21-269: Vector Analysis

Spring 2026

Course Information

Discussion Board / Mailing List

Office Hours

Time Place Person
Mon 3:00--4:00pm WEH 8115 Gautam Iyer
Mon 8:00--9:00pm WEH 8215 Isabella Wu
Tue 3:00--4:00pm WEH 8215 Ishin Shah
Fri 4:00--5:00pm WEH 8215 Isabella Wu

Homework

Homework is due every Wednesday at 10:00 AM on Gradescope, one hour before class starts. (Use this invite code if you’re not signed up). Please read the late homework policy

  • All homework and solutions are here.
  • Your grades and statistics are here.

Handouts

Exam Dates

  • Midterm 1: Wed, Feb 11 (5th week), closed book, in class.
  • Midterm 2: Wed Mar 25th (10th week), closed book, in class.
  • Final: Comprehensive, closed book, in class. The time of the final will be announced by the registrar here. Be aware of their schedule before making your travel plans.

Grading

Your scores on exams and homework will each be converted to common scale using cutoffs announced after each exam. Your exam average will be computed as the maximum of your converted midterm / final scores with the following weights:

  • 30% each midterm, 40% final.
  • 30% better midterm, 70% final.

Your overall grade will be the minimum of your exam average, and your homework average.

Course description.

This course is an honors introduction to vector analysis (multi-variable differential and integral calculus). Topics covered may vary, but will at least include a through foundation of multi-variable differential calculus (continuity, differentiation, inverse, implicit function theorems, Taylors theorem, Lagrange multipliers). Certain topics on integration in $\R^d$ will be also be covered, and the topics covered will vary depending on the time available. Irrespective of the specific topics covered, a core component of this class is being able to write, long, involved proofs clearly and correctly.

Tentative Syllabus

  • Functions of several variables, regions and domains, limits and continuity.
  • Sequential compactness.
  • Partial derivatives, linearization, Jacobian.
  • Chain rule, inverse and implicit functions and geometric applications.
  • Higher derivatives, Taylor’s theorem, optimization, vector fields.
  • Multiple integrals and change of variables, Leibnitz’s rule.
  • Line integrals, Green’s theorem.
  • Path independence and connectedness, conservative vector fields.
  • Surfaces and orientability, surface integrals.
  • Divergence theorem and Stokes’s theorem.

Learning Objectives

  1. Understand the foundations of analysis in $\R^n$.
  2. Develop logical thinking / problem solving skills.
  3. Be able to write involved, mathematically rigorous proofs in an abstract setting.

Pre-requisites

  • A rigorous and through course on one variable calculus
  • Familiarity with writing proofs
  • Linear algebra
  • Official pre-requisites: 21122 and (21127 or 15151) and (21241 or 21242).

References

The material covered in this course is standard and can be found in many good references. As a result my notes above are extremely brief, and will only have statements of definitions and theorems. Proofs and intuition will be done in class; if you miss class I suggest reading this from one of many standard references. The specific choice of topics we cover may not be done similarly, with the same notation, or in the same order in the references below.

  • Introduction to Multivariable Mathematics by Leon Simon (This covers the differential calculus portion of this class.)
  • Calculus on Manifolds by Spivack.
  • Lecture Notes on Multivariable Calculus by Barbara Niethammer and Andrew Dancer. Currently the book can be found online here, but the link may change as time progresses. (This covers the differential calculus portion of this class.)
  • Principles of Mathematical Analysis by W. Rudin (Excellent “classic”.)

Class Policies

Lectures

  • If you must sleep, don’t snore!
  • Be courteous when you use mobile devices

Homework

  • The primary purpose of homework is to help you understand the material!
    • You may collaborate, use AI/online resources, etc. freely, as long as you learn from the solutions you turn in, and you write up the solutions yourself independently.
    • Turning in solutions you don’t understand will be treated as a violation of academic integrity.
    • In order to ensure academic integrity is maintained, we may randomly quiz some students about their homework solutions.
  • Recommendations.
    • Start the homework early. You learn a lot more by thinking about a problem for longer. (Also, most students won’t be able to do the homework in one evening.)
    • A significant portion of your exams will consist of homework problems.. You do not have access to the internet, friends and AI during exams, so make sure you thoroughly understand homework problems, and are capable of solving modified homework problems quickly in a closed book exam setting.
    • Go over homework problems you didn’t get correct, and understand correct solutions thoroughly.
    • According to the grading policy, in order to earn grade X in this class, you will have to earn grade X both on the homework, and on the exams. The grading scale for the homework is generous; most students get a better grade on the homework than they do on exams. Focus on using the homework to learn and understand the material better, so that you can do better on exams.
  • Logistics.
    • All homework must be scanned and turned in via Gradescope. (Everyone who was registered for this class on day 1 was automatically added to Gradescope. If you joined later, use this invite code to add yourself.)
    • Please take good quality scans; homework that’s too hard to read won’t be graded. I recommend using a good scanning app that adjusts the contrast of your images for readability. (I’ve had good luck with Adobe Scan, and Google Drive.)
  • Solutions.
    • Nearly perfect student solutions may be scanned and posted here, with your identifying information removed.
    • If you don’t want any part of your solutions used, please make a note of it in the margin of your assignment.

Exams

  • Logistics.
    • The final time will be announced by the registrar here. Be aware of their schedule before making your travel plans.
    • No calculators, computational aids, or internet enabled devices are allowed for closed book exams.
    • No makeup midterms will be given. (The grading policy allows you to miss one midterm without affecting your grade.)
  • Format.
    • Exams will typically be composed of the following:
      • 25%: Homework problems.
      • 25%: Results done in class.
      • 25%: Small modifications of homework problems / results from class.
      • 25%: Ph.D. Thesis Interesting problems.
    • Questions on an exam are chosen randomly. If a particularly long homework problem / proof from class gets chosen, the exam question will typically be a simplification or a step in the proof.

Academic Integrity

  • All students are expected to follow the academic integrity standards outlined here.
  • There will be zero tolerance for academic integrity violations, and any violation will result in an automatic R. Examples of academic integrity violations include (but are not limited to):
    • Not writing up solutions independently and/or plagiarizing solutions.
    • Turning in solutions you do not understand.
    • Receiving assistance from another person during an exam.
    • Providing assistance to another person taking an exam.
  • All academic integrity violations will further be reported to the university, and the university may chose to impose an additional penalty.

Accommodations for Students with Disabilities

If you have a disability and have an accommodations letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with me as early in the semester as possible. I will work with you to ensure that accommodations are provided as appropriate. If you suspect that you may have a disability and would benefit from accommodations but are not yet registered with the Office of Disability Resources, I encourage you to contact them at access@andrew.cmu.edu.

Student Wellness

As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and/or reduce your ability to participate in daily activities. CMU services are available, and treatment does work. You can learn more about confidential mental health services available on campus here. Support is always available (24/7) from Counseling and Psychological Services: 412-268-2922.

Faculty Course Evaluations

At the end of the semester, you will be asked to fill out faculty course evaluations. Please fill these in promptly, I value your feedback. As incentive, if over 75% of you have filled out evaluations on the last day of class, then I will release your grades as soon as they are available. If not, I will release your grades at the very end of the grading period.