Course Syllabus
MATH 46/136 SPRING 2023
Introduction to Applied Mathematics
From the ORC description for 46: "This course introduces a wide variety of mathematical tools and methods used to analyze phenomena in the physical, life, and social sciences. This is an introductory course and is accessible to undergraduate and graduate students in mathematics and other scientific disciplines who have completed the prerequisites. Topics include dimensional analysis and scaling, perturbation analysis, calculus of variations, integral equations, and eigenvalue problems." The description for 136 emphasizes partial differential equations, which will be addressed briefly in the last part of the course (and which students can also choose to address with their selected readings).
Instructor: Peter J. Mucha, Kemeny 240, peter.j.mucha@dartmouth.edu, https://mucha.host.dartmouth.edu.
Prerequisites: MATH 22/24 and MATH 23.
Lecture Information: 10A (TR 10:10–12:00 and F 3:30–4:20), Kemeny 307.
We will hold regular class meetings in the 10AX hour except where indicated. There will be no class on the following dates: Tuesday April 4, Friday April 14, Tuesday May 9, Friday May 12, Tuesday May 16, Friday May 19.
Office Hours: Thursdays 12:00–1:45 in Kemeny 240, and by appointment.
For questions outside office hours, I strongly encourage you to post your question on our class Slack team instead of email. You are much more likely to get a quicker response from me over Slack. It is also very likely that others will have the same or similar question.
Course Objectives:
The course introduces a variety of essential methods of applied mathematics. Our aim is to develop students' conceptual understanding of these methods and the ability to implement them to solve problems. Calculations will be performed by hand and through computational symbolic algebra.
Textbook:
Applied Mathematics by J. David Logan is recommended but not required. Lectures and course materials will be mostly self-contained, guided by the topics in the text. From the Fourth Edition, we will cover most of Chapters 1 and 3, possibly along with limited parts of Chapters 4–6 (italics added as course progressed). If you instead have access to the Third Edition, this more or less corresponds to the first half of Chapter 1, Chapter 2, and then parts of Chapters 3. 4 and 6.
Assignments, Exams and Grading:
The course grade for MATH 46 will be determined by written homework (50%), the midterm exam (25%) and the final exam (25%).
Homework assignments will include short "daily assignments" and longer "weekly assignments". Typically, "daily assignments" will be due at the next class meeting and are worth 2 points each — the first for demonstrated good effort and the second for correctness. No fractional points will be awarded. The "weekly assignments" will have longer times to their due dates, and will be worth more points. All assignments will be managed through Canvas. Unless otherwise specified, all daily assignments are due by the start of the corresponding 10A/10AX class period and all weekly assignments are due by the end of the day (11:59pm) on the due date.
There will be one midterm exam, in class on Tuesday May 2.
The final exam is Sunday June 4 at 11:30am.
The course grade for MATH 136 will include the above-described homework (50%). MATH 136 students will engage in two independent reading assignments (25% each), e.g., working through related journal articles or sections of other books. These readings are in lieu of the two MATH 46 exams. Each reading and the form of its assessment (written or oral) must be selected in collaboration with and approved by the instructor no later than one week before the corresponding exam date.
Attendance Policy:
Because of the natural way that lectures will hop around material from the textbook and other sources, I strongly encourage you to attend class whenever possible. However, for the health and safety of our class community please do not attend class when you are sick, nor when you have been instructed by Student Health Services to stay home. If possible, please let me know in a timely way if you are sick and missed class or believe you might miss an upcoming class, and I will work with you to obtain information about the missed material.
Similarly, some students may wish to take part in religious observances that occur during this academic term. Dartmouth has a deep commitment to support students’ religious observances and diverse faith practices. If you have a religious observance that conflicts with your participation in the course, please notify me well in advance to discuss appropriate accommodations.
In short, be responsible.
Academic Honesty:
The Academic Honor Principle is an essential tenet of the Dartmouth community. Collaboration is strongly encouraged in this course. At the same time, ultimately, all assignments submitted must represent your own understanding of the material. Be generous and honest in your citing any assistance and input from your fellow students. You are encouraged to look at the Writing Program's Sources and Citations page. All work on the exams must be carried out independently without input from others.
Student Accessibility and Accommodations:
Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Apply for Services webpage; student.accessibility.services@dartmouth.edu; 1-603-646-9900) and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.
Mental Health:
The academic environment at Dartmouth is challenging, our terms are intense, and classes are not the only demanding part of your life. There are a number of resources available to you on campus to support your wellness, including the undergraduate deans, Counseling Center, and Student Wellness Center. I encourage you to use these resources to take care of yourself throughout the term, and to speak to me if you experience any difficulties.
Diversity and Inclusion:
I strongly agree with the sentiments expressed in this sample syllabus statement on diversity from Monica Linden, Senior Lecturer in Neuroscience at Brown University. In linking to this statement, rather than copying it, I aim to both acknowledge the source and, in asking you to follow the link, highlight the message therein.
Title IX:
At Dartmouth, we value integrity, responsibility, and respect for the rights and interests of others, all central to our Principles of Community. We are dedicated to establishing and maintaining a safe and inclusive campus where all have equal access to the educational and employment opportunities Dartmouth offers. We strive to promote an environment of sexual respect, safety, and well-being. In its policies and standards, Dartmouth demonstrates unequivocally that sexual assault, gender-based harassment, domestic violence, dating violence, and stalking are not tolerated in our community.
The Sexual Respect Website at Dartmouth provides a wealth of information on your rights with regard to sexual respect and resources that are available to all in our community.
Please note that, as a faculty member, I am a mandatory reporter obligated to share disclosures regarding conduct under Title IX with Dartmouth's Title IX Coordinator. Confidential resources are also available, and include licensed medical or counseling professionals (e.g., a licensed psychologist), staff members of organizations recognized as rape crisis centers under state law (such as WISE), and ordained clergy (see https://dartgo.org/titleix_resources).
Should you have any questions, please feel free to contact Dartmouth's Title IX Coordinator. Their contact information can be found on the sexual respect website at: https://sexual-respect.dartmouth.edu.
Course Schedule:
Listed sections refer to the fourth edition of the Logan textbook. Additional references will be included as needed. Please note that topics listed in the future are tentative and will likely shift as time permits.
| Week | Topics |
|---|---|
| 1 |
Dimensional Analysis and Scaling
See also Lin & Segel (available online), especially Chapter 1, "What is Applied Mathematics?", and Chapter 6 on dimensional analysis and scaling. For additional references for dimensional analysis and scaling, look at almost any book by G. I. Barenblatt. |
| 2 |
Regular Perturbation Methods
See also Perturbation Methods by E. J. Hinch. See these instructions for obtaining Maple on your computer. |
| 3 |
Examples
|
| 4 |
Perturbation of Eigenvectors & Singular Perturbation
|
| 5 |
Boundary Layers
|
| 6 |
Multiple Scales
|
| 7 |
Optimal Asymptotic Truncation
|
| 8 |
Asymptotic Expansion of Integrals
|
| 9 |
Continue approximations of integrals and more examples from the literature
|
| 10 |
|
Course Summary:
| Date | Details | Due |
|---|---|---|