MATH 2531 — Calculus III
University of New Mexico · 4 credit hours
Calculus III extends differentiation and integration to functions of several variables, with an introduction to geometry and vector calculus in three dimensions.
This is a semester-independent course overview. Official meeting times, assignment dates, grading rules, and university policies are provided in the syllabus for each offering.
Course syllabus
1. Course information
Course: MATH 2531 — Calculus III
Institution: University of New Mexico
Credits: 4
The course develops three-dimensional geometry, differentiation of vector-valued and multivariable functions, multiple integration, and introductory vector calculus.
2. Learning objectives
By the end of the course, students should be able to:
- Work with vectors and describe lines, planes, and common surfaces in three dimensions.
- Differentiate and integrate vector-valued functions and interpret velocity, acceleration, and curvature.
- Compute partial and directional derivatives, gradients, and tangent planes.
- Find and classify extrema, including constrained extrema using Lagrange multipliers.
- Set up and evaluate double and triple integrals in appropriate coordinate systems.
- Evaluate line integrals and apply the fundamental theorem for line integrals and Green’s theorem.
3. Course materials and prerequisites
Prerequisite: MATH 1522 (Calculus II) or its equivalent.
Textbook: James Stewart, Calculus, 9th edition (used in a previous coordinated offering; the assigned edition may vary).
Course resources: Lecture notes, assignments, announcements, and grades are distributed through UNM Canvas for each semester.
4. Assessment and grading
The following is an illustrative grading breakdown from a previous coordinated offering, not a fixed policy for every semester.
| Assessment | Weight |
|---|---|
| Weekly homework | 15% |
| Quizzes | 10% |
| Three midterm exams | 50% |
| Cumulative final exam | 25% |
Homework: Regular problem sets emphasize complete, clearly explained solutions. Written submissions are typically collected through Canvas as PDFs.
Exams: Timed midterm exams assess computational skill, conceptual understanding, and mathematical reasoning. The final exam is cumulative. Exact exam rules and arrangements are given in the current semester syllabus.
5. General course schedule
The order below is a topic sequence rather than a week-by-week calendar. Individual instructors may adjust the pacing.
| Order | Topic |
|---|---|
| 1 | Vectors, lines, planes, and surfaces |
| 2 | Vector-valued functions |
| 3 | Functions of several variables |
| 4 | Multiple integrals |
| 5 | Vector calculus |
Lecture materials
Lecture notes and examples will be added as they are developed. The structure below already uses the same sequential, single-column layout as my other course pages, with space for one illustration per topic.
1. Vectors, lines, planes, and surfaces
Coordinates, vectors, dot and cross products, lines, planes, and quadric surfaces.
Topics covered
- Cartesian coordinates and three-dimensional space
- Vectors and vector operations
- Dot products, angles, and projections
- Cross products and their geometric interpretation
- Equations of lines and planes
- Quadric surfaces and their traces
Lecture materials in development
Notes, examples, and figures for this topic will be added here.
2. Vector-valued functions
Space curves, derivatives, arc length, curvature, and motion in three dimensions.
Topics covered
- Vector-valued functions and space curves
- Limits, derivatives, and integrals of vector-valued functions
- Tangent vectors and unit tangent vectors
- Arc length, curvature, and normal vectors
- Motion in space: velocity and acceleration
Lecture materials in development
Notes, examples, and figures for this topic will be added here.
3. Functions of several variables
Limits, partial derivatives, gradients, tangent planes, optimization, and Lagrange multipliers.
Topics covered
- Functions of several variables; graphs and level sets
- Limits and continuity
- Partial derivatives and higher-order partial derivatives
- Tangent planes and linear approximations
- The multivariable chain rule
- Directional derivatives and the gradient vector
- Local and absolute extrema
- Constrained optimization with Lagrange multipliers
Lecture materials in development
Notes, examples, and figures for this topic will be added here.
4. Multiple integrals
Double and triple integration, including polar, cylindrical, and spherical coordinates.
Topics covered
- Double integrals over rectangular and general regions
- Double integrals in polar coordinates and applications
- Triple integrals in Cartesian coordinates
- Triple integrals in cylindrical and spherical coordinates
Lecture materials in development
Notes, examples, and figures for this topic will be added here.
5. Vector calculus
Vector fields, line integrals, conservative fields, Green's theorem, curl, and divergence.
Topics covered
- Vector fields
- Line integrals of scalar and vector functions
- Conservative vector fields and the fundamental theorem for line integrals
- Green's theorem
- Curl and divergence
Lecture materials in development
Notes, examples, and figures for this topic will be added here.