Math 245 Engineering Mathematics VI
SECTION: 001 Quarter: Spring 2004 Time: 8:00-9:15 CLASSROOM: GTM 317
INSTRUCTOR: Dr. Kanno OFFICE NUMBER: GTM 350
OFFICE HOURS: M 9:30-10:30, 3:00-4:00 T&R 9:30-10:30, 1:30-2:30 W&F 7:30-8:00,9:15-10:45
PHONE: 257-3329 E-MAIL: WEB: www.latech.edu/~jkanno
PREREQUISITE COURSE: MATH 244, Engineering Mathematics V
COURSE GOALS: The instructional objectives are included in the text.
TEXTBOOKS:
1) Differential Equations and Statistics, B. Schröder, compile date: April 9, 2003, or, MATH 245 update disseminated in Summer 2003, Fall 2003 or Winter 2003-04, compile date: July 3, 2003
2) Calculus: Concepts and Contexts, by James Stewart
3) http://www.eng.uml.edu/Dept/Chemical/onlinec/onlinec.htm (Math Methods for Engineers)
COURSE OUTLINE: Schröder, Modules LT, BVP, PDE, SoN, SoF and SER
ATTENDANCE REGULATIONS: Class attendance is regarded as an obligation as well as a privilege. All students are expected to attend regularly and punctually; failure to do so may jeopardize a student’s scholastic standing and may lead to suspension from the university.
STUDENTS NEEDING SPECIAL ACCOMODATIONS: Students needing testing or classroom accommodations based on a documented disability are encouraged to discuss the need with me as soon as possible.
HOMEWORK POLICY: Homework assignments are included in the course outline. On homework hand-ins clearly identify the problems by page and number. Key HW will be informed at the end of class. Passed due HW can’t be accepted with any reason. Two worst HW including zero grades will be dropped in the end.
MINUTE PAPER: You will write the following at the end of each class. 1. The most important/useful thing we discussed in class from your point of view. 2.The question(s) that remain uppermost in your mind at the end of class about today’s material.
EXAMINATIONS: There will be THREE exams, in addition to a comprehensive final exam at the end of the quarter . If you miss an exam, you must notify the instructor prior to the exam either in person or by phone. When you return, it is your responsibility to arrange for a makeup exam.
GRADE DETERMINATION POLICY: The standard ten-point grading scale will be used for this class: A = 90% - 100%; B = 80% - 89%; C = 70% - 79%; D = 60% - 69%; F = 0% - 59%. The course grade will be calculated as follows:
Exams (3×20% each) 60%
Homework 10%
Minute paper 4%
Final 26%
Total 100%
GRADE APPEAL: In the event of a question regarding an exam grade or final grade, it will be the responsibility of the student to retain and present graded materials which have been returned for student possession during the quarter.
HONOR CODE: In accordance with the Academic Honor Code, students pledge the following: Being a student of higher standards, I pledge to embody the principles of academic integrity. For details refer to http://www.latech.edu/tech/students/honor-code.pdf.
