Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

Instructor Name Aime FournierInstructor Office AB-4126Instructor E-mail aime.fournier@ucdenver.edu

Department Mathematical and Statistical SciencesCollege Liberal Arts & SciencesWebsite https://ucdenver.instructure.com/courses/345258

Class Location North 1323Class Meeting Time Monday, Wednesday 9 am–10:50 amOffice Hours Monday, Wednesday 11 am–noon or by appointmentTerm Spring

COURSE OVERVIEW

I. Welcome!

Linear algebra concerns spaces of vectors x e.g., an ordered set of n real numbers x1, x2, . . . xn,and transformations Ax of x that are linear i.e., they preserve addition and multiplication.Differential equations are essential for understanding and applying the laws that govern nat-ural (or artificial) systems in which some quantities’ variations depends on other quantities’variations. For example, in one space dimension, Newton’s Second Law for a rigid object ofconstant mass m relates the 2nd derivative of its position x with respect to time t to theexternal force

f = ma, a =dv

dt, v =

dx

dt

acting on it. When a number n of derivatives appear in a differential equation, then linearalgebra of order n may be applied to the system. For example, if the force has the formula

f = −kx− cv + f0

then we will study how to write and solve a matrix differential equation[1 00 m

]d

dt

[xv

]=

[0 1−k −c

] [xv

]+

[0f0

]that governs the system of order n = 2. There are countless applications of linear algebra anddifferential equations, including understanding and engineering the systems they describe.In this course we will study how to find solutions of ordinary differential-equation systems(those with exactly one independent variable such as t), or obtain information about thosesolutions even if we cannot find them explicitly.

II. University Course Catalog description

Presents the essential ideas and methods of linear algebra and differential equations, em-phasizing the connections between and the applications of both subjects. The course isdesigned for students in the sciences and engineering. Note: No co-credit with MATH 3200and MATH 3191. Semester Hours: 4 to 4. See Listing of core courses.

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Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

III. Short description of the course

The major topics addressed in this course are the following.

• Systems of differential equations that can be explicitly integrated, including separablesystems

M1(x1)dx1 = M2(x2)dx2 = · · · = Mn(xn)dxn = f(t)dt

and linear systems

d

dt

x1

x2...xn

+ A

x1

x2...xn

=

f1f2...fn

, A =

a11 a12 · · · a1na21 a22 · · · a2n...

.... . .

...an1 an2 · · · ann

.

• Operations on matrices A, such as

– matrix algebra,

– reduction to echelon form,

– calculation and applications of eigenvalues and (generalized) eigenvectors, and

– counting the existence and uniqueness of solutions x of Ax = b when b and xmay have different numbers of rows, and

– finding basis vectors for solution spaces when Ax = 0.

• Stability analysis of the nonlinear system

d

dt

x1

x2...xn

=

f1(x1, x2, . . . , xn)f2(x1, x2, . . . , xn)

...fn(x1, x2, . . . , xn)

≈

∂f1∂x1

∂f1∂x2

· · · ∂f1∂xn

∂f2∂x1

∂f2∂x2

· · · ∂f2∂xn

......

. . ....

∂fn∂x1

∂fn∂x2

· · · ∂fn∂xn

∣∣∣∣∣∣∣∣∣x=x∗

x1 − x∗1x2 − x∗2

...xn − x∗n

in neighborhoods of its fixed points x∗, and graphical analysis when n = 2.

IV. Course objectives & learning outcomes

• To translate differential-equation problems expressed in words, into mathematical no-tation, and vice-versa.

• To apply differential-equation methods in science or engineering contexts.

• To solve differential-equation systems explicitly when possible.

• To find and characterize the stability of fixed points of systems even without explicitsolutions.

• To interpret and use of (context-dependent) mathematical notation and concepts cor-rectly, such as:

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Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

– condition, instance, substitution, supposition;

– arbitrary, constant, fixed, parameter, value, variable;

– assignment, definition, equation, equivalence, expression, formula, identity, law,theorem;

– coefficient, exponent, factor, product, quotient, ratio, term; and

– dependence, function, (in)determinancy, operation, solution.

