain a number greater than or equal to 5, divide by 5 and replace the number by the remainder, which will always be in {0, 1, 2, 3, 4}. For example: 2 + 4 = 6 = 1 (modulo 5); (3)(4) = 12 = 2 (modulo 5), etc. This, in fact, defines a Field with only 5 members.
a) Construct a 5 x 5 addition table for this Field.
b) Construct a 5 x 5 multiplication table for this Field.
c) Construct a 5 x 2 table listing the additive inverse and the multiplicative inverse for each member of the Field (for which they are defined).
d) Prove that the set {0, 1, 2, 3} with addition and multiplication defined "modulo 4" is, in fact, NOT a Field. (Hint: Repeat steps a), b), and c). What goes wrong?)
However there is a Finite Field with 4 elements with a different addition and multiplication table. Type "finite field of order 4" into Wolfram Alpha to see it.
Due Mon 3/10
HW 9A (Welcome to Complex Analysis)
FCA Page 5 Problems 7, 9, 11, 13, 14, 17, 19, 20, 21, 23, 24, 25; Page 13 Problems 7, 9, 11, 13, 16
Due Mon 3/10 (but do as much as possible for Fri 3/7)
HW 8 (Linear Homogeneous ODE's with Constant Coefficients - Characteristic Equation has only Real Roots, Some Possibly Multiple)
ODE Page 220 Problems 14, 15, 16, 18, 19, 20, 21, 22
Due Thu 3/6
HW 7 (Linear Homogeneous ODE's with Constant Coefficients - Characteristic Equation has only Real Roots, All Distinct)
ODE Page 220 Problems 1-9 (all)
Due Wed 3/5
HW MFO (Mixed First Order ODE's)
ODE Page 104 Problems Any/All. You are not responsible for finding integrating factors and/or substitutions for types of ODE's that we did not cover.
Due Exam 1 Review Day (Thu 3/13)
HW 6 (Exact First Order ODE's)
ODE Page 79 Problems 5-17 odd
Due Fri 2/28 (But start now! Do at least one problem for Wed 2/26 and complete as much as possible for Thu 2/27)
HW 5 (Bernoulli Equation)
Read Lesson 11D (ODE Page 95-96) on the Bernoulli Equation. Then do:
ODE Page 97 Problems 3, 7, 8, 22
Due Mon 2/24
HW 4 (Linear First Order ODE's)
ODE Page 97 Problems 1, 2, 5, 6, 9, 11, 13, 14, 17, 24 (Hint: Problem 24 is linear if x is regarded as the dependent variable)
Due Fri 2/14
HW 3 (Homogeneous First Order ODE's)
ODE Page 61 Problems 3, 5, 6, 8, 12-15 (solve explicitly for y in 12-15) (Hint: Problem 3 is much easier if x is regarded as the dependent variable and you let u = x/y)
Due Wed 2/12
HW 2B (Separable First Order ODE's)
ODE Page 55 Problems 10, 12, 18, 19, 20, 21
Due Mon 2/10
HW 2A (Separable First Order ODE's)
ODE Page 55 Problems 2, 3, 5, 7, 11, 17
Due Fri 2/7
HW 1 (Verifying Solutions to ODE's)
ODE Page 27 Problems 1-3; Page 37 Problems 1-4
Due Thu 2/6
Note: All Test Statistics are before adding make-up points (if applicable).
Exam 1
Solutions (Note: There is an error in the final step of the domain calculation in Problem 5.)
Student Work
Exam 2
Student Work
Exam 3
Solutions (Note: The answer to Problem 7c) should, of course, include a "ln 2" term.)
Student Work