Schedule of Classes MATH 245
Day / Topic / Homework3/10/04 Wed / Schröder LT1: Introducing the Laplace Transform / LT.1 #1a,d, 2a, 4b,d, 5a, d
3/12 Fri / Schröder LT1: Introducing the Laplace Transform / LT.1 #6b,6d, 8, 11, 13, 14, 16
3/15 Mon
/ Schröder LT2: Systems of Linear Equations / LT.2 #1a,c,h, 2, 3,3/17 Wed
/ Schröder LT2: Systems of Linear Equations / LT.2 #4a, 6a,6bi3/19 Fri / Schröder LT3: Expanding the Transform Table / LT.3 #1a,b, 2a,g, 3a,b,f, 4a,b,d,f,
3/22 Mon / Schröder LT3: Expanding the Transform Table
Schröder LT4: Discontinuous Forcing Terms / LT.3 #5a, 6a, 7a, 13, 14, 15
LT.4 #1a, 2, 3a,b, 4b,
3/24 Wed / Schröder LT4: Discontinuous Forcing Terms / LT.4 #5a,b,j,k, 6a, 9a-d, 10
3/26 Fri / Schröder LT5: Convolutions / LT.5 #1a, 2b, 3a, 4a,b, 5a, 6
3/29 Mon / Exam 1: Schröder LT.1-LT.5
3/31 Wed / Schröder BVP.1: Examples of Boundary Value Problems
Review: Linear Differential Equations with constant coefficients. / BVP.1 # 1a,c,d,e, 2, 3
LDλ.1 #1a, 1c, 2g
4/2 Fri / Schröder BVP.2: Solving Boundary Value Problems
Review: Separable Differential Equations. / LT.1 #18
BVP.2 #1b,d,e,g,h, 6a,b,c
FIR.1 #1c,d,j
4/5 Mon / Schröder PDE1: Examples of PDEs
/ PDE.1 #2a,b, 3b,c,e, 5,6, 11a
4/7 Wed /
Schröder PDE2: Initial and Boundary Conditions
Schröder PDE3: A Simple Start: Separation of Variables / PDE.2 #1a,d, 2a-cPDE.3 #1b,d,
4/14 Wed / Schröder PDE3: A Simple Start: Separation of Variables
(start project PDE.4) / PDE.3 #1f,g, 2,4,8
4/16 Fri / Schröder MF.1: Proofs by induction / MF.1 #1b,d,g, 2a,c
4/19 Mon / Schröder SoN.1: Sequences of numbers, / SoN.1 #1c,e, 2c, 3b,c, 6b,d,e
4/21 Wed /
Exam 2: Schröder BVP.1,2, PDE.1-PDE.3, MF.1
4/23 Fri / Schröder SoN.2: Limits of Sequences,(Schröder SoN.3: The Exact Definition of the Limit) / SoN.2 # 1a,b,c,d,e, 2a,d, 3a
SoN.3 #1b, 3a
4/26 Mon / Schröder SoN.4: When Does a Sequence NOT Converge / SoN.4 #2a,e, 3a,b, 4c
4/28 Wed / Schröder SoN.5: Recursively Defined Sequences
Schröder SoN.6: Limits of Recursive Sequences. / SoN.5 # 1a,c, 2b, 3, 4a,c, 5a,c, 7 SoN.6 #1a,b, 2
4/30 Fri / Schröder SoN.6: Limits of Recursive Sequences.
Schröder MF.2: Summation Notation / SoN.6 #3a,5,7a
MF.2 #1a,c,g, 2a,c,e, 4a,c,e
5/3 Mon / Schröder NUM.1.1: Taylor Polynomials / NUM.1 #1a,b, 2b,d,f, 3a,c
5/5 Wed / Schröder NUM.1.2: Error Estimates for Taylor Polynomials
Schröder NUM.2: Power Series / NUM.1 #6a,b,d, 7b,c,d
NUM.2 #1b,g,h, 2a,3b,c,d
5/7 Fri / Schröder SER1: Expansions About Regular Points / SER.1 #1b, 2a,b,c,i, 3a, 4a,b,
5/10 Mon /
Exam 3: Schröder SoN.1-6, MF.1,2, NUM.1,2
5/12 Wed /Schröder SER1: Expansions About Regular Points
Schröder SER2: Expansions About Singular Points
/ SER.1 #5a,b,j,k, 6aSER.2 #1a,b, 2a,b
5/14 Fri / Schröder SER2: Expansions About Singular Points / SER.2 #3a,b, 4, 5a
5/17 Mon / Schröder SER3: Reduction of Order / SER.3 #1a,b,c,d,e
5/19 Wed / Schröder NUM.3: Multiplying and Dividing (Power) Series
Schröder NUM.4: Fourier Series (wrap up) / NUM.3 #1a,b, 2a,b
NUM.4 #2a,c
5/21 Fri / Final: Cumulative
Course Objectives for MATH 245, Engineering Mathematics VI
Course objectives. The framework for our course objectives is given by the ABET2000 engineering criteria, which list the following desired outcomes for engineering graduates.