V. Prerequisites

This course assumes that students have taken MATH 2411 or equivalent. Students whohave a grade of B− or better in MATH 2411 pass this course at a much higher rate. We willspend the first two days reviewing and testing students’ understanding of required resultsfrom basic algebra and single-variable calculus, like the chain rule

d

dxf(g(x)) = f ′(g(x))g′(x)

and its inverse differentiation corollary

f(g(x)) = x ⇔ g′(x) =1

f ′(g(x)),

derivatives and integrals of transcendental functions, first and second parts

d

dx

∫ x

a

f(u)du = f(x)

and ∫ x

a

f ′(u)du = f(x)− f(a)

of the fundamental theorem of calculus, partial fraction expansion, product rule, and quotientrule. Adept facility with such results is required to do well in this course.

VI. Course credits Max hours: 4 Credits.

VII. Required text

C. Henry Edwards and David E. Penney, 2010: Differential Equations and Linear Algebra,3/E. Pearson, 792 pp. ISBN: 9780136054252.

Read the textbook sections to be covered before the class: even if you may not understandeverything, it makes the class much easier.

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Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

VIII. Supplementary materials

• Textbook website

• MATH 2411 textbook website

• Wikipedia on Diff. Eqs.

• Wolfram Alpha is a powerful tool to verify symbolic calculations, especially differentialequations, but no substitute for learning how to do it by hand.

• If a Problem needs a direction field —a visual display of scaled (dt, dx) segments inthe (t, x) plane, use GeoGebra or another tool.

IX. Course schedule See Table 1

EVALUATION

X. Assignments

Homework will consist of the textbook Assignment Problems in the schedule (§IX). Solutionsmust be written up neatly, and will be marked up to provide feedback, but graded only interms of completion, 0 or 1 (no fractions). Doing homework is vital to learning the materialin this class. Expect to spend ≈12–15 hours/week on the homework and studying. Successwill not come from only listening to lectures or reading the book. Spend your time wisely:if you find yourself working on one problem for more than 10 minutes without progressing,then move on to another problem and return to the challenging problem later. If you’regetting stuck on all the problems, then it is time to:

• talk with a classmate,

• come to office hours, or

• start or reply to a Canvas Discussion to engage with either Prof. Fournier or a class-mate.

Also see the Intellectual Wellness and Academic Success web site.

XI. Basis for Final Grade

For student i the raw Final Grade gi will be computed from Table 2 as

0 ≤ gi =7∑

j=1

wjsijpj≤ gmax ≤ 100%.

If the mean grade gmean < 75% then every grade will be curved upward according to

gi < gi = gmean +gmax − gmean

gmax − gmean

(gi − gmean) ≤ gmax, (1)

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Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

Date Topic Reading Assignments1/20 Calculus, Mathematical Notation1/25,27 First-Order Differential Eqs. §§1.1–3 Calculus Quiz

1.1: 45, 481.2: 22, 42, 441.3: 18, 27

2/1,3 Separable and Linear Eqs. §§1.4–5 1.4: 17, 26, 59,1.5: 20, 31–32, 38

2/8,10 Mathematical Models §§2.1–3 2.1: 132.2: 10, 23–242.3: 2, 4, 13

2/15,17 Linear Systems §§3.1–3 3.1: 10, 343.2: 12, 243.3: 3, 35

2/22,24 Matrix Operations §§3.4–6 First-Order Quiz3.5: 31–323.6: 7, 8, 41, 56–57

2/29, 3/2 Vector Spaces §§4.1–4 4.2: 154.3: 234.4: 3–5, 15, 27–28

3/7,9 Higher-Order Linear Eqs. §§5.1–2 5.1: 32, 515.2: 18, 20, 31, 32, 36

3/14,16 Mechanical Vibrations §§5.3–4 Linear Algebra Quiz5.3: 33, 525.4: 13, 24, 27, 30, 38

3/21,23 Spring Break3/28,30 Nonhomogeneous Eqs. §§5.5–6 5.5: 43, 47, 57, 63

5.6: 15, 18, 264/4,6 Eigenvalues and Eigenvectors §§6.1–3 6.1: 14, 27

6.2: 32–33, 386.3: 35–37

4/11,13 Linear Differential-Eq. Systems §§7.1–3 Higher-Order Quiz7.1: 6–8, 10–127.2: 18, 277.3: 8, 17

4/18,20 Matrix Exponentials §§8.1–2 8.1: 3, 4, 8, 27, 338.2: 18, 31

4/25,27 Nonlinear Systems §§9.1–2, 9.4 Linear Systems Quiz9.1: 9, 159.2: 13, 15, 209.4: 14–16

5/2,4 Review5/9,11 Final Exam TBD

Table 1: Schedule —see Canvas for modifications. Quiz dates are underlined. No electronicaids will be permitted during quizzes without approval by Prof. Fournier. Textbook Problemsmust be handed in Wednesday the week after assignment.

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Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

j Assessment sj Points Possible pj Percent wj of Final Grade1 Calculus quiz 100 52 First-order quiz 100 143 Linear-algebra quiz 100 144 Higher-order quiz 100 145 Linear-systems quiz 100 146 Final Exam 100 297 N ≈ 13 Homework sets N 10

Table 2: Assessment.

which favors smaller no less than larger gi, as long as

0 ≤ gmax − gmax ≤ gmean − gmean.

After assigning

gmean = max (gmean, 75%) ,

gmax = gmax + b (gmean − gmean)

for a constant

0 ≤ b ≤ min

(1,

100%− gmax

gmean − gmean

)to be determined, letter grades will be assigned based on thistable.

94% ≤ gi A90% ≤ gi < 94% A−87% ≤ gi < 90% B+84% ≤ gi < 87% B80% ≤ gi < 84% B−77% ≤ gi < 80% C+74% ≤ gi < 77% C70% ≤ gi < 74% C−67% ≤ gi < 70% D+64% ≤ gi < 67% D60% ≤ gi < 64% D−

gi < 60% F

XII. Grade Dissemination

Raw sij grades will be posted in Canvas. During the course, when requested, current gradestatus will be estimated using eq. 1 with gi replaced by(

j∑k=1

wksikpk

+ w7rij

)/(j∑

l=1

wl + w7

), (2)

where 0 ≤ rij ≤ 1 is the average homework score of student i until the most recent assessmentj. Homework will be returned in class. Incomplete homework will be so marked, and countfor 0.

COURSE PROCEDURES

XIII. Course Policies

Attendance Class attendance is recommended, but will not be specifically counted towards stu-dents’ grades. A few absences are acceptable, but advance notice to Prof. Fournier

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Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

would be courteous. Class participation will not be specifically counted towards stu-dents’ grades, but is strongly recommended, to help learn the material from differentpoints of view, and to provide context for evaluating students’ written work. UC Den-ver Student Attendance and Absences Policy: http://www.ucdenver.edu/faculty_

staff/employees/policies/Policies%20Library/OAA/StudentAttendance.pdf.

Late Work Late homework will not be accepted without advance approval from Prof. Fournier.There are no make-ups for quizzes or the final exam.

Extra Credit None.

Incompletion The current university policy concerning incomplete grades will be followed in thiscourse. Incomplete grades are given only in situations where unexpected emergenciesprevent a student from completing the course; students have up to one year (threesemesters) to complete course requirements. Your instructor is the final authority onwhether you qualify for an incomplete. Incomplete work must be finished within thetime allowed or the “I” will automatically be recorded as an “F” on your transcript.

Resubmission Work may not be resubmitted.

Group Work Working together on homework is encouraged —but don’t simply copy another’s work!

XIV. Technology and Media

E-mail E-mail is preferred over Canvas Conversations. Students should use their universitye-mail accounts to communicate with Prof. Fournier about this course.

Canvas Canvas will be used mainly for its Discussions feature, including comments, sugges-tions and modifications regarding: due dates, homework, lectures, quizzes, other andschedule items. Raw sij will be tabulated in the Grades section, but the curved gradesgi (eqs. 1, 2) may be presented in Discussions (in anonymous form) and by e-mail asindividuals request them.

Laptops Laptop use is encouraged outside of class. A student wishing to use a laptop duringclass e.g., to look up something relevant to the class discussion, must ask Prof. Fournierto approve the use.

Mobile Devices See Laptops. Mobile devices must be silenced during all class meetings. Phone con-versations and texting during class are not permitted. Emergency conversations andtexting must be taken outside the class.

Calculators Calculators of any kind are permitted but not during quizzes or the Final Exam.

Clickers Clickers are not used in this course.

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Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

XV. Student Expectations

Civility My commitment is to create a climate for learning characterized by respect for eachother and the contributions each person makes to class. I ask that you make a similarcommitment.

Late Arrivals Regarding late arrivals and early departures, advance notice to Prof. Fournier regardingwould be courteous.

Religion Students are expected to notify Prof. Fournier in advance if they intend to miss classbecause of religious observances.

CARE The Campus Assessment, Response & Evaluation Team (http://www.ucdenver.edu/CARE) addresses health and safety needs of the campus community. The purpose of theteam is to assess if individuals pose a risk to themselves or others, to intervene whennecessary and, more generally, to identify and provide assistance to those in need. Theteam takes a preventive approach to risk assessment by offering resources, referrals,and support to both the concerning individual and those impacted by their behavior.

E-Cigarettes E-cigarettes are prohibited indoors and ≤25 feet from any entrance (http://www.ucdenver.edu/faculty_staff/employees/policies/Policies%20Library/Admin/Smoke-Free.

pdf).

UNIVERSITY POLICIES

XVI. Disability Access

CU Denver is committed to providing reasonable accommodation and access to programsand services to persons with disabilities. Students with disabilities who want academicaccommodations must register with the Disability Resources and Services (DRS) in AcademicBuilding 1, #2116, Phone: 303-315-3510, Fax: 303-315-3515. Prof. Fournier will be happyto provide approved accommodations, once you provide him with a copy of DRSs letter.

XVII. Academic Honesty

Cheating or other academic misconduct will result in no credit for the homework, quiz, orexam in question, plus lowering the overall class grade by 10%, and will be reported to the de-partment chair for further administrative action. For suggestions on ways to avoid academicdishonesty, please see the Academic Honesty Handbook at http://www.ucdenver.edu/

faculty_staff/faculty/center-for-faculty-development/Documents/academic_honesty.

pdf. Also see below.

XVIII. Important Dates

See the College of Engineering and Applied Science Drop Policy Also see below.

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� Academic Dishonesty: Students are required to know, understand, and comply with the CU Denver Academic Dishonesty Policy as detailed in the Catalog and on the CLAS website. Academic dishonesty consists of plagiarism, cheating, fabrication and falsification, multiple submission of the same work, misuse of academic materials, and complicity in academic dishonesty. If you are not familiar with the definitions of these offenses, go to http://www.ucdenver.edu/academics/colleges/CLAS/faculty-staff/policies/Pages/DefinitionofAcademicDishonesty.aspx. This course assumes your knowledge of these policies and definitions.

Failure to adhere to them can result in possible penalties ranging from lowering a grade on an assignment to dismissal from the University; so, be informed and be careful. If this is unclear to you, ask me.The College of Liberal Arts and Sciences (CLAS) Ethics Bylaws allow theinstructor to decide how to respond to an ethics violation, whether bylowering the assignment grade, lowering the course grade, and/or filingcharges against the student with the Academic Ethics Committee.

� Students who have complaints about the course or instructor should: 1) meet with the instructor face-to-face; 2) if not satisfied, meet with the Associate Chair of the math department/unit, Prof. Steve Billups or the Chair, Prof. Jan Mandel; 3) if not satisfied, appeal to the Associate Dean. No step in this process may be skipped. See "Procedures for Student Grievances about Courses or Faculty, CLAS.”

� Missing an Exam: If circumstances arise that prevent you from attending an exam, please contact me ahead of time as I will be much more lenient. Unexplained absences will require hard evidence such as a death certificate, hospital paperwork, etc.

Other resources for this courseThe Learning Resources Center

� The Learning Resource Center is where students go to get help or insight with class assignments, course-loads, and study skills. The Center also helps with arranging tutoring sessions, which take place in the days or evenings.

North Classroom Building (NC) Room 2006(303) 556-2802Monday - Thursday 9am-7pm Friday 9am-5pm

http://www.ucdenver.edu/life/services/LRC/Pages/default.aspxProgram Access for Persons with Disabilities

� The University of Colorado Denver is committed to providing reasonable accommodations and access to programs and services to persons with disabilities. Students should contact the Disabilities Resources Offices.

North Classroom building 2514;

Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

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Phone # 303-556-3450, TTY 303-556-4766. Monday – Friday 8am – 5pm

http://www.ucdenver.edu/academics/colleges/CLAS/faculty-staff/faculty-resources/teaching/supporting-students/Pages/students-disabilities.aspx

Academic Success and Advising Center

� This office serves as the first point of contact for students who are pre-business, pre-engineering, or who have not declared a major in CLAS or CAM. In addition, the center provides general information and resource referral to all students.

North Classroom Building (NC) Room 2024 Phone # 303-352-3520

Career Center

� The Career Center offers a full array of services that prepare students for career success, such as resume help, internship and career counseling and they have a large career library.

Tivoli Student Union Room 267 Phone # 303-556-2250

Definition of Academic DishonestyStudents are expected to know, understand, and comply with the ethical standards of the University. In addition, students have an obligation to inform the appropriate official of any acts of academic dishonesty by other students of the University. Academic dishonesty is defined as a student's use of unauthorized assistance with intent to deceive an instructor or other such person who may be assigned to evaluate the student’s work in meeting course and degree requirements. Examples of academic dishonesty include, but are not limited to, the following:

Plagiarism: Plagiarism is the use of another person’s distinctive ideas or words without acknowledgment. The incorporation of another person’s work into one’s own requires appropriate identification and acknowledgment, regardless of the means of appropriation. The following are considered to be forms of plagiarism when the source is not noted:

1. Word-for-word copying of another person's ideas or words.2. The mosaic (the interspersing of one’s own words here and there while, in essence, copying another's work).3. The paraphrase (the rewriting of another’s work, yet still using their fundamental idea or theory).4. Fabrication of references (inventing or counterfeiting sources).5. Submission of another’s work as one's own.6. Neglecting quotation marks on material that is otherwise acknowledged.

Acknowledgment is not necessary when the material used is common knowledge.

Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

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Cheating: Cheating involves the possession, communication, or use of information, materials, notes, study aids or other devices not authorized by the instructor in an academic exercise, or communication with another person during such an exercise. Examples of cheating are:

1. Copying from another's paper or receiving unauthorized assistance from another during an academic exercise or in the submission of academic material.2. Using a calculator when its use has been disallowed.3. Collaborating with another student or students during an academic exercise without the consent of the instructor.

Fabrication and Falsification: Fabrication involves inventing or counterfeiting information, i.e., creating results not obtained in a study or laboratory experiment. Falsification, on the other hand, involves deliberately alternating or changing results to suit one’s needs in an experiment or other academic exercise.

Multiple Submissions: This is the submission of academic work for which academic credit has already been earned, when such submission is made without instructor authorization.

Misuse of Academic Materials: The misuse of academic materials includes, but is not limited to, the following:

1. Stealing or destroying library or reference materials or computer programs.2. Stealing or destroying another student’s notes or materials, or having such materials in one’s possession without the owner’s permission.3. Receiving assistance in locating or using sources of information in an assignment when such assistance has been forbidden by the instructor.4. Illegitimate possession, disposition, or use of examinations or answer keys to examinations.5. Unauthorized alteration, forgery, or falsification.6. Unauthorized sale or purchase of examinations, papers, or assignments.

Complicity in Academic Dishonesty: Complicity involves knowingly contributing to another’s acts of academic dishonesty.

Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

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Spring 2016 CLAS Academic Policies

The following policies, procedures, and deadlines pertain to all students taking classes in the College of Liberal Arts and Sciences (CLAS). They are aligned with the Official University Academic Calendar:

http://www.ucdenver.edu/student-services/resources/Registrar-dev/CourseListings/Pages/AcademicCalendar.aspx

•   Schedule verification: It is each student’s responsibility to verify that their official registration and schedule of classes is correct in their Passport ID portal before classes begin and by the university census date. Failure to verify schedule accuracy is not sufficient reason to justify late adds or drops. Access to a course through Canvas is not evidence of official enrollment.

•   E-mail: Students must activate and regularly check their official CU Denver e-mail account for university related messages. •   Administrative Drops: Students may be administratively dropped from a class if they never attended or stopped attending, if the course

syllabus indicates that the instructor will do this. Students may be administratively dropped if they do not meet the requisites for the course as detailed in course descriptions.

•   Late adds and late withdrawals require a written petition, verifiable documentation, and dean’s approval. CLAS undergraduate students should visit the CLAS Advising Office (NC1030) and graduate students should visit the Graduate School (12th floor LSC) to learn more about the petition process and what they need to do to qualify for dean’s approval.

•   Waitlists: The Office of the Registrar notifies students at their CU Denver e-mail account if they are added to a class from a waitlist. Students are not automatically dropped from a class if they never attended, stopped attending, or do not make tuition payments. After waitlists are purged, students must follow late add procedures to be enrolled in a course. Students will have access to Canvas when they are on a waitlist, but this does not mean that a student is enrolled or guaranteed a seat in the course. Students must obtain instructor permission to override a waitlist and this is only possible when there is physical space available in a classroom, according to fire code.

Important Dates and Deadlines All dates and deadlines are in Mountain Time (MT).

•   January 19, 2016: First day of classes. •   January 24, 2016: Last day to add or waitlist a class using the Passport ID portal. •   January 24, 2016: Last day to drop a class without a $100 drop charge--this includes section changes. •   January 25, 2016: All waitlists are purged. Students should check their schedules in their Passport ID portal to confirm in which classes

you are officially enrolled. •   January 26-Feburary 3, 2016, 5 PM: To add a course students must obtain instructor permission using the Instructor Permission to

Enroll Form and bring it to the CLAS Advising Office (NC 1030) or have their instructor e-mail it to CLAS_Advising@ucdenver.edu . •   February 3, 2016: Census date.

o   2/3/16, 5 PM: Last day to add full term classes with instructor approval. Adding a class after this date (late add) requires a written petition, verifiable documentation, and dean’s approval. After this date, students will be charged the full tuition amount for additional classes added – College Opportunity Fund hours will not be deducted from eligible student’s lifetime hours.

o   2/3/16, 5 PM: Last day to drop full term classes with a financial adjustment on the Passport ID portal. After this date, withdrawing from classes requires instructor signature approval and will appear on student’s transcript with a grade of ‘W’. After this date, a complete withdrawal (dropping all classes) from the term will require the signature of the dean and no tuition adjustment will be made. Students should consult appropriate service offices (e.g. international status, Financial Aid (loans, grants, and/or scholarships) or Veteran’s Student Services) before withdrawing from course(s) to determine any impact for continued enrollment and funding.

o   2/3/16, 5 PM: Last day to apply for Spring 2016 graduation. Undergraduates must make an appointment and see their academic advisor before this date to apply. Graduate students must complete the Intent to Graduate and Candidate for Degree forms.

o   2/3/16, 5 PM: Last day to request No Credit or Pass/Fail grade for a class using a schedule adjustment form. o   2/3/16, 5 PM: Last day to petition for a reduction in Ph.D. dissertation hours.

•   February 4-April 4, 2016, 5 PM: To withdraw from a course, students must obtain instructor permission using the Schedule Adjustment Form and must bring the signed form to the Office of the Registrar. To add a course, students must petition through College/School undergraduate advising offices or the Graduate School, as appropriate.

•   March 21-27, 2016: Spring break- no classes, campus open. •   April 5, 2016: The Office of the Registrar now requires both the instructor’s signature and a CLAS advisor’s/dean’s signature on a

Schedule Adjustment Form to withdraw from a class. Students should consult their home college advising office for details. •   April 18, 5 PM: Deadline for undergraduate CLAS students to withdraw from a course without filing a late withdrawal petition. Contact

CLAS Advising (NC 1030 – 303-556-2555). •   May 14, 2016: End of semester. •   June 24, 2016: Final grades available on the Passport ID portal and on transcripts (tentative).

Please contact an academic advisor if you have questions or concerns.

 

Math 3195–004 Linear Algebra and Differential Equations COURSE SYLLABUS

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