Course Syllabus
MATH 46 FALL 2025
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."
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: 9L (MWF 8:50–9:55 and R 9:05–9:55), Haldeman 028.
We will hold occasional class meetings in the 9LX hours on Thursdays (please hold the time). Many of these Thursdays are already listed in the weekly course schedule below, where we will continue to update class meeting days (dates subject to possible change, so please continue to hold this time on Thursdays).
Office Hours: Typically Thursdays 11–12 and Fridays 2–3 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 in the #general channel of 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:
Our primary textbook is Mathematics Applied to Deterministic Problems in the Natural Sciences by C. C. Lin and L. A. Segel, 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 Part B (Chapters 6–11) followed by as much of Part A (Chapters 1–5) as we can fit into the term.
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, or your favorite LLM after you specify the preferred series solution form and if you can get it to do the algebra correctly for you) 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 (30%), two midterm exams (20% each), and the final exam (30%).
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 9L/9LX class period and all weekly assignments are due by the end of the day (11:59pm) on the due date.
There will be two midterm exams, in class on Wednesday October 8 and Wednesday November 5.
The final exam, which will be comprehensive, is Sunday November 23 at 8am.
The midterms and final are closed-book, closed-notes, closed-technology exams. While you are encouraged to use technology on the homework assignments — indeed, some questions will require you to use technology — it is essential that you also develop enough practice and comfort with the material to be able to perform well in these closed-everything exam settings.
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.
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 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 Honor Principle:
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. Please also see the Arts and Sciences Academic Honor Policy for Undergraduates.
Collaboration is strongly encouraged in this course. At the same time, ultimately, all submitted work 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:
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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. You should be prepared to promptly submit the full prompts and output (in the form of screenshots, copy and paste, PDF, etc.) upon request.
- You should note on your submitted assignment which model(s) you used and comment briefly on how they helped. The Generative AI policy includes sample acknowledgement statements, though I would suggest that you err on the side of greater detail when making such acknowledgements (in this course and in general).
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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.
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 (sexual-respect.dartmouth.edu) provides a wealth of information on your rights and obligations with regard to sexual respect and resources that are available to all in our community. As a faculty member, I am obligated to share disclosures regarding conduct under Title IX with Dartmouth’s Title IX Coordinator.
Should you have any questions, please feel free to contact Dartmouth’s Title IX Coordinator Kristi.Clemens@Dartmouth.edu and deputies if appropriate.
Socioeconomic Differences and Financial Difficulty:
The materials for this course have been deliberately chosen to be fully freely available to you.
Disclaimer:
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 MWF |
Dimensional Analysis and Scaling
Office Hours Friday 2:30–3:30 For additional references for dimensional analysis and scaling, look at almost any book by G. I. Barenblatt. |
|
2 MWF |
Regular Perturbation Theory
Office Hours Thursday 11–12 and Friday 1–2 See these instructions for obtaining Maple on your computer. |
|
3 MWF |
Singular Perturbation & Initial Layers
Office Hours Friday 2-3 (not on Thursday) |
|
4 MWR |
Boundary Layers
Office Hours Tuesday 2:15–3:15 (not Thursday or Friday) |
|
5 WRF |
Biochemical Kinetics
Office Hours Thursday 11–12 and Friday 2-3 |
|
6 MWF |
Return of the Simple Pendulum
Office Hours Thursday 11–12 and Friday 2-3 |
|
7 MF |
Random Processes and Asymptotic Series
Office Hours Friday 2-3 (not Thursday) |
|
8 MWF |
Asymptotic Expansion of Integrals
Office Hours Tuesday 11–12 and Friday 2-3 Asymptotic expansions of integrals are covered in greater depth (than in Lin & Segel) in Applied Mathematics by J. David Logan and Advanced Mathematical Methods for Scientists and Engineers by Carl M. Bender and Steven A. Orszag; but our coverage this week is self contained so that you do not need to consult those books unless you are interested in going into greater depth. (Full disclosure: Orszag was one of my Ph.D. advisors, but I didn't get to take this class from him!) |
|
9 WRF |
Partial Differential Equations and Fourier Series
Office Hours Thursday 11–12 and Friday 2-3 For additional reference, consider "Part C: Fourier Analysis, Partial Differential Equations (PDEs)" of Advanced Engineering Mathematics by Erwin Kreyszig (online access via the library). |
|
10 M |
Fourier Transform
Office Hours Thursday 11–12 and Friday 2-3 |
Course Summary:
| Date | Details | Due |
|---|---|---|