Engineering programs must demonstrate that their graduates have:
1) An ability to apply knowledge of mathematics, science, and engineering,
2) An ability to design and conduct experiments, as well as to analyze and interpret data,
3) An ability to design a system, component, or process to meet desired needs,
4) An ability to function on multi-disciplinary teams,
5) An ability to identify, formulate, and solve engineering problems,
6) An understanding of professional and ethical responsibility,
7) An ability to communicate effectively,
8) The broad education necessary to understand the impact of engineering solutions in a global and societal context,
9) A recognition of the need for, and an ability to engage in life-long learning,
10) A knowledge of contemporary issues,
11) An ability to use the techniques, skills, and modern engineering tools necessary for engineering practice.
This course focuses mainly on outcomes 1, 2, 3, 5, and 7. Outcome 9 is implicit in that we are investigating a field (ordinary and partial differential equations) that is very wide spread and which cannot be totally absorbed even in several courses. The primary instructional objectives for this class as they relate to the above are as follows:
Objective 1. At the end of this course the student will be able to represent functions as series and determine how closely a polynomial approximates the sum of the series. (1,3)
Objective 2. At the end of this course the student will be able to solve differential equations using elementary techniques (separable or linear constant coefficient equations), series or Laplace transforms. (1)
Objective 3. At the end of this course the student will be able to solve partial differential equations using separation of variables. (1,2)
Objective 4. At the end of this course the student will be able to use (ordinary and partial) differential equations to model engineering phenomena such as for example circuits and elementary heat transfer. (1,5)
Assessment.
Achievement of the objectives will be demonstrated by satisfactory performance on homework and examinations.
Specific objectives and how they relate to the primary objectives are as follows.
At the end of MATH 245 the student will be able to
Exam 1:
1. State the definition of the Laplace transform. (2)
2. Compute the Laplace transform of a given function (2)
· Using the definition of the Laplace transform.
· Using a table of Laplace transforms.
· Using theorems about Laplace transforms.
3. Compute the Laplace transform of a differential equation. (2)
4. Compute the inverse Laplace transform of a given function or equation. (2)
5. Model time-limited phenomena using unit step functions. (4)
6. Model instantaneous transfers using Dirac delta functions. (4)
7. Solve linear differential equations with constant coefficients using Laplace transforms. (2,3)
· By hand (using only a transform table)
· Using a MathCAD template to be written by the student
8. Solve systems of linear differential equations with constant coefficients. (2)
9. Model spring-mass systems and LRC circuits using differential equations. (4)
10. Solve boundary value problems involving ordinary linear differential equations with constant coefficients (2,3)
Exam 2:
11. Reproduce the heat and the wave equation (4)
12. Model simple heat transfer and wave phenomena (4)
13. Solve partial differential equations with initial and boundary conditions using separation of variables or recognize that the approach will not work. (3)
14. Compute the limit of a sequence or show that no limit exists, (1)
15. Prove that a sequence converges or diverges (1)
16. For a convergent sequence and a given e, compute an index N such that all terms beyond the Nth term are closer than e to the limit.
17. Compute the sum of a series or show that no limit exists; tools include but are not limited to (1)
· The ratio test,
· The comparison test,
· The p-series test.
18. Estimate the difference between a partial sum of a series and its sum. (1)
19. Approximate the sum of a series to a given accuracy. (1)
20. Reproduce the power series representations of sine, cosine and the natural exponential function. (1)
21. Multiply and divide power series (1)
22. Compute the Taylor polynomials/series of a sufficiently often differentiable function. (1)
23. Estimate the difference between a Taylor polynomial approximation of a function and the function. (1)
24. Approximate integrals of power series to a given accuracy. (1)
25. Compute Fourier series (1)
Rest for final:
26. Solve ordinary differential equations about ordinary points using power series techniques. (2,3)
27. Solve ordinary differential equations about singular points using power series techniques. (2,3)
28. Reduce the order of a linear differential equation when one solution is given (2)
29. Recognize and solve the Bessel and Legendre equations. (2)