Course Syllabus
MATH 46 SPRING 2024
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."
Our course this term is also available as 136. ORC description for 136 emphasizes partial differential equations, which we might touch on very briefly. Please note that a new version of 136 will be taught as a separate course in Fall 2024 and I encourage interested students from 46 to sign up to take it!
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: 11 (MWF 11:30–12:35 and T 12:15–1:05), Kemeny 242.
We will hold regular class meetings in the 11X hour on Tuesdays, except where indicated. There will be no class on the following dates (dates listed in italics are still tentative):
- Friday April 5,
- Monday April 8 (go try to see the eclipse!),
- Tuesday April 9,
- Tuesday April 16 (office hours 12:00–12:50),
- Tuesday April 23 (extra office hours during X-hour),
- Tuesday April 30 (extra office hours during X-hour),
- Friday May 10,
- Tuesday May 21,
- Monday May 27 (Memorial Day is a reading day), and
- Tuesday May 28.
The course schedule below will also list which days we plan to have class each week, but this plan is subject to possible change.
Office Hours: Mondays 2-3 and Tuesdays 1:30-2:30 in Kemeny 240, and by appointment. (See the schedule below for deviations from these times.)
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:
Mathematics Applied to Deterministic Problems in the Natural Sciences by C. C. Lin and L. A. Segel is available online. Lectures and course materials will be mostly self-contained, but drawn heavily from this text, along with some other references as needed. Many of the homework problems will come directly from this text. We will aim to cover most of Chapters 1–3 and 6–11.
Software:
This class includes intensive use of symbolic computer algebra. We will use Maple in class and in the homework solutions. You are welcome to use another computer algebra system (e.g., Mathematica) if you are already well versed in its use, though of course you risk running into syntax and logic problems that I cannot help you solve as easily. See these instructions for obtaining Maple.
Importantly, if you are using Maple off campus, you may need to use the campus VPN to connect to the license server. Please be especially careful to save your work before changing locations, because if you lose connection to the license server, your session may crash.
Assignments, Exams and Grading:
The course grade will be determined by written homework (55%), the midterm exam (20%) and the final exam (25%).
Homework assignments will include short "daily assignments" and longer "weekly assignments". Typically, "daily assignments" will be due within 48 hours. Most problems will be worth 2 points each — the first for demonstrated good effort and the second for correctness. No fractional points will be awarded (except perhaps in extreme circumstances). 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 11/11X 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 Wednesday May 1.
The final exam is Sunday June 2 at 11:30am.
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. Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible—before the end of the second week of the term at the latest—to discuss appropriate course adjustments.
In short, be responsible.
Academic Honesty:
The faculty, administration, and students of Dartmouth College acknowledge the responsibility to maintain and perpetuate the principle of academic honor, and recognize that any instance of academic dishonesty is considered a violation of the Academic Honor Principle.
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.
Generative Artificial Intelligence:
As machine learning continues to advance, large language models (LLMs), such as ChatGPT, MathGPT, and other Generative Artificial Intelligence (GAI) technologies are becoming more widespread. These models can at times be useful tools to accelerate productivity and understanding. The use of such technologies is permitted for the assignments in our course, so long as the following guidelines are adhered to:
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When using an LLM or other GAI to aid in completion of an assignment, all prompts and output should be saved and submitted as part of the assignment. This may be in the form of a screenshot, copy and paste, PDF, etc.
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The work that you submit should reflect your own understanding of the assignment.
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Copying the output from an LLM or other GAI and handing it in as your own work is not permitted, similarly to how copying a peer’s work and submitting it as your own is not allowed.
Examples of situations where you might find it useful to use GAI in your work include when you know what kind of calculation you want to do but you don’t know all of the details or syntax for how to code it, or you forgot an idea or concept from a previous class that is needed for the current item you are working on. Many other reasonable examples are surely possible; you are strongly encouraged to share your experiences using such tools. Please be aware that in many cases these technologies can give answers to a prompt that are completely incorrect (and sometimes wildly so). As such, you should always be skeptical of any GAI output you see and verify the veracity of the information contained within. If you have any questions about the use of GAI in the class, please reach out to the instructor.
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 is challenging, our terms are intensive, 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 Counseling Center which allows you to book triage appointments online, the Student Wellness Center which offers wellness check-ins, and your undergraduate dean. The student-led Dartmouth Student Mental Health Union and their peer support program may be helpful if you would like to speak to a trained fellow student support listener. If you need immediate assistance, please contact the counselor on-call at (603) 646-9442 at any time. Please make me aware of anything that will hinder your success in this course.
Diversity and Inclusion:
Science should be open to all. Science benefits from diverse perspectives. I aim to create a learning environment that supports and encourages diversity of experiences, perspectives, identities, and thoughts. If something was said in class (by anyone) that makes you feel uncomfortable, please talk to me about it.
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 community members have equal access to Dartmouth's educational and employment opportunities. We strive to promote an environment of sexual respect, safety, and well-being. Through the Sexual and Gender-Based Misconduct Policy (SMP), Dartmouth demonstrates that sex and gender-based discrimination, sex and gender-based harassment, sexual assault, dating violence, domestic violence, stalking, etc., are not tolerated in our community.
For more information regarding Title IX and to access helpful resources, visit Title IX's website (sexual-respect.dartmouth.edu). As a faculty member, I am required to share disclosures of sexual or gender-based misconduct with the Title IX office.
If you have any questions or want to explore support and assistance, please contact the Title IX office at 603-646-0922 or TitleIX@dartmouth.edu. Speaking to Title IX does not automatically initiate a college resolution. Instead, much of their work is around providing supportive measures to ensure you can continue to engage in Dartmouth's programs and activities.
Socioeconomic Differences and Financial Difficulty:
Our community is composed of students from a variety of financial backgrounds. Socioeconomic diversity can be invisible, and you may be experiencing financial difficulties related to the cost of textbooks, materials, or other necessities for our class of which I am not aware.
If you encounter financial challenges related to this class, there may be sources of support for you. If you feel comfortable sharing your experience with me, you may. You may also consider meeting with a financial aid officer to discuss options, reaching out to the First-Generation Office if you are a first-generation student, browsing the Funding Resources page, or, for unexpected expenses, applying to the Barrier Removal Fund through the Financial Aid tile in DartHub.
Course Schedule:
Listed sections refer to the Lin & Segel 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 MTWF |
Chapter 6: Dimensional Analysis and Scaling
For additional references for dimensional analysis and scaling, look at almost any book by G. I. Barenblatt. |
|
2 MTW |
Chapter 7: Regular Perturbation Theory
|
|
3 WF |
Series solutions and an application of perturbations
|
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4 MWF |
Chapter 9: Singular Perturbation & Initial Layers
|
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5 MWF |
Boundary Layers & Chapter 10: Biochemical Kinetics
|
|
6 MWF |
Model Problems & Midterm
|
|
7 MTW |
Chapter 11: Return of the Simple Pendulum
|
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8 MTWF |
Chapter 3: Random Processes and Asymptotic Expansion of Integrals
|
|
9 MWF |
Chapter 3: Random Processes and PDEs
|
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10 W only |
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Course Summary:
| Date | Details | Due |
|---|---|